Cluj-Napoca, Romania, September 6–9, 2018

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2 0 1 8

N A A T

International Conference on Numerical Analysis and

Approximation Theory

Fourth Edition

Cluj-Napoca, Romania, September 6–9, 2018

BOOK OF ABSTRACTS AND PROGRAM

This edition is organized jointly with Tiberiu Popoviciu Institute of Numerical Analysis,

Romanian Academy; www.ictp.acad.ro

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Honorary Chairs

Professor Gheorghe Coman

Professor Petru Blaga

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The meeting is devoted to aspects of approximation of functions, integral and differential operators, linear approximation processes, splines, numerical analysis, statistics, stochastic processes, wavelets.

Scientific Committee:

Octavian Agratini (Romania) Francesco Altomare (Italy) George Anastassiou (USA) Carlo Bardaro (Italy) Emil C˘atina¸s (Romania) Teodora C˘atina¸s (Romania) Ioan Gavrea (Romania) Heiner Gonska(Germany) Mircea Ivan (Romania) Giuseppe Mastroianni (Italy) Gradimir Milovanovi´c (Serbia) Ioan Ra¸sa (Romania)

Nicolae Suciu (Romania) Gancho Tachev (Bulgaria)

Webpage: math.ubbcluj.ro/naat2018/

E-mail: [email protected]

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Invited speakers:

Francesco Altomare (Italy) Carlo Bardaro (Italy)

Maria Garrido-Atienza (Spain) Heiner Gonska (Germany)

Aaron Melman (USA)

Gradimir Milovanovi´ c (Serbia) Cihan Orhan (Turkey)

Wolfgang L. Wendland (Germany)

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Organizing Committee:

Octavian Agratini Teodora C˘atina¸s Ioana Chiorean Hannelore Lisei Sanda Micula Natalia Ro¸sca Agoston R´´ oth Ildik´o Somogyi

Anna So´os (Vice-rector) Radu Trˆımbita¸s

Volunteers: Anamaria Biri¸s, Ioana S¸omˆıtc˘a Special thanks to Diana S¸otropa.

The organizers would like to acknowledge the substan- tial support received fromthe Rectorate of Babe¸s- Bolyai University, for which they are grateful.

This conference was partly funded by grant 39/20.08.

2018 of the National Authority for Scientific Research/the Ministry of Research and Inno- vation and sponsored by the Association of Ma- thematicians and Computer Scientists of Cluj (AMIC).

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Aaron Melman Ulrich Abel 16²⁰-16⁴⁰ C˘alin Gheorghiu Heinz-Joachim Rack 16⁴⁰-17⁰⁰ Petr Hasil Robert Vajda 17⁰⁰-17²⁰ Nicu¸sor Minculete Dan Miclau¸s 17²⁰-17⁴⁰ Flavian Georgescu Mehmet Unver 17⁴⁰-18⁰⁰ Daniel Pop Mohd Ahasan 18⁰⁰-18²⁰ Flavius P˘atrulescu Adonia Opri¸s

19⁰⁰ CONFERENCE DINNER

Grand Hotel Napoca, 1, Octavian Goga St.

web: http://hotelnapoca.ro/

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Saturday, September 8

Plenary Talks

Chairman:Cihan Orhan 9⁰⁰-9⁵⁰ Maria Garrido-Atienza 9⁵⁰-10⁴⁰ Aaron Melman

10⁴⁰-11¹⁰ COFFEE BREAK

Contributed Talks

Ioan Gavrea Mircea Ivan 11¹⁰-11³⁰ Alexandru Mitrea Marius Birou 11³⁰-11⁵⁰ Julian Dimitrov Silviu Urziceanu 11⁵⁰-12¹⁰ Maria Cr˘aciun Ildiko Somogy 12¹⁰-12³⁰ Diana Otrocol Vicuta Neagos 12³⁰-12⁵⁰ Alina Baias Larisa Cheregi 12⁵⁰-13¹⁰ Tugba Yurdakadim Augusta Ratiu

*****

September 9

9⁰⁰ EXCURSION

Sunday,

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LIST OF PARTICIPANTS

Participants Page

Ulrich Abel 32

Ana Maria Acu 35

Mohd Ahasan 36

Francesco Altomare 18 Alina Ramona Baias 38, 80

Carlo Bardaro 20

Marius Birou 39

Mirella Cappelletti Montano 40

Emil C˘ atina¸s 42

Ganna Chekhanova 46

Larisa Cheregi 43

Maria Cr˘ aciun 44

Madalina Dancs 38

Markus Dietz 46

Julian Dimitrov 47

Sorin Gal 49

Maria Garrido-Atienza 22

Bogdan Gavrea 51

Ioan Gavrea 52, 73

Flavian Georgescu 53

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Participants Page C˘ alin Gheorghiu 54

Heiner Gonska 23

Wilfried Grecksch 55

Petr Hasil 56

Margareta Heilmann 57

Laura Hodis 74

Sever Hodis 38

Harun Karsli 59

Davod Khojasteh Salkuyeh 61 Szil´ ard Csaba L´ aszl´ o 63

Vita Leonessa 64

Alexandra-Ioana M˘ adut¸a 74

Aaron Melman 24

Dan Micl˘ au¸s 66

Radu Miculescu 67

Gradimir V. Milovanovi´ c 26

Nicusor Minculete 68

Alexandru Mitrea 69

Camelia Liliana Moldovan 75

Carmen Violeta Muraru 71

Vicuta Neago¸s 72, 43

Adonia Augustina Opri¸s 73

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Participants Page

Cihan Orhan 28

Diana Otrocol 74

Radu P˘ alt˘ anea 75 Flavius P˘ atrulescu 76 Ion P˘ av˘ aloiu 42

Daniel Pop 78

Dorian Popa 80

Heinz-Joachim Rack 82 Voichit¸a Adriana Radu 83

Ioan Ra¸sa 85, 57

Augusta Rat¸iu 87

Ralf Rigger 88

Bj¨ orn Schmalfuß 90

Andra Silaghi 72

Ildik´ o Somogyi 91 Hans-J¨ org Starkloff 93, 46

Nicolae Suciu 94

Emre Ta¸s 95, 118

Radu Trˆımbita¸s 96

Mehmet Unver 97

Silviu-Aurelian Urziceanu 99

Gumrah Uysal 101

Robert Vajda 104

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Participants Page Rodrigo Vjar Asem 106

Michal Vesel 108

Adrian Viorel 110 Wolfgang Wendland 30

Pawel Wozny 114

Ralf Wunderlich 116 Tugba Yurdakadim 118, 95

Note: In the sequel the symbol * designates the author giving the talk.

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PLENARY TALKS

POSITIVE APPROXIMATION PROCESSES AND INITIAL-BOUNDARY VALUE

DIFFERENTIAL PROBLEMS

Francesco Altomare

Department of Mathematics, University of Bari, Italy [[email protected]]

MSC 2010: 41A36, 47D06, 47D07, 35K65, 35B65

Keywords: Approximation by positive operators, positive C₀-semigroup of operators, initial boundary-value differential problem.

The talk will be centered about a topic concerning three in- terrelated subjects: positive approximating operators, pos- itiveC0-semigroups of operators and initial-boundary value evolution problems.

The main aim is to discuss a series of results concerning those sequences (Ln)n≥1 of bounded linear operators on a Banach spaceE whose iterates converge to aC₀-semigroup (T(t))t≥0 of operators on E.

To such a semigroup it is naturally associated its in- finitesimal generator A : D(A) → E which, in turn, gives rise to an abstract Cauchy problem (initial-boundary value problem) whose solutions can be given, al least from a the- oretical point of view, by the semigroup itself.

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Thus, if it is possible to determine the operator A and its domainD(A),then the initial sequence (L_n)n≥1becomes the key tool to approximate and to study (especially, from a qualitative point of view) the solutions of the Cauchy problem.

