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Rev. Anal. Num´er. Th´eor. Approx., vol. 31 (2002) no. 2, pp. 135–141 ictp.acad.ro/jnaat

SEQUENCES OF LINEAR OPERATORS

RELATED TO CES `ARO - CONVERGENT SEQUENCES

MIRA-CRISTIANA ANISIUand VALERIU ANISIU

Abstract. Given a Ces`aro-convergent sequence of real numbers (an)n∈N, a se- quence (ϕn)n∈N of operators is defined on the Banach spaceR(I, F) of regular functions defined onI= [0,1] and having values in a Banach spaceF,

ϕn(f) = 1 n

n

X

k=1

akf nk .

It is proved that if, in addition, the sequence |a1|+...+|an n|

n∈N is bounded, then ϕn(f) converges to a·R1

0 f, wherea= limn→∞a1+...+an

n .The converse of this statement is also true. Another result is that the supplementary condition can be dropped if the operators are considered on the spaceC1(I, F).

MSC 2000. 47B38, 26E60.

Keywords. Linear operators, Ces`aro-convergent sequences.

1. INTRODUCTION

Let (an)n∈N be a sequence of real numbers. It will be called Ces`aro-con- vergentif the sequence of its Ces`aro (arithmetic) means is convergent, i.e.

n→∞lim

a1+. . .+an

n ∈R.

For x∈ R, bxc will denote the greatest integer number nx (the integer part ofx).

Given the interval I = [0,1] and a Banach space F 6= {0}, we denote by B(I, F) the Banach space of bounded functions f :IF endowed with the sup norm. The spaceB(I, F) contains as a subspace the set of “step-functions”

E(I, F) ={f :IF :∃t0, . . . , tnI, t0 = 0< t1 < . . . < tn = 1,∃ukF so that f|(t

k−1,tk) = uk, k = 1, . . . , n}. In fact each f ∈ E(I, F) is a finite sum of functions having the form χ[α,β]·u, where 0≤αβ ≤1, u∈F and χ[α,β]is the characteristic function of the interval [α, β].We denote byR(I, F) the Banach space ofregular functions (which admit side limits at each tI), endowed with the uniform norm kfk = supt∈[0,1]kf(t)k. We mention that R(I, F) is the closure in B(I, F) of the subspace E(I, F), and it contains the

“T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68–1, 3400 Cluj-Napoca, Ro- mania ([email protected]).

“Babe¸s-Bolyai” University, Faculty of Mathematics and Computer Science, st. Kog˘alni- ceanu 1, 3400 Cluj-Napoca, Romania, e-mail: [email protected].

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Banach space of continuous functionsC(I, F). More details on these spaces of functions are to be found in [3, p. 137].

We define a sequence of operators associated to (an)n∈N, namely ϕn : R(I, F)→F, n∈N

(1) ϕn(f) = n1

n

X

k=1

akfkn.

Proposition 1. The operatorϕn is linear and continuous, and its norm is given by

(2) kϕnk= n1

n

X

k=1

|ak|.

Proof. The linearity is straightforward. Becausefkn≤ kfk,it follows

(3) kϕn(f)k ≤n1

n

X

k=1

|ak|· kfk,

hence ϕn is also continuous. To obtain the norm of ϕn, we use the inequality (3) and the function

f0(t) =

( (signak)u, fort= kn, k = 1, . . . , n

0, otherwise,

where uF and kuk = 1. We have f0 ∈ E(I, F) ⊆ R(I, F), kf0k = 1 and ϕn(f0) = n1 Pn

k=1

|ak|·u, hence the equality (2) follows.

2. MAIN RESULTS

We are interested in finding conditions on the sequence (an)n∈N in order to obtain the convergence of the sequence of linear operators (1). The theo- rem below guarantees the convergence of (ϕn(f))n∈

Nfor each regular function f ∈ F(I, F). Beside the condition of Ces`aro-convergence for (an)n∈N, the boundedness of a certain sequence related to this is imposed.

Theorem 2. Let there be given a regular function f ∈ R(I, F) and a se- quence (an)n∈N of real numbers satisfying the conditions:

1. (an)n∈N is Ces`aro-convergent to a(limn→∞ a1+...+an n =a);

2. the sequence|a1|+...+|an n|

n∈N

is bounded.

