View of On interpolation operators - IV (Estimates in the neighbourhood of nodal points for differentiable functions)

J\TAT}IEI,IATICA

-

^REVUI] D'ANALYSE NUNIIiRI QUE

ET

DIi

TIIÉORIE DE I-'APPROXIMATION

Í,'ANTTI,YSE NUTIÉRIQUE

ET I,A THÉORIE DD

L2APPROXIÛTA.TION Tome 14,

No 2,

lÐ86, pp. ¹³⁷

-145

ON INTDRPOITATION OPDRATORS-IY

(ESTTMA1ES

IN TEE

NEIGHBOURTTOOD

Or¡ NODAL

POtrNTS FOR,

DII'I'ERENTIABI-,8

FUNCTIONS)

I{. ll. SRIVr\S]'AVA and lì. B. SAXIìNA (Dar cs Salaan-r) (Lucl<nolv)

l. Introduction and

^tesults.

Man¡' constructive

a,pproâch.es have been mad.e

after the

celebratecl

Telyakovskii [8]

^and. Gopengauz's

[1]

-bheorems

that

were established

through

classical methotls. Contritrutions

by

Saxena, Freud,

Vertesi [2]

^and,

of

course, ma,ny others

in

i;his clirec- .bion are not beyond comprehension. Recenl,ly,

in

a series of papers l4t ^5r

6]

we

formulated certain

iclentities rvhich formed

the

ac1,ual basis

lor

^the

.construction of interpolatory

polynomials

leading to the

reprotluction ,of Telyakovshii-Gopengauz's

theorem

_18]

for functions

whose

first

deri-

vatives are

continuous.

Following

an idea

put forth by

lVleir

[3]' iij ^is

^{possible now}

to

con-

struct a

^sequence of

positive linear interpolary

^operators

^rvhich

^satisfy Teþakovkii-Gopengauz's

theorem

(and,

of

course, some

other type

^also

in which

estimates of the differences can be expressed"

in

tcrms of the nt¡ds .,of nodal points).

It ^is worthwhile to mention that

^1,hese estirnates

reflect

l,he

fact that the

operators are

interpolatory:

2. The

oparator

Q"*(Í'; ^{n) (m} 2

^0).

^Lct

^ø

:

cosfr

frho:

^cos¿¡;¿

'(2.1)

lnc - --

t¡o:--)lt,:U.n

11'

:anclforh:O¡2tt,-L

.(2.2)

sin ^(ú

-

^{f*n) cos}

|-A - ^rr,l

gr '(ü)

:

znsinf,$-tr,)

*lt " ^',8,

^cos

ⁱ

^(¿

-

^t,n)

^-t

cos n'(t_,tr,^))

:3 - ^{ç. 151'1}

If

^there is nô danger òf confusion,

rie'sltll_lwrit"

^tn,

^ntr

g*(ú) etc.

in

place

ît

^tr"r-

^nri,9¡,

^{(1j) õtc.}

^{froù ilow}

onwarcìs' Denote

by

2n-l

¡u (¿)

7 l\

rz¡o+q¡n(¿)

: I ^9l'*o

^(t')

'l ON INT-E.R'POLATION OPERATORS _ rV

1iJ0

,,'n

r^lrrr,

tnr' rur:j

¿ :

:

1 r:i' haue

.fot.r e

_ll-1,1.1

.(2.10) _{lQll,U; a)-f{,,)} (n)l <

^c,,,

{ry\"-u^,r*¡(@)

^,,)

- ^0,

^nI

tive constant ancl

l;,Ám) (

.) _{is Ure}

_usual

and (2.2)

that the

follorving

iclentity

_{holcls :}

Qn,,(Ï,ø) =

^1,

_{,l =} I

*

lcosú

-

^cos

^t,l-+t-r

rzn,.r,t(t) I<. B: SRIVASTAVA, ñ.; B. SAXENå'

,tr+2

:I

^{cos 2vt¿ü} a,n a,bsolutc nosi

c,ontin¡i1,)' of ^-¡t,,t .

trom (1.6)-(2.4)

á.38 2

(2.3)