The principal ideas and some of the more recent results on such functional analytic approach to study these kinds of problems, will be discussed in the context of continuous function spaces by also assuming that the operators L_n, n≥1, are positive. Moreover, particular attention will be devoted to the important case when the approximating operators are constructively generated by a given positive linear operator T : C(K) → C(K) which, in turn, allows to determine the differential operator (A, D(A)) as well,K being a compact subset of R^d, d ≥ 1, having non-empty interior.

Initial-boundary value evolution problems corresponding to these particular settings, occur, for instance, in the study of diffusion problems arising from different areas such as biology, mathematical finance and physics.

For more details and for several other aspects related to the above outlined theory, the reader is referred to the monograph [1].

REFERENCES

[1] F. Altomare, M. Cappelletti Montano, V. Leonessa and I.

Ra¸sa, Differential Operators, Markov Semigroups and Pos- itive Approximation Processes Associated with Markov Op- erators, de Gruyter Series Studies in Mathematics, Vol. 61, De Gruyter, Berlin, 2014.

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PALEY-WIENER THEOREMS FOR MELLIN TRANSFORMS AND THE EXPONENTIAL SAMPLING

Carlo Bardaro

Department of Mathematics and Computer Sciences, University of Perugia, Italy

[[email protected]]

MSC 2010: 26D10, 30D20, 44A05

Keywords: Mellin transforms, Paley-Wiener spaces, Bernstein spaces, band-limited functions, polar-analytic functions.

The exponential sampling formula, introduced by a group of physicists and engineers during the end of seventieth, was studied in a rigorous form, through the Mellin transform theory, by P.L. Butzer and S. Jansche during the ninetees.

This formula is formally equivalent to the classical Shan- non sampling formula of signal analysis, valid for Fourier bandlimited functions. Indeed, by a formal change of vari- able and change of function it is possible to obtain one formula from the other. However, this equivalence is only formal. Indeed, the structure of the Paley-Wiener space of all continuous functions in L²(R) which are bandlimited, is characterized by the famous Paley-Wiener theorem of Fourier analysis which states that a (Fourier) bandlimited function has an extension to the complex field as an entire function of exponential type (the Bernstein space). In Mellin transform setting this is not true. Indeed, it is shown that a (non trivial) Mellin bandlimited function cannot be extended as an entire function over C. As proved in [1], it has an extension as an analytic function to the Riemann

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surface of the (complex) logarithm with certain exponential type conditions (Mellin-Bernstein spaces). In this talk, we discuss some further version of the Paley-Wiener theorem in Mellin setting, which avoid the use of Riemann surfaces and analytic branches, employing a new concept of analyt- icity (see [2]). Another version characterizes the space of all the functions whose Mellin transform decays exponentially at infinity involving Hardy-type spaces (see [3]). Applica- tions to exponential sampling theory are described.

REFERENCES

[1] C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, On the Paley-Wiener theorem in Mellin transform setting, J.

Approx. Theory, 207 (2016), pp. 60-75.

[2] C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, A fresh approach to the Paley-Wiener theorem for Mellin transforms and the Mellin-Hardy spaces, Math. Nachr., 290 (2017), pp. 2759-2774.

[3] C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, A generalization of the Paley-Wiener theorem for Mellin transforms and metric characterization of function spaces, Frac- tional Calculus and Applied Analysis, 20 (2017), pp. 1216- 1238.

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STOCHASTIC DIFFERENTIAL

EQUATIONS DRIVEN FRACTIONAL BROWNIAN MOTION

Mar´ıa J. Garrido-Atienza

Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Spain

[[email protected]]

In this talk we are concerned with the study of the existence and uniqueness of solutions of stochastic (partial) differential equations driven by a fractional Brownian motion (fBm), as well as their longtime behavior. We will analyze different approaches and consider both the cases of an fBm with Hurst parameter H ∈(1/2,1) andH ∈(1/3,1/2].

This talk is based on some joint works with Duc Hoang Luu (Max-Planck Institut of Leipzig), A. Neuenkirch (Uni- versity of Mannheim) and B. Schmalfuss (University of Jena).

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CHLODOVSKY-BERNSTEIN

POLYNOMIAL OPERATORS - PAST AND PRESENT

Heiner Gonska

University of Duisburg - Essen, Germany [[email protected]]

The talk will be on a long-neglected polynomial approximation process for continuous functions defined on the real half-line. It was introduced by the Russian mathematician Igor Nikolaevich Chlodovsky. Although the first publication on it appeared in 1937 already, progress on investigating it was interrupted for more than half a century. As Butzer and Karsli said in 2009, this approximation process is not so easy to handle.

Some progress was made over the last ten years. We will present some of these latest developments including very recent results.

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EIGENVALUE LOCALIZATION FOR MATRIX POLYNOMIALS

Aaron Melman

Department of Applied Mathematics, Santa Clara University, CA, USA

[[email protected]]

MSC 2010: 12D10, 15A18, 15A42, 15A54, 26C10, 30C10, 30C15, 47A56, 65F15

Keywords: Bound, localization, eigenvalue, scalar, polynomial, matrix polynomial.

We survey a number of well-known and less well-known results ([5], [6]) on the location of scalar polynomial zeros and their generalization to localization results for the eigenvalues of matrix polynomials. Such polynomials occur in polynomial eigenvalues problems, which can be found in a wide range of engineering applications. We include results for matrix polynomials expressed in generalized bases, cov- ering all classical orthogonal bases, such as Hermite, Leg- endre, Chebyshev, etc.

Finally, we show how some of the aforementioned results for scalar polynomials lead to extensions of the Enestr¨om- Kakeya theorem for polynomials with positive coefficients.

REFERENCES

[1] A. Melman, Bounds for eigenvalues of matrix polynomials with applications to scalar polynomials, Linear Algebra Appl. 504 (2016), pp. 190-203.

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[2] A. Melman,Improvement of Pellet’s theorem for scalar and matrix polynomials, C. R. Math. Acad. Sci. Paris, 354 (2016), pp. 859-863.

[3] A. Melman, Improved Cauchy radius for scalar and matrix polynomials, Proc. Amer. Math. Soc., 146 (2018), pp.

613624.

[4] A. Melman, Eigenvalue bounds for matrix polynomials in generalized bases, Math. Comp., 87 (2018), pp. 1935-1948.

[5] G.V. Milovanovi´c, D.S. Mitrinovi´c, and Th. Rassias, Top- ics in polynomials: extremal problems, inequalities, zeros, World Scientific Publishing Co., Inc., River Edge, NJ, 1994.

[6] Q.I. Rahman, and G. Schmeisser, Analytic Theory of Poly- nomials, London Mathematical Society Monographs. New Series, 26. The Clarendon Press, Oxford University Press, Oxford, 2002.

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SOME CLASSES OF ORTHOGONAL

POLYNOMIALS IN THE COMPLEX PLANE AND APPLICATIONS

Gradimir V. Milovanovi´ c

Mathematical Institute of the Serbian Academy of Sciences and Arts, Serbia

[[email protected]]

MSC 2010: 33C45, 33C47, 65D25, 65D30, 78A30

Keywords: Complex orthogonal polynomials, recurrence rela- tions, zeros, numerical differentiation, quadrature formula.

We consider a few classes of polynomials orthogonal in the complex plane with respect to the Hermitian and Non- Hermitian inner products, as well as some applications of such polynomials.

The first class of such complex polynomials was introduced and studied in [1], [2] and [3]. The inner product is not Hermitian and defined by

(f, g) = Z

Γ

f(z)g(z)w(z)(iz)⁻¹dz,

where z 7→w(z) is a complex weight function holomorphic in the half disk D₊ ={z ∈C| |z|< 1, im z >0}. Polyno- mials orthogonal on the radial rays in the complex plane is the second class of polynomials in our investigation. This class was introduced in [4] (see also [7], [5], [6]).

Beside some analysis of such kinds of orthogonality and an electrostatic interpretation of zeros of the polynomials on the radial rays, we give several applications of these

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polynomials in numerical integration and numerical differentiation.