Then the sequencen(f))n∈

N is convergent and

(3)

(4) lim

n→∞ϕn(f) =a· Z 1

0

f.

Proof. At first we shall prove (4) for functionsf of the form (5) f =χ[α,β]·u, where 0≤αβ ≤1, u∈F.

We have ϕn(f) =

1 n

X

k∈N αn≤k≤βn

ak

·u= 1

n X

k∈N k≤βn

ak

·u1

n X

k∈N k<αn

ak

·u.

Ifα = 0 the conclusion follows obviously.

Forα >0 we denotean= a1+...+an n and we write the two sums in the above formula as

X

k∈N k≤βn

ak=bβnc ·abβnc, X

k∈N k<αn

ak=bαnc ·abαncabαnc·θn,

where

θn=

1, forαn∈N 0, otherwise.

We finally obtain

ϕn(f) =bβnc

n ·abβnc−bαnc

n ·abαnc+ abαnc n ·θn

·u.

We have lim

n→∞abαnc = a; but ann = an1−n1an−1, hence lim

n→∞

an

n = 0. It follows that in this case

n→∞lim ϕn(f) = (βa−αa)·u=a· Z 1

0

f.

We consider now the general casef ∈ R(I, F).The sequence |a1|+...+|an n|n∈

N

being bounded, let us chooseM such that |a1|+...+|an n|M for eachn∈N; let also ε >0 be an arbitrary constant. From the definition of the space R(I, F) it follows the existence of the functions fi, i = 1, . . . , p of the type described in (5), withfPpi=1fi

< ε. We have ϕn(f)−a

Z 1 0

f =ϕn f

p

X

i=1

fi+

p

X

i=1

ϕn(fi)−a Z 1

0

fia Z 1

0

f

p

X

i=1

fi.

The norm ofϕn,as given by (2), is kϕnk= |a1|+...+|an n|,hence

(6) ϕn f

p

X

i=1

fi

≤ kϕnk ·f

p

X

i=1

fi

M ·ε.

Taking into account the first part of the proof, for each i= 1, . . . , pthere exists ni ∈ N so that ϕn(fi)−a·R01fi < εp for nni. It follows that for

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n≥ max

i=1,...,pni we have

(7)

p

X

i=1

ϕn(fi)−a· Z 1

0

fiε.

But

(8) a

Z 1 0

f

p

X

i=1

fi≤ |a| ·ε,

and the inequalities (6), (7) and (8) imply that

ϕn(f)−a· Z 1

0

fM·ε+ε+|a| ·ε, forn≥N.

It follows that the conclusion holds also for the general casef ∈ F(I, F).

Remark 1. The Ces`aro-convergence of (an)n∈N in Theorem 2 does not necessarily imply the boundedness of |a1|+...+|an n|n∈

N. For example, let the sequence be given by

an=

n, nodd

−√

n−1, neven.

Then

an=

1/√

n, n odd 0, n even, hence lim

n→∞an= 0,but lim

n→∞

|a1|+...+|an|

n = lim

n→∞|an|=∞.

The condition of Ces`aro-convergence imposed to the sequence (an)n∈N in Theorem 2 is a natural one and cannot be relaxed, neither the boundedness of the sequence |a1|+...+|an n|n∈

N.In fact, Theorem 2 does admit the following converse:

Theorem3. Letn)n∈Nbe the sequence (1)of linear operators associated to the sequence of real numbers (an)n∈N. If lim

n→∞ϕn(f) exists for every f ∈ C(I, F)⊆ R(I, F),then:

1. (an)n∈N is Ces`aro-convergent to a( lim

n→∞

a1+...+an

n =a);

2. the sequence |a1|+...+|an n|n∈

N is bounded.

Proof. The first conclusion follows by taking f(t) = u for each tI, with uF \ {0}.In this caseϕn(f) = a1+...+an nu.

The norm of the operators ϕn in the space C(I, F) is the same as in (1).

Indeed, in the proof of Proposition 1, the function f0 can be modified to a continuous and piecewise affine one which takes also the values (signak)u on the points nk, k = 1, . . . , n. From the principle of uniform boundedness

[4, p. 66] the second conclusion follows.