(2.6)

(2.8) where (2.e)

rt hct'c _{c,,, is} modulus _of

lVo

obsclrr.e (2.r]_) wlrere ^{ø.,n ts a,l'e} polynomials

in

^m

of

d'cgree

12m' f ³

^{such l'hat}

(2.4)

Ù'12

(Zn)z''+s

: _Ð

o,n.

v:0

tzn+ctn(t)

is a

polynomial

in

^coßÙ

of

^d'eg'r'ee

"¿"¡llt,f,$*T-jT _äTïi.T-

^?-

Ji'i

^:

i.e., Let it

be

c¡

, ^i.e.t

(2.ú) 1> ¡,'(f))c,v'

Let

us also ^assume

Fo,,o(fr)

:

gï'"*n (t) Fr,r,(û)

:

_9';:+o (t)

g,,,(u):

^qzn+A(t)

-l

q'rTiIî(t);

[ü

is ryorl'c,rrn.ar,irrg

_{tJrat (2.10)}

_{is the}

_stronger

losulls

csrabtishiq vcrs

:-,!;,f"'ri;ïäËirpels

^14,

^{;ì, 6l}

tlral; rve

hrr,r'c

a constlucl,ii,c and

Jrmptc

proof

_oî

gauz's thco.cm' u-" riòiì' .iåiät-i"å¿

_{rhe compricar}

ou.

asscrtions.

rlt â col_rseqrcr*"}-Þ.i-0) ;,;'ä.;

flr'rnonnu _3:^-

L*,f(,,ù e c[_1,1]

aytcl

_0<u<I be giaett,

_{then fot.}

rrn'.r¡ rt.cttu,r,ctl nu,mher tr., ru'e ltaae ttitifir:m.Ly r:r,

¡l]i

¡-

- _",

(2.r2)

_tQli,I

_(f ; ⁿ⁾ _

lt,) (,r,)i

:

o

{J. ^ra?,,I 1

,

o,,,,,,,

(#),,,:o,n

for

v

:

nt,

lve 'emarr' 7t*(2:!z) e-rpticiry exhibits the interpolatory nat're :'i,"li""n"n?'fi

",*f o"Í*ify3;"*Jt"#, j¿11T,J""iJu"'Tåh;;;i"-i-ö.:

3' rn ,re

forlowing ìve p-rg.,e

u f:y.

asserúions gir.en

in

bhe

form

of lemrna

which we

neccì

ío

"ÁtãÈii*t

äo" ,n.oru^..

LDMMÄ

I:

We haae u,niforml,g

in, l_1,11

,t 1, ^{E tr-}

ür|,,+t-u er,*

(n) : , (T-,)"',

^'-u,u

: olñ,

ancl,

(3.2)

.'Ë ø - ^n,.fntt-v

^tq':,k@)t

:

o

(lfI=

*,1''J-1-2v

¡:o l.-" ₎

o,"n

,'fl;fifrliÏflr?,:T"t?#:""mations

^bv

^putting

^ø

:

^cos

t,

ø, =:sss ¿o

f

^W

- arlrn+r-,er,,,(n):

#, 'l- _l.o.

_Í

-

^cos t,.lr,+t-,, q1,,*4 (t)

7

fzn-t

: t

^'

(r)

[

^{,,Ðr i cos}

^I -

^{cos l/- l''+ 1- '/ ozu r-+ (t)}

Ìa:t¡ ⁿ -X

and-

(2.7J h,*@): #; k:03'

Then for a,ny function / ^given on [-1,1], rve define the

operator Qn,*(f, æ) as follows

Q n^(1, ø)

: _tr

Tt,',,gr,*

(fi) , ^{m )'}

^o

lì-O

(u - ^n)"

tù

{lrr,ur: \

v:0 v! ¡t"r (o¡).

fn ^the

following'we Prove

Tsnonn¡r !'Qn* ff,

^æ)

^is

a rationøI Junctiort' ^{oJ ord'er}

{(znt'l+)

^n'

f

^m'

Qm'

! ^a)n)

^and'

(i) 8,*(1, ^üù :.f(r) lç:

^O,¡t'

øù

^AgL

(î,

ur)

:

;f{u)(ø¡),v

: T-,

n't'

h+j

I<. B. SIII1/I\STAVÁ" Il. B. SA>|DNÀ ^a) ON II\FTERPOLATION OPERÁ,TOR.S * T\/

140 747

\\,herc

j is

tlefined b¡' (3.3)

ì\,Iaking use of

It-t,¡ç _' -L

Ztt'

"ooot1r"rî*r:r';lî"ffi1!.t#*Jho

second parh

of

rho lernrna wo trâ,vrr, on iìc_

(3.7)

qÍ:.1,(â)\

-f I

^P,,,*

v"("1

^(a) J^ft"t

: å('r)r're,(') (--1- [.,'

Norv' _orving

to the

rep-eated use of À:[arl<or,ts irrcrlrLa,lity _ftrr, tÌrr¡ rlt-.r,iyiìti't¡

of the

polynomial, wdhave.;

- ^{- "-} *'-

lcost

-

^cos

^{t* | < 2}

^{sin ¿}

sin-l-l ¿l-

^¿¡,,1 1-z sinz..t;-

t' -

_'-1

a,ntl

(ø

l- ^{Þ)' 4}

^{ZP-t (a,'}

)-

^bo)

(3.8)

ft is

obscrvccl l,h¿rú

(3.e)

'l'hus

.n'c .have (3.10)

lp,{iJ

(ø)l

_< ^{IL !L}

a,nd tlenoting (3,4)

[r-7

^{ip^.,,}

^(r)l

't)¡, ==-

)

^{r, gir.}

!

(tr

-

^1',,)

lV,r'(r))r,-,, I g ø, Iurl>-

2í.'-1,

^r;

: _ll, - ^{.il ,k +} i

^',

k-

^{0,2 n' -=L}

j :

0,2

tt -'1

rve

at

once obl,ain

(3.5) tr

ll ^lr,,

-

#t,.1"')1-nQrr"n,

(fr) (

2"'-"

À..=0

sin f ^{,r)-l -r 2rL -1 1} lqî'.),,(r,)i

4

^{2u €,,,} _{1t r, ,r, (u')}

2 tt,

x

lo* In ^{+ s-t'v}

,li -=0 On ¿rccount of (8.10), (S.9), (B.g) :rnct (i3.?),

l.o

obLain (lì.2).

LnMITA 2.

Itor

t¡

e _{l_1,1]}

a,t¿d, v

:.

0, i:,i

tt

.X

lø-ø¿-lar-l-t-, qr,*lr.t)

-O1n -

ar,. l)ør,¡. 1-.u

, ,),: _{Oit r +}

_lt;

/ú:0 ñ

,ä1, -

ã;,.lut'tt-', q12,,Ø))

-

^O

1r - *,t)ut.t.t-2,,r.: OjrI, *

^l¡.

lhe proof

_{depenrts upon}

the

follorvinp; inequ:r,lit;ies :

sinú

< 2 sinÊ+.-L\

anrr \z I

(3.11)

_{lsiu rrf}

_| ç ^2rsirr t

l, -

^,,.t.

2

We

m¿r,ko use

of

Lcmrli¿r,

1 to

¿rccourplish t,ho ¡l,oof.

4- Proof

tllo

r,rreo*em

2 lor

v

-

^-0.

o'

aocornt r¡f ûhc

itle^tity

(2.11), rve have

frorn

_1Z.O¡, (Z.B) ancf

-þ.?)

ihc

ictentii¡.