REFERENCES

[1] W. Gautschi, G.V. Milovanovi´c,Polynomials orthogonal on the semicircle, J. Approx. Theory, 46 (1986), pp. 230–250.

[2] W. Gautschi, H.J. Landau, G.V. Milovanovi´c,Polynomials orthogonal on the semicircle. II, Constr. Approx., 3 (1987), pp. 389–404.

[3] G.V. Milovanovi´c, Complex orthogonality on the semicircle with respect to Gegenbauer weight: theory and applications, In: Topics in Mathematical Analysis (Th. M. Rassias, ed.), pp. 695–722, Ser. Pure Math., 11, World Sci. Publ., Teaneck, NJ, 1989.

[4] G.V. Milovanovi´c,A class of orthogonal polynomials on the radial rays in the complex plane, J. Math. Anal. Appl., 206(1997), pp. 121–139.

[5] G.V. Milovanovi´c,Orthogonal polynomials on the radial rays and an electrostatic interpretation of zeros, Publ. Inst. Math.

(Beograd) (N.S.), 64(78)(1998), 53–68.

[6] G.V. Milovanovi´c,Orthogonal polynomials on the radial rays in the complex plane and applications, Rend. Circ. Mat.

Palermo (2) Suppl., 68(2002), 65–94.

[7] G.V. Milovanović, P.M. Rajković, Z.M. Marjanović,Zero dis- tribution of polynomials orthogonal on the radial rays in the complex plane, Facta Univ. Ser. Math. Inform., 12(1997), 127–142.

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THE SCOTTISH CAFE AND STATISTICAL CONVERGENCE

Cihan Orhan

University of Ankara, Turkey

[[email protected]]

Many valuable problems and solutions are originated in a cafe so-called ”Scottish Cafe” in Lviv around 1930’s. These problems are listed in the well-known book The Scottish Book, Mathematics from the Scottish Cafe. The participants of this informal meetings includes well-known math- ematicians such as Stefan Banach, Hugo Steinhaus, Stanis- law Mazur, W. Orlicz, J.P. Schauder, M. Kac, S. Kacmarz, S. Saks, S. Ulam etc.

Here is the one problem posed by S. Mazur (July 22, 1935):

”A sequence (xn) is asymptotically convergent to L if there exits a subsequence of density one convergent to L.

In the domain of all sequences this notion of convergence is not equivalent to any Toeplitz (regular method). How is it in the domain of bounded sequences?” Mazur also made some comments that indicates the solution of this problem is negative (see, The Scottish Book, Second Edition, page 55, problem 5).

In our talk we will provide a positive answer to Mazur’s problem and his claim has to be false. In order to prove our result we first note that the notion of asymptotic convergence of Mazur is equivalent to the notion of statistical convergence. We show that statistical convergence is always boundedly (as well as over the space of uniformly

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integrable sequences) equivalent to a nonnegative regular matrix method which is boundedly multiplicative. We also characterize the set of bounded multipliers of multiplicative methods. This talk is mainly based on my joint paper with M.K. Khan (Matrix characterization of A-statistical convergence; J. Math. Anal. Appl. 335 (2007), 406-417).

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ON NEUMANN’S METHOD AND DOUBLE LAYER POTENTIALS

Wolfgang. L. Wendland

IANS & Simtech University Stuttgart, Germany [[email protected]]

MSC 2010: 31A10, 45F15, 65N30

Keywords:Boundary integral equations, boundary element methods.

Neumann’s classical integral equations with the double layer boundary potential is considered on different spaces of boundary charges such as continuous data,L² and energy trace spaces on the domain’s boundary for interior and exte- rior boundary value problems of elliptic partial differential equations. Corresponding known results for different classes of boundaries are discussed in view of collocation and Galerkin boundary element methods.

REFERENCES

[1] O. Steinbach, W.L. Wendland, On C. Neumann’s method for second-order elliptic systems in domains with non–smooth boundaries, J. Math. Anal. Appl., 262 (2001), pp. 733–748.

[2] M. Costabel, Some historical remarks on the positivity of boundary integral operators, In: Boundary Element Analy- sis (M. Schanz, O. Steinbach eds.), Springer–Verlag Berlin 2007, pp.1–27.

[3] W.L. Wendland, On the double layer potential, Operator Theory: Advances and Appl. 193 (2009), pp. 319–334.

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[4] G.C. Hsiao, O. Steinbach, W.L. Wendland, Boundary element methods: Foundation and error analysis, In: Encyclo- pedia of Computational Mechanics. Second Edition (E. Stein, R. de Borst, T.J.R. Hughes eds.), 2217 John Wiley & Sons Ltd. pp. 1–62.

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CONTRIBUTED TALKS

ASYMPTOTIC PROPERTIES AND OPERATOR NORMS OF GAUSS-

WEIERSTRASS OPERATORS AND THEIR LEFT QUASI INTERPOLANTS

Ulrich Abel

Department Mathematik, Naturwissenschaften und Datenverarbeitung, Technische Hochschule Mittelhessen, Wilhelm-Leuschner-Strasse 13, 61169 Friedberg, Germany [[email protected]]

MSC 2010: 41A36, 41A45, 47A30

Keywords:Approximation by positive operators, operator norm.

The Gauß–Weierstraß convolution operators Wn (n = 1,2, 3, . . .) are defined by

(W_nf) (x) = q

n π

Z ∞

−∞

f(t) exp −n(t−x)²

dt (1) (see, e.g., [6, SS 5.2.9]). They are positive linear approximation operators which are applicable to the class L_c(R) of all locally integrable real functionsf onRsatisfying the growth condition f(t) = O(e^ct²), as t → ±∞, for some c >0, provided that n > c. If f is continuous we have

x→∞lim (Wnf) (x) = f(x)

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uniformly on compact subsets ofR.

The complete asymptotic expansion of (Wnf) (x) asntends to infinity appears to be a special case of the results [3] on a more general operator defined by Altomare and Milella [5].

In 2014 Sablonni`ere [7] defined quasi-interpolants of W_n and studied their basic properties including the operator norm for bounded functions with respect to the sup-norm.

In particular, he proved asymptotic expansions for polynomials and functions having a bounded higher order derivative on the whole line.

In the first part we report recent results [2] by presenting the complete asymptotic expansions of (W_nf)(x) and the left quasi-interpolant (Wn^[r]f)(x) as n tends to infinity, for functions belonging to the class Lc(R) which are assumed to be only locally smooth. The corresponding results for the Favard operators which are the discrete version of W_n can be found in [4, 1].

Finally, in the second part, we consider the operator norms of W_n and Wn^[r] when acting on various function spaces.

REFERENCES

[1] U. Abel, Asymptotic expansions for Favard operators and their left quasi-interpolants, Stud. Univ. Babe¸s-Bolyai Math. 56 (2011), pp. 199–206.

[2] U. Abel, O. Agratini, R. P˘alt˘anea, A complete asymptotic expansion for the quasi-interpolants of Gauß–

Weierstraß operators, Mediterr. J. Math. (2018) 15:156, https://doi.org/10.1007/s00009-018-1195-8

[3] U. Abel, M. Ivan, Simultaneous approximation by Altomare operators, Proceedings of the 6th international conference on functional analysis and approximation theory, Ac- quafredda di Maratea (Potenza), Italy, Sept. 24–30, 2009,

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Palermo: Circolo Matem`atico di Palermo, Rend. Circ. Mat.

Palermo, Serie II, Suppl. 82 (2010), pp. 177–193.

[4] U. Abel, P. L. Butzer, Complete asymptotic expansion for generalized Favard operators, Constr. Approx. 35 (2012), pp. 73–88.

DOI: 10.1007/s00365-011-9134-y.

[5] F. Altomare, S. Milella, Integral-type operators on continuous function spaces on the real line, J. Approx. Theory 152 (2008), pp. 107–124.

http://dx.doi.org/10.1016/j.jat.2007.11.002

[6] F. Altomare, M. Campiti, Korovkin-type Approximation Theory and its Applications, de Gruyter Studies in Mathe- matics 17, W. de Gruyter, Berlin, New York, 1994.