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Remark 2. Using a principle of condensation of singularities [2], one can prove that the convergence in (4) does not hold for “typical” continuous func- tions. Even stronger principles of condensation of singularities [1] may be

applied.

In what follows we shall prove that for the class of continuous functions having also a continuous derivative, the condition of boundedness of the se- quence |a1|+...+|an n|n∈

N is no longer necessary. In this setting, the principle of uniform boundedness does not work, because C1(I, F) endowed with the uniform norm is not a Banach space. The norm ofϕnis still the same. In this case we have

Theorem 4. Let there be given a function f ∈ C1(I, F) and a sequence (an)n∈N of real numbers which is Ces`aro-convergent toa( lim

n→∞

a1+...+an

n =a).

Then

(9) lim

n→∞ϕn(f) =a· Z 1

0

f.

Proof. We writeϕn(f) successively as ϕn(f) = n1

n

X

k=1

kak−(k−1)ak−1fnk

= n1

n

X

k=1

kakfknn1

n−1

X

k=1

kakfk+1n

=

n−1

X

k=1

akknfnkfk+1n +anf(1).

We bring now into the scene the continuous functionggiven byg(t) =tf0(t) and express ϕn(f) in the form

ϕn(f) =−n−1P

k=1

akknfk+1n fnkn1f0(kn)1nn−1P

k=1

akknf0(nk) +anf(1) (10)

=−n−1P

k=1

akknfk+1n fnk1nf0(nk)1n Pn

k=1

akg(kn) +n1anf0(1) +anf(1).

Applying Theorem 2 for the functiongand for the sequence (an)n∈N conver- gent to a, for which obviously lim

n→∞

a1+...+an

n = a and |a

1|+...+|an| n

n∈N is bounded (because of the convergence of (an)n∈N) we get

n→∞lim

1 n

n

X

k=1

akgnk=a· Z 1

0

g=a·f(1)a· Z 1

0

f

(the last equality is a consequence of an integration by parts). The function f0 being uniformly continuous on I, given ε > 0 and n sufficiently large, we

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obtain as a consequence of a mean theorem [3, p. 154]

fk+1n fkn1nf0kn 1n sup

t∈(kn,k+1n )

f0(t)−f0kn< nε, hence

n−1

X

k=1

akknfk+1n fknn1f0kn

n−1

X

k=1 M ε

n2 k= n−12n M εM ε, where M is a upper bound for the convergent sequence (|an|)n∈

N. It follows that

n→∞lim

n−1

X

k=1

akknfk+1n fnkn1f0nk = 0.

We take the limit in (10) and get the conclusion.

As an application of Theorem 2 we obtain a somehow surprising result, proved directly for differentiable functions with bounded derivative in [5]: For each a∈[0,1],there exist εn∈ {0,1}such that

n→∞lim

1 n

n

X

k=1

εkfnk=a· Z 1

0

f, ∀f ∈ R(I, F).

To prove this equality, we choose εn =an=b(n+ 1)ac − bnac, n∈Nwhich satisfyεn∈ {0,1} and limn→∞ a1+...+an n =a.

Open question. It would be interesting to find out if the conclusion of Theorem 2 also holds for a class of functions more general than the regular ones as, for example, the Riemann integrable real-valued functions. For the class of Lebesgue integrable functions the result does not hold, as the function of Dirichlet typef :IF =R,

f(t) =

arbitrary, t∈[0,1]∩Q

0, otherwise

shows.

REFERENCES

[1] Anisiu, V.,A principle of double condensation of singularities usingσ-porosity, “Babe¸s- Bolyai” Univ., Fac. of Math., Research Seminaries, Seminar on Math. Analysis, Preprint Nr.7, pp. 85–88, 1985.

[2] Cobzas¸, S.andMuntean, I.,Condensation of singularities and divergence results in approximation theory, J. Approx. Theory,31, pp. 148–153, 1981.

[3] Dieudonn´e, J.,Fondements de l’analyse moderne, Paris, Gauthier-Villars, 1963.

[4] Dunford, N. andSchwartz, J. T.,Linear Operators. Part 1: General Theory, John Wiley & Sons, New York, 1988.

[5] Trif, T., On a problem from the Mathematical Contest, County Stage, Gazeta Mate- matic˘a CVI (11), pp. 394–396, 2001 (in Romanian).

Received by the editors: January 14, 2002.

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