[7] P. Sablonni`ere, Weierstrass quasi-interpolants, J. Approx.

Theory 180(2014), pp. 32–48.

http://dx.doi.org/10.1016/j.jat.2013.12.003

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DIFFERENCES OF POSITIVE LINEAR OPERATORS

Ana Maria Acu

Department of Mathematics and Informatics, Lucian Blaga University, Sibiu, Romania

[[email protected]]

MSC 2010: 41A25, 41A36

Keywords:First modulus of continuity, positive linear operators.

The results obtained are motivated by the recent results which give a solution to a problem proposed by A. Lupa¸s in [1]. One of the questions raised by him was to give an estimate for

Bn◦Bⁿ−Bⁿ◦Bn =:Un−Sn,

whereB_n are the Bernstein operators andBn are the Beta operators. We introduce new inequalities for such differences of positive linear operators and their derivatives in terms of moduli of continuity. This is a joint work with Ioan Ra¸sa from Technical University of Cluj-Napoca, Ro- mania.

REFERENCES

[1] A. Lupa¸s,The approximation by means of some linear positive operators, in Approximation Theory (M.W. M¨uller et al., eds), Akademie-Verlag, Berlin, 1995, 201-227.

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THE DUNKL GENERALIZATION OF STANCU TYPE Q-SZ ´ ASZ-MIRAKJAN- KANTOROVICH OPERATORS AND SOME APPROXIMATION RESULTS

M. Mursaleen and Mohd. Ahasan*

Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

[[email protected]; [email protected]]

MSC 2010: 41A25, 41A36, 33C45

Keywords:q-integers,q-exponential functions,q-hypergeometric functions, Dunkl’s analogue, Sz´asz operators, Stancu type q- Sz´asz-Mirakjan-Kantorovich operators, rate of convergence, modulus of continuity and Peetre’sK-functional.

In this paper, a Dunkl type generalization of Stancu type q-Sz´asz-Mirakjan-Kantorovich positive linear operators of the exponential function is introduced. With the help of well-known Korovkin’s theorem, some approximation properties and also the rate of convergence for these operators in terms of the classical and second-order modulus of continuity, Peetre’s K-functional and Lipschitz functions are investigated. Further, some approximation results for bi- variate Stancu type q-Sz´asz-Mirakjan-Kantorovich operators are obtained.

REFERENCES

[1] S.N. Bernstein,Démonstration du théoréme de Weierstrass fondée sur le calcul des probabilités, Commun. Soc. Math.

Kharkow, 2(13)(1912), pp. 1-2.

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[2] H.M. Srivastava, M. Mursaleen, A. Alotaibi, Md. Nasiruzza- man and A.A.H. Al-Abied, Some approximation results involving theq-Sz´asz-Mirakjan-Kantrovich type operators Via Dunkls generalization, Math. Meth. Appl. Sci., 40(15)(2017), pp. 5437-5452.

[3] O. Sz´asz,Generalization of S. Bernsteins polynomials to the infinite interval, J. Res. Natl. Bur. Stand., 45(1950), pp.

239-245.

[4] S. Sucu, Dunkl analogue of Sz´asz operators, Appl. Math.

Comput., 244(2014), pp. 42-48.

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FUNCTIONS WITH CONVEX IMAGES UNDER BERNSTEIN OPERATORS

Alina Baias

^1,∗

, Madalina Dancs

²

, Sever Hodis

³

1,2,3 Department of Mathematics, Technical University of Cluj-Napoca, Romania

[[email protected]],[dancs [email protected]], [[email protected]]

MSC 2010: 26A51, 26D05

Keywords:Bernstein operators, convex function, log-convex function.

The preservation of convexity under the Bernstein operators B_n is exhaustively investigated in literature.

In this talk we present some non-convex functionsf for whichB_nf is convex. Inequalities for such functions are also discussed.

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QUANTITATIVE RESULTS FOR THE ITERATES OF SOME KING TYPE OPERATORS

Marius-Mihai Birou

Department of Mathematics, Technical University of Cluj Napoca, Romania

[[email protected]]

MSC 2010: 41A25, 41A36

Keywords:Iterates, linear positive operators, convergence.

In this article we define the -q variant of some King type operators which fix the functions e₀ and e₂ +αe₁, α > 0.

We study the rates of convergence for the iterates of these operators using the first and the second order modulus of continuity. We show that the convergence is faster in the case of -q operators (q < 1) than in the classical case (q= 1).

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ON THE POSITIVE SEMIGROUPS

GENERATED BY FLEMING-VIOT TYPE DIFFERENTIAL OPERATORS

Mirella Cappelletti Montano

Department of Mathematics, University of Bari, Italy [[email protected]]

MSC 2010: 35K65, 41A36, 47D06, 47D07

Keywords:Bernstein-Durrmeyer operators with Jacobi weights, Fleming-Viot type differential operator, positive semigroup, approximation of semigroups.

Joint work with Francesco Altomare and Vita Leonessa ([1]).

In this talk, we introduce, in the framework of function spaces defined on the d-dimensional hypercube of R^d, d ≥ 1, a class of polynomial type positive linear operators, which generalize the Bernstein-Durrmeyer operators with Jacobi weights on [0,1] ([3, 2]). By means of them, we study some degenerate second-order elliptic differential operators, often referred to as Fleming-Viot type operators, showing that their closures generate positive semigroups both in the space of all continuous functions and in weighted L^p-spaces. In addition, we show that those semigroups are approximated by iterates the above mentioned Bernstein-Durrmeyer operators.

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REFERENCES

[1] F. Altomare, M. Cappelletti Montano and V. Leonessa,On the positive semigroups generated by Fleming-Viot type differential operators, Comm. Pure Appl. Anal., to appear.

[2] H. Berens and Y. Xu,On Bernstein-Durrmeyer polynomials with Jacobi weights, in: C. K. Chui (Ed.), Approximation Theory and Functional Analysis, Academic Press, Boston, 1991, pp. 25-46.

[3] R. P˘alt˘anea, Sur un opérateur polynomial défini sur l’en- semble des fonctions intégrables, Univ. Babe¸s-Bolyai, Cluj- Napoca, 83-2 (1983), pp. 101-106.

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ON A NEWTON-HERMITE-STEFFENSEN TYPE METHOD

Ion P˘ av˘ aloiu

¹

, Emil C˘ atina¸s

^2,∗

1,2Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

[[email protected]], [[email protected]]

MSC 2010: 65H05

Keywords:Inverse interpolatory iterative methods, Newton method, local convergence, monotone convergence.

We consider the solving of nonlinear equations inR, and we introduce an inverse interpolatory iterative method of Her- mite type, for which the nodes are given using the Newton method:

yn =xn− _f^f(x0(xⁿn⁾)

x_n+1 =y_n− _f^f0^(y(yⁿn⁾) − ^[y_fⁿ0^,y(yⁿn^,x)[yⁿn^;f]f,xn²;f^(y]ⁿ²⁾.

The convergence order of the resulted method is 5, and the efficiency index is higher than in the case of the Newton and the Steffensen methods.

Under some natural conditions, the generated iterates converge monotonically to the solution, and one may obtain larger convergence sets than the usual attraction balls.

REFERENCES

[1] I. P˘av˘aloiu, E. C˘atina¸s, On a robust AitkenNewton method based on the Hermite polynomial, Appl. Math. Comput., 287-288 (2016), pp. 224-231.

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ON THE POMPEIU MEAN-VALUE THEOREM

Larisa Cheregi

^1,∗

, Vicuta Neagos

²

1,2 Department of Mathematics, Technical University of Cluj-Napoca, Romania

MSC 2010: 26A24

Keywords: Pompeiu mean-value theorem, compact set, affine support.

We generalize the Pompeiu mean-value theorem by replac- ing the graph of a continuous function with a compact set.

REFERENCES

[1] T. Boggio, Sur une proposition de M. Pompeiu, Mathemat- ica, Timi¸soara23(1948), pp. 101–102.

[2] M. Ivan, A note on a Pompeiu-type theorem, Mathematical analysis and approximation theory, Burg, Sibiu (2002), pp.

129–134.

[3] D. Pompeiu, Sur une proposition analogue au th´eor`eme des accroissements finis, Mathematica22(1946), pp. 143–146.

[4] O.T. Pop, D. B˘arbosu, A mean-value theorem and some applications, Didactica Matematica31(1)(2013), pp. 47–50.

[5] P.K. Sahoo, T. Riedel,Mean value theorems and functional equations, World Scientific Publishing Co., Inc., River Edge, NJ (1998).

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A BOSE-EINSTEIN CONDENSATE DARK MATTER MODEL TESTED WITH THE SPARC GALACTIC ROTATION CURVES DATA

Maria Cr˘ aciun

^1,∗

and Tiberiu Harko

²

1 Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

2 Department of Physics, Babes-Bolyai University, Cluj-Napoca, Romania;

School of Physics, Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China;

Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom [[email protected]]

MSC 2010: 83F05, 85A40

Keywords:Cosmology, galactic rotation curves, dark matter models.

In the present study we investigate the properties of the galactic rotation curves in the Bose-Einstein Condensate dark matter model, with quadratic self-interaction, by u- sing more than one hundred galaxies from the recently pub- lished Spitzer Photomery & Accurate Rotation Curves (SPARC) data.

REFERENCES

[1] F. Lelli, S.S. McGaugh, J.M. Schombert, SPARC: Mass models for 175 disk galaxies with Spitzer Photometry and Accurate Rotation Curves, The Astronomical Journal, 152 (2016), article id. 157.

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[2] X. Zhang, M.H. Chan, T. Harko, S.D. Liang, C.S. Leung, Slowly rotating Bose Einstein condensate galactic dark matter halos, and their rotation curves, The European Physical Journal C, 78 (2018), pp. 1-20.

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ON A STOCHASTIC ARC FURNACE MODEL

Markus Dietz*, Anna Chekhanova, Hans-J¨ org Starkloff

Faculty of Mathematics and Computer Science, Technische Universit¨at Bergakademie Freiberg, Saxony, Germany [[email protected]]

MSC 2010: 34F05

Keywords: Electric arc furnace, random differential equation, Ornstein-Uhlenbeck process.

One approach of modeling of an electric arc furnace (EAF) is by the power balance equation which results in a nonlinear ordinary differential equation.

In real-world data can be observed that arc-current and arc-voltage vary randomly in time (cf. [1]), for example they oscillate with a randomly time-varying amplitude and a slight shiver. Therefore it is better to model them as stochastic processes and then solve a random differential equation.

Here we want to propose one example for a modulation by using the Ornstein-Uhlenbeck process and present some results which we gained by studying this model.

This is a joint work with Anna Chekhanova and Hans- J¨org Starkloff.

REFERENCES

[1] D. Grabowski, J. Walczak, M. KlimasElectric arc furnace power quality analysis on a stochastic arc model, 978-1-5386- 5186-5/18/$31.00 2018 IEEE.

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AN APPROACH TO DEALING WITH

UNCERTAINTY IN NUMERICAL MODELS

Julian Dimitrov

Department of Mathematics, University of Mining and Geology, Sofia, Bulgaria

[[email protected]]

Keywords: Valuation of dependences, stability of numerical algorithms, principle of uncertainty, relative metric.

The article discusses a method for valuation of the functional dependencies applied for estimation of some numerical models. The method for valuation of dependencies is applied similar to ”functional stability analysis” method, which is intended to detect instability in numerical algorithms [1].

The uncertainty principle is a fundamental concept [2], [3]. In this paper, we propose a new approach for dealing with such uncertainty, which combines the uncertainties in the values of parameters and transformation uncertainty that describes local degree of correlation at a given point of domain.

For optimal evaluation of the dependencies, we use relative distance in space of input parameters. The used mathematical theory is based on the theory of functional analysis on metric spaces, where the metric gives an estimation of the error [4]. The p-relative distance was introduced by Ren-Cang [5]. In [6] was introduced the main results ofM- relative distances.

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We use calculations with semi-logarithmic derivative en- suring accordance with the relative distance and we estab- lish relevant properties. A valuation is applied for an ana- lytical expression.

REFERENCES

[1] N.J. Higham, Accuracy and Stability of Numerical Algo- rithms, SIAM, Philadelphia (2002).

[2] H.-W. Bandemer,Mathematics of Uncertainty - Ideas, Meth- ods, Application Problems, Springer (2006).

[3] E. Hofer,The Uncertainty Analysis of Model Results, Springer (2018).

[4] A. Mennucci,On asymmetric distance, Anal. and Geom. in Metric Spaces (2013)

[5] R.-C. Li,Relative perturbation theory I: Eigenvalue and Sin- gular Value Variations, SIAM J. Matrix Anal. Appl. 19(4) (1998)

[6] P. Hasto, A new weighted metric: the relative metric, I, J.

Math. Anal. Appl. (2002).

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APPROXIMATION BY MAX-PRODUCT OPERATORS OF KANTOROVICH TYPE

Lucian Coroianu

¹

and Sorin G. Gal

^2,∗

1,2 Department of Mathematics and Computer Science, University of Oradea, Romania

MSC 2010: 41A35, 41A25, 41A20

Keywords: Max-product operators, max-product operators of Kantorovich kind, uniform approximation, shape preserving properties, localization results, max-product Kantorovich-Choquet operators.

We associate to various linear Kantorovich type approximation operators, nonlinear max-product operators for which we obtain quantitative approximation results in the uniform norm, shape preserving properties and localization results.

REFERENCES

[1] B. Bede, L. Coroianu and S.G. Gal, Approximation and shape preserving properties of the Bernstein operator of max-product kind, Intern. J. Math. Math. Sci. vol. 2009, Article ID 590589, 2009.

[2] B. Bede, L. Coroianu and S.G. Gal, Approximation by Max-Product Type Operators, Springer, 2016. xv+458 pp., ISBN: 978-3-319-34188-0;

[3] L. Coroianu and S.G. Gal, Approximation by truncated max-product operators of Kantorovich-type based on generalized (ϕ, ψ)-kernels, submitted.

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[4] L. Coroianu and S.G. Gal,Approximation by max-product operators of Kantorovich-Choquet type based on generalized (ϕ, ψ)-kernels, in preparation.

[5] L. Coroianu and S.G. Gal, Classes of functions with improved estimates in approximation by the max-product Bern- stein perator, Anal. Appl.9(2011), 249-274.

[6] L. Coroianu and S.G. Gal,Localization results for the Bern- stein max-product operator,Appl. Math. Comp.,231(2014) 73-78.

[7] S.G. Gal,Shape-preserving approximation by real and complex polynomials, Birkh¨auser, Boston, Basel, Berlin, 2008.

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ON A CONVEXITY PROBLEM

Bogdan Gavrea

Department of Mathematics, Technical University of Cluj-Napoca, Romania

[[email protected]]

MSC 2010: 26D15, 26D10, 46N30

Keywords: Linear positive operators, convex functions, Bern- stein operators.

The work presented here is a continuation of what was done in [2] and it is strongly connected to the work done in [1]. Applications are given for Mirakyan-Favard-Sz´asz, Baskakov and Sz´asz-Schurer type operators.

REFERENCES

[1] U. Abel, I. Ra¸sa, A sharpening of a problem on Bernstein polynomials and convex functions, Math. Inequal. Appl., 21 (2018), pp. 773-777.

[2] B. Gavrea,On a convexity problem in connection with some linear operators, J. Math. Anal. Appl., 461 (2018), pp. 319- 332.

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ON THE DECOMPOSITION OF SOME LINEAR POSITIVE OPERATORS

Ioan Gavrea

Department of Mathematics, Technical University, Cluj-Napoca, Romania

[[email protected]]

MSC 2010: 41A35, 41A10, 41A25

Keywords:Linear operators, Beta operators, positive operators.

In this talk we consider the decomposition of some dis- cretely linear positive operators. Our results generalize the results obtained in [1].

REFERENCES

[1] M. Heilmann, I. Rasa, On the decomposition of Bernstein operators, Numer. Funct. Anal. Optimiz., 36 (2015), pp. 72- 85.

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INVARIANT MEASURES ASSOCIATED TO ϕ - MAX - IFSs WITH PROBABILITIES

Flavian Georgescu

Department of Mathematics and Computer Science, University of Pite¸sti, Romania, Pite¸sti, Arge¸s, Romania [[email protected]]

MSC 2010: 28A80, 37C70, 54H20

Keywords:ϕ-max-contraction, comparison function, iterated function system with probabilities, Markov operator, fixed point, invariant measure.

We prove that the Markov operator associated to an iterated function system consisting ofϕ-max-contractions with probabilities has a unique invariant measure whose support is the attractor of the system.

REFERENCES

[1] F. Georgescu, R. Miculescu, A. Mihail, A study of the attractor of aϕ-max-IFS via a relatively new method, J. Fixed Point Theory Appl., (2018) 20:24, https://doi.org/10.1007/

s11784-018-0497-6.

[2] J. E. Hutchinson,Fractals and self similarity, Indiana Univ.

Math. J., 30 (1981), pp. 713–747.

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SPECTRAL COLLOCATION SOLUTIONS TO A CLASS OF NONLINEAR BVPs ON THE HALF LINE

C˘ alin-Ioan Gheorghiu

Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

[[email protected]]

MSC 2010: 65L10, 65L60, 65L70, 65N35

Keywords:Collocation, Laguerre-Gauss-Radau, Kidder problem, nonlinear Schr¨odinger, radially symmetric solutions.

The existence, uniqueness and regularity of solutions to the boundary value problems

1

p(x)(p(x)u⁰(x))⁰ =q(x)f(x, u(x), p(x)u⁰(x)), x∈(0,∞) αu(0)−βu⁰(0) =r, limx→∞u(x) = 0,

(1) are established. We assume α > 0, β ≥ 0 and r is a given constant and f, p and ¹_q are continuous. Our aim is to accurately approximate the solutions of these problems, as well as to some PDEs reducible to (1), by a high order Laguerre-Gauss-Radau collocation method (see [1], Ch. 2).

REFERENCES

[1] C.I. Gheorghiu, Spectral Collocation Solutions to Problems on Unbounded Domains,Casa C˘art¸ii de S¸tiint¸˘a, Cluj-Napoca, 2018 (see also https://ictp.acad.ro/gheorghiu)

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PARAMETER ESTIMATIONS FOR A

LINEAR PARABOLIC FRACTIONAL SPDE WITH JUMPS

Wilfried Grecksch

^1,∗

, Hannelore Lisei, Jens Lueddeckens

1Institute of Mathematics, Martin-Luther-University of Halle-Wittenberg, Germany

[[email protected]]

MSC 2010: 60H15, 62F12, 60G22

Keywords: Stochastic partial differential equations, parameter estimations.

A drift parameter estimation problem is studied for a linear parabolic stochastic partial differential equation driven by a multiplicative cylindrical fractional Brownian motion with Hurst index h ∈]1/2,1[ and a multiplicative Poisson process with values in a Hilbert space. Equations are introduced for the Galerkin approximations of the mild solution process. A mean square estimation criterion is used for these equations. It is proved that the estimate is unbiased and weakly consistent for the original problem.

REFERENCES

[1] I. Cialenko, Parameter estimation for SPDEs with multiplicative fractional noise, Stochastics and Dynamics, 10 (2010) no. 4, pp. 561-576.

[2] J. Lueddeckens, Fraktale stochastische Integralgleichungen im White-Noise-Kalk¨ul,Dissertation Martin-Luther-Univer- sit¨at Halle-Wittenberg, April 2017.

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HALF-LINEAR DIFFERENTIAL

OPERATORS IN OSCILLATION THEORY

Petr Hasil

Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Czech Republic

[[email protected]]

MSC 2010: 34C10, 34C15

Keywords:Riccati technique, half-linear equation, oscillation theory, oscillation constant, conditional oscillation.

This is a joint work with M. Vesel´y. We consider half- linear differential equations given by operators with one- dimensional p-Laplacian. The main subject of this talk is to present results concerning the conditional oscillation of such equations, i.e., we find a border value which sepa- rates oscillatory equations from non-oscillatory ones and we explicitly determine this borderline (depending on the equations coefficients). For some of the presented results, we refer to [1, 2].

REFERENCES

[1] P. Hasil, M. Vesel´y,Oscillation and non-oscillation of half- linear differential equations with coefficients determined by functions having mean values, Open Math., 16 (2018), pp.

507–521.

[2] P. Hasil, M. Vesel´y, Oscillation and non-oscillation results for solutions of perturbed half-linear equations, Math. Meth- ods Appl. Sci., 41 (2018), pp. 3246–3269.

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REPRESENTATIONS FOR K-TH ORDER KANTOROVICH MODIFICATIONS OF LINKING OPERATORS

Margareta Heilmann

^1,∗

, Ioan Ra¸sa

1School of Mathematics and Natural Sciences, University of Wuppertal, Gaußstraße 20, D-42119 Wuppertal, Germany [[email protected]]

MSC 2010: 41A36, 41A10, 41A28

Keywords:Linking operators, Baskakov-Durrmeyer type operators, Kantorovich modifications of operators.

In our talk we investigate linking operators acting on the unbounded interval [0,∞). Linking operators for the Sz´asz- Mirakjan case were defined by P˘alt˘anea in [3] and for Bas- kakov type operators by Heilmann and Ra¸sa in [2]. If the linking paprameterρis a natural number we present a representation for Kantorovich variants of arbitrary order in terms of the Baskakov and Sz´asz-Mirakjan basis functions, respectively. This leads to a simple proof of convexity properties which also solves an open problem mentioned in [1].

At the end of our talk we present a conjecture concerning a limit to B-splines which is connected to the above mentioned representation.

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REFERENCES

[1] K. Baumann, M. Heilmann, I. Ra¸sa,Further results forkth order Kantorovich modification of linking Baskakov type operators, Results Math. 69 (3), (2016), 297-315.

[2] M. Heilmann, I. Ra¸sa, k-th order Kantorovich modification of linking Baskakov type operators, in: Recent Trends in Mathematical Analysis and its Applications, Proceedings of the Conference ICRTMAA 2014, Rorkee, India, December 2014, (ed. P. N. Agrawal et al.), Proceedings in Mathematics

& Statistics 143, 229-242.

[3] R. P˘alt˘anea, Modified Sz´asz-Mirakjan operators of integral form, Carpathian J. Math., 24(3) (2008), 378-385.

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SOME APPROXIMATION PROPERTIES OF URYSOHN TYPE OPERATORS

Harun Karsli

Department of Mathematics, University of Bolu Abant Izzet Baysal, Turkey

[karsli [email protected]]

MSC 2010: 41A25, 41A35, 47G10, 47H30

Keywords:Urysohn operators, linear positive operators, approximation.

In the present work, our aim is generalization and extension of the theory of interpolation of functions to functionals by means of Urysohn type operators. In accordance with this purpose, we introduce and study a new type of Urysohn type operators. In particular, we investigate the convergence problem for operators that approximate the Urysohn type operator. We construct our operators by using a nonlinear forms of the kernels together with the Urysohn type operator values instead of the sampling values of the function.

REFERENCES

[1] C. Bardaro, I. Mantellini,On the reconstruction of functions by means of nonlinear discrete operators. J. Concr. Appl.

Math. 1 (2003), no. 4, pp. 273–285.

[2] C. Bardaro, J. Musielak, G. Vinti, Nonlinear Integral Op- erators and Applications, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 9, xii + 201 pp., 2003.

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[3] C. Bardaro, G. Vinti,Urysohn integral operators with homogeneous kernel: approximation properties in modular spaces, Comment. Math. (Prace Mat.) 42 (2002), no. 2, pp. 145-182.

[4] C. Bardaro, H. Karsli and G. Vinti, Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems, Applicable Analysis, Vol. 90, Nos. 3–4, March–

April (2011), pp. 463–474.

[5] C. Bardaro, H. Karsli and G. Vinti, On pointwise convergence of linear integral operators with homogeneous kernels, Integral Transforms and Special Functions, 19(6), (2008), pp. 429-439.

[6] D. Costarelli, G. Vinti,Degree of approximation for nonlinear multivariate sampling Kantorovich operators on some functions spaces, Numer. Funct. Anal. Optim. 36 (2015), no. 8, pp. 964–990.

[7] D. Costarelli, G. Vinti, Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing, Numer. Funct. Anal. Optim. 34 (2013), no. 8, pp. 819–844.

[8] I.I. Demkiv,On Approximation of the Urysohn operator by Bernstein type operator polynomials, Visn. L’viv. Univ., Ser.

Prykl. Mat. Inform., Issue 2, pp. 26 - 30.

[9] H. Karsli,Approximation by Urysohn type Meyer-K¨onig and Zeller operators to Urysohn integral operators, Results Math.

72 (2017), no. 3, pp. 1571–1583.

[10] H. Karsli,Some convergence results for nonlinear singular integral operators, Demonstratio. Math., Vol. XLVI No 4, (2013), pp. 729-740.

[11] V.L. Makarov, and I.I. Demkiv,Approximation of the Ury- sohn operator by operator polynomials of Stancu type, Ukra- inian Math Journal, 64(3)(2012), pp. 356 - 386.

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A NEW METHOD FOR SOLVING

COMPLEX SYMMETRIC SYSTEMS OF LINEAR EQUATIONS

Davod Khojasteh Salkuyeh

Department of Mathematics, University of Guilan, Rasht, Iran [e-mail [email protected]]

MSC 2010: 65F10

Keywords: Complex, symmetric, convergence, symmetric positive definite.

We consider the system of linear equations (W+iT)u=b, where W, T ∈ R^n×n are symmetric positive semidefinite matrices with at least one of them being positive definite andi=√

−1. Lettingb =p+iqandu=x+iy, we propose a new iterative method for two-by-two block real equivalent

form

W −T

T W

x y

= p

q

,

of the system. Convergence of the proposed method is studied and the numerical results of the method are compared with those of the MHSS [1], the PMHSS [2], the TTSCSP [3], the CRI [4] methods.

REFERENCES

[1] Z.-Z. Bai, M. Benzi, F. Chen,Modified HSS iterative methods for a class of complex symmetric linear systems, Com- puting 87 (2010), pp. 93-111.

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[2] Z.-Z. Bai, M. Benzi, F. Chen, On preconditioned MHSS iteration methods for complex symmetric linear systems, Nu- mer. Algor. 56 (2010), pp. 93-317.

[3] D.K. Salkuyeh, T.S. Siahkolaei,Two-parameter TSCSP method for solving complex symmetric system of linear equa-

tions, Calcolo (2018), pp. 55: 8.

https://doi.org/10.1007/s10092-018-0252-9.

[4] T. Wang, Q. Zheng, L. Lu, A new iteration method for a class of complex symmetric linear systems, J. Comput.

Appl. Math., 325 (2017), pp. 188-197.

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CONVERGENCE RATES FOR AN

INERTIAL ALGORITHM OF GRADIENT TYPE ASSOCIATED TO A SMOOTH NONCONVEX MINIMIZATION

Szil´ ard Csaba L´ aszl´ o

[[email protected]]

MSC 2010: 90C26, 90C30, 65K10

Keywords:Inertial algorithm, nonconvex optimization, Kurdyka- Lojasiewicz inequality, convergence rate.

We investigate an inertial algorithm of gradient type in connection with the minimization of a nonconvex differentiable function. The algorithm is formulated in the spirit of Nes- terov’s accelerated convex gradient method. We show that the generated sequences converge to a critical point of the objective function, if a regularization of the objective function satisfies the Kurdyka- Lojasiewicz property. Further, we provide convergence rates for the generated sequences and the objective function values formulated in terms of the Lojasiewicz exponent.

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ON THE INTERPLAY BETWEEN

MARKOV OPERATORS PRESERVING POLYNOMIALS AND CONVEX COMPACT SUBSETS OF R

^d

Vita Leonessa

Department of Mathematics, Computer Science and Economics, University of Basilicata, Potenza, Italy [[email protected]]

MSC 2010: 47B65, 47D07

Keywords: Markov operators, polynomial preserving property, second-order elliptic differential operator, Markov semigroup.

LetK be a convex compact subset ofR^d, d≥1, with non- empty interior. In this talk we are interesting to all positive linear operators T acting on the spaceC(K) of continuous functions on K which leave invariant the polynomials of degree at most 1 and which, in addition, map polynomials into polynomials of the same degree m, m≥2.

In particular, we discuss the existence of operators T whenK is a non-trivial strictly convex subset ofR^d,d≥2, discovering, among other things, a characterization of ellip- soids and balls ofR^d. A discussion of the above polynomial preserving property in the setting of product spaces, as well as for convex convolution products of positive linear operators, is also presented.

Such a polynomial preserving property play a central role in studying the possibility to get a representation/

approximation formula for semigroups generated by certain differential operators, associated with T, in terms of

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constructively defined linear positive operators associated with the same T (see, e.g. [1, 4]). Anyway, we point out that this is not the only case where a property of this kind is required (see, e.g. [3]). Moreover, in the literature there are many classes of positive linear operators that satisfy it.

All results presented are contained in the joint work [2].

REFERENCES

[1] F. Altomare, M. Cappelletti Montano, V. L., I. Ra¸sa, On differential operators associated with Markov operators, J.

Funct. Anal.266 (2014), pp. 3612–3631.

[2] F. Altomare, M. Cappelletti Montano, V. L., I. Ra¸sa, On Markov operators preserving polynomials, J. Math. Anal.

Appl.415(2014), pp. 477–495.

[3] O. Agratini, A sequence of positive linear operators associated with an approximation process, Appl. Math. Comput.

269(2015), pp. 23–28.

[4] F. Altomare, M. Cappelletti Montano, V. L., I. Ra¸sa, El- liptic differential operators and positive semigroups associated with generalized Kantorovich operators, J. Math. Anal.

Appl.458(2018), pp. 153–173.

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SOME NEW RESULTS CONCERNING THE CLASSICAL BERNSTEIN QUADRATURE FORMULA

Dan Micl˘ au¸s

Department of Mathematics and Computer Science, Technical University of Cluj-Napoca, North University Center at Baia Mare, Romania

[[email protected]]

MSC 2010: 41A36, 41A80.

Keywords: Bernstein operator, convex function, divided differ- ence, Popoviciu theorem, remainder term.

In this talk, we intend to highlight an applicative side of the classical Bernstein polynomials, in contrast to the well- known theory of the uniform approximation of functions.

We will present some new results concerning the classical Bernstein quadrature formula

Z b a

F(x)dx≈ _n+1^b−a

n

X

k=0

F

a+ ^k(b−a)_n ,

which can be found in [1] and in another recent paper submitted for publication.

REFERENCES

[1] D. Micl˘au¸s, L. Pi¸scoran, A new method for the approximation of integrals using the generalized Bernstein quadrature formula, Appl. Math. Comput., (accepted for publication 2018).

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A NEW ALGORITHM THAT GENERATES THE IMAGE OF THE ATTRACTOR OF A GIFS

Radu Miculescu

Department of Mathematics and Computer Science, University of Bucharest, Romania, Academiei Street 14, 010014, Bucharest, Romania

[[email protected]]

MSC 2010: 28A80, 37C70, 41A65, 65S05, 65P99

Keywords:Generalized infinite iterated function system (GIFS), attractor, deterministic algorithm, grid algorithm.

We provide a new algorithm (called the grid algorithm) designed to generate the image of the attractor of a generalized iterated function system and we compare it with the deterministic algorithm regarding generalized iterated function systems presented by P. Jaros, L. Ma´slanka and F. Strobin in [Algorithms generating images of attractors of generalized iterated function systems, Numer. Algor., 73 (2016), 477-499].

REFERENCES

[1] P. Jaros, L. Ma´slanka, F. Strobin, Algorithms generating images of attractors of generalized iterated function systems, Numer. Algor., 73 (2016), pp. 477-499.

[2] A. Mihail, R. Miculescu, Applications of Fixed Point The- orems in the Theory of Generalized IFS, Fixed Point The- ory Appl. Volume 2008, Article ID 312876, 11 pages doi:

10.1155/2008/312876.

[3] A. Mihail, R. Miculescu, Generalized IFSs on Noncompact Spaces, Fixed Point Theory Appl. Volume 2010, Article ID 584215, 11 pages doi: 10.1155/2010/584215.

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ON SEVERAL INEQUALITIES IN AN INNER PRODUCT SPACE

Nicu¸sor Minculete

Transilvania University, Romania [[email protected]]

MSC 2010: 46C05, 26D15, 26D10

Keywords:Inner product space, Cauchy-Schwarz inequality.

The aim of this presentation is to prove new results related to several inequalities in an inner product space. Among these inequalities we will mention inequality Cauchy-Sch- warz’s inequality. Also we obtain some applications of these inequalities.

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ON THE UNBOUNDED DIVERGENCE OF SOME INTERPOLATORY PRODUCT

INTEGRATION RULES

Alexandru Mitrea

[[email protected]]

MSC 2010: 41A10, 65D32

Keywords: Product integration, Condensation of singularities, superdense set.

Based on some principles of Functional Analysis, this paper emphasizes the topological structure of the set of unbounded divergence for some interpolatory product quadrature formulas on Jacobi and equidistant points of the interval [−1,1], associated with the Banach space of all s-times continuously differentiable functions and with a weighted Banach space of absolutely integrable functions of order p >1.

REFERENCES

[1] H. Brass, K. Petras,Quadrature Theory. The Theory of Nu- merical Integration on a Compact Interval, American Math- ematical Society, Providence (2011).

[2] S. Cobzas, I. Muntean,Condensation of singularities and divergence results in Approximation Theory, J. Approx. The- ory, 31 (1981), pp. 138-153.

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[3] A.I. MitreaOn the dense unbounded divergence of interpolatory product integration on Jacobi nodes, Calcolo, vol.55, No.1, Art.10 (2018), pp. 1-15.

[4] I.H. Sloan, W.E. Smith,Properties of interpolatory product integration rules, SIAM J. Numer. Anal., 19 (1982), pp. 427- 442.

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APPROXIMATION PROPERTIES OF CERTAIN BERNSTEIN-STANCU TYPE OPERATORS

Carmen Violeta Muraru

Department of Mathematics and Informatics, Vasile Alecsandri University of Bacau, Romania

[[email protected]]

MSC 2010: 41A36, 41A25

Keywords:Bernstein-Stancu operator, q-integers, rate of convergence, moduli of continuity.

In this paper we introduce and investigate a new operator of Bernstein-Stancu type, based on q polynomials. We study approximation properties for these operators based on Korovkin type approximation theorem and study some direct theorems. Also, the study contains numerical con- siderations regarding the constructed operators based on Maple algorithms.

This is a joint work with Ana-Maria Acu from Lucian Blaga University of Sibiu, Romania, Ogun Dogru from Gazi University, Turkey and Voichita Adriana Radu from Babes- Bolyai University, Romania.

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ON THE FIRST MEAN-VALUE THEOREM FOR INTEGRALS

Vicuta Neagos

^1,∗

, Andra-Gabriela Silaghi

²

1,2Department of Mathematics, Technical University of Cluj-Napoca, Romania

[[email protected]], [fun [email protected]]

MSC 2010: 26A24

Keywords:Positive linear functionals, mean-value theorems, integrals of Riemann and Lebesgue type.

We improve the classical First Mean-value Theorem for In- tegrals and obtain related results.

REFERENCES

[1] R.G. Bartle, A modern theory of integration, Graduate Studies in Mathematics, vol. 32, American Mathematical Society, Providence, RI (2001).

[2] P. Khalili, D. Vasiliu,An extension of the mean value theorem for integrals, Internat. J. Math. Ed. Sci. Tech. 41(5)(2010), pp. 707–710.

[3] M. Monea, An integral mean value theorem concerning two continuous functions and its stability, Int. J. Anal., Art. ID 894,625, 4 (2015).

[4] P.K. Sahoo, T. Riedel,Mean value theorems and functional equations, World Scientific Publishing Co. Inc., River Edge, NJ (1998).

[5] S.G. Wayment, An integral mean value theorem, Math.

Gaz., 54(389) (1970), pp. 300–301.

Cluj-Napoca, Romania, September 6–9, 2018

International Conference on Numerical Analysis and

Approximation Theory

Cluj-Napoca, Romania, September 6–9, 2018

BOOK OF ABSTRACTS AND PROGRAM

Honorary Chairs

Scientific Committee:

Invited speakers:

Francesco Altomare (Italy) Carlo Bardaro (Italy)

Maria Garrido-Atienza (Spain) Heiner Gonska (Germany)

Aaron Melman (USA)

Gradimir Milovanovi´ c (Serbia) Cihan Orhan (Turkey)

Wolfgang L. Wendland (Germany)

Organizing Committee:

CONTENTS

CONFERENCE SCHEDULE

Thursday, September 6

Plenary Talks

Contributed Talks

Plenary Talk

Contributed Talks

Friday, September 7

Plenary Talks

Contributed Talks

Plenary Talk

Contributed Talks

Saturday, September 8

Plenary Talks

Contributed Talks

September 9

Sunday,

LIST OF PARTICIPANTS

Participants Page

Ulrich Abel 32

Ana Maria Acu 35

Mohd Ahasan 36

Francesco Altomare 18 Alina Ramona Baias 38, 80

Carlo Bardaro 20

Marius Birou 39

Mirella Cappelletti Montano 40

Emil C˘ atina¸s 42

Ganna Chekhanova 46

Larisa Cheregi 43

Maria Cr˘ aciun 44

Madalina Dancs 38

Markus Dietz 46

Julian Dimitrov 47

Sorin Gal 49

Maria Garrido-Atienza 22

Bogdan Gavrea 51

Ioan Gavrea 52, 73

Flavian Georgescu 53

Participants Page C˘ alin Gheorghiu 54

Heiner Gonska 23

Wilfried Grecksch 55

Petr Hasil 56

Margareta Heilmann 57

Laura Hodis 74

Sever Hodis 38

Harun Karsli 59

Davod Khojasteh Salkuyeh 61 Szil´ ard Csaba L´ aszl´ o 63

Vita Leonessa 64

Alexandra-Ioana M˘ adut¸a 74

Aaron Melman 24

Dan Micl˘ au¸s 66

Radu Miculescu 67

Gradimir V. Milovanovi´ c 26

Nicusor Minculete 68

Alexandru Mitrea 69

Camelia Liliana Moldovan 75

Carmen Violeta Muraru 71

Vicuta Neago¸s 72, 43

Adonia Augustina Opri¸s 73

Participants Page

Cihan Orhan 28

Diana Otrocol 74

Radu P˘ alt˘ anea 75 Flavius P˘ atrulescu 76 Ion P˘ av˘ aloiu 42

Daniel Pop 78

Dorian Popa 80

Heinz-Joachim Rack 82 Voichit¸a Adriana Radu 83