(4.1) _l@) *

_Q*o,

_(1; rl : É

_Å:o ^U@)

-

^,r,*

^(*)l

^{gr.,, (n).}

h+r 1 ^{1il-i 7 -v 2¡t-1}sin2,¡ i.2-2v

I t, -

t,,) gîr*. ^o(t)

I Or¡-v

1't 2tu

,å

²

| )u,r,-.{(îiu)

h+j

l¡¡ ì- 1 -v gin¿+ 1- ^v f) ì

+ I

(4n)"'+t-'

sin t ^{rn-t- I -v ]_ 1} srn

t

'¿+1-v 1

5

_{10, lo'+s-tu}

⁺

Zilï+1-\

X

_aftu+z

'll' n

+

t

u#)-'"{'*#-}

SIN \r¡¡.rl-v I 1 ì- I

ì l1 + -- [ç-'

/ ì- t2rr+7-vla _a'ì*'

n T[ete \\te ha,ve made use of

lq'l(¿)l (

^1'

anc[

(3.6) ^lsin

^{ri'ú l}

^<

^l

^z'l

^{sin Ú l}

Thus

from

(3.5) owing

to ^(3'4),

we have (3'1)'

742 K. B, SRIVASTAVA, R. B, SAXE.NA'

We

makc frequenl; use of

(4.2) l\i - ^E

lll

(, -

rh)v./0,l (¡nr)

:

0

(ln -*,1"'to,(nt) ^(lc;-e;'l)

v'=0

which ^{is þasecl} on l,ho

finite

Taylor's expansion. l]Iaking the transforma,tiolr

b)' nsing # :

cos

t¡ frt,:

cos

fr

^and-

using

(2.9)

-

^(2.6),

u'c obtain flottl

(4.1)

.f

(r) _--

^(ì

^,,,

(l

₎

n)

-=

tr

(,/("os

t) -

^{g*,,n) gi"*n} _A,n (Ú)^(¿)

1

,ft,4

A

sinn

nt -- !\

sino ^r¿f

60

⁹⁰

- -L¡-r_r 11 _{ú ntf_}

sin' /¿ú

- * [¡*

^sina

^øf * ,* sit J ^t

^{t (zn)?}

A direct

compul,ation

gives the inequality

1 > ¡ ^{B^(t)}

>- '874'

Now, our

main

aim is to

show

(4.4)

_l8:i),(1,

_n) - ^l$(n)l : o

ON IhI"TENPOLA"TION OPERATONS _ _{rV 143}

+

¡+j

+ -+

^{(./ (cos}

^t) -

^!J1,,,) q\'"*n

(t) - ^Tt * ^T,

^.

À'(¿)

fn

orcler 1,o esl,inrate

thc

stlmm¿r,nds

?,

ancì-

l'r,

tt'c m¿lke uSe of lernma ^1,.

(3.1)

for

^.¡

:- Q,

(4.2).

.I.n

_i,he

simiiar Îilshion, we

cân pt'ove

theolcm 3 fol

^v

: 0

n'hele' we lìa,\,c

to

use

lcmma 2 for

^v

:

0.

We ttistinguish the

follorving trvo

czrscs separatol¡r ^;

c'1asc

r: ^\\rtron U+

" þ#!L

a¡rd

.¿r,sc

rr: when]/t - ^n' -,

^Ia'

-

^{er, lå .}

n, -

_ITr-*

'lo illustrate the validity of

l,heorem 2 ancl 3

for

arbitlalY_ v,

let

us

take

¡hã

particular

^case

of ni -: Z.lġc ha'e from

16l i,he iclenl,il,y

" : I _I

rgogz+srrt

-

^?1680r¿5

)'

^{7-9712 ns}

 r,

(Í) : I _h.o

^{çÎ' (¿)}

7I(Zrt)zl"

--5280r¿

l- 'L28 \ -3I- Fl

soq,sga?r7 -l- 26880 rtl

-

^{22L76 n'ï}

)-7Ð20n- #)

^cos2rtl -f- (sozzo?ù? -l- ^430081¿6

-

^3168r¿

l- ^28- ) _"u*

^4tl|

^-)

^(2lt6n1

*

^{1792 tÚ}

!2E

l

1-2464tt3

|

^it28r,

- ^-

_r.28

^l--

^{cos 6mt -l- --:1}

12IJ ^-

^{cos 8r¿l J}

I

fl'he version

of

theorem B reacls as follows :

(4.5) te[i,l

ff,, n) -.[t\t(n)t : o(#)'-"o,,,, (ry),v: ^L,2

1o

prove theorem 1

for

v

:

Lr 2 tve require the following pí (ú,,)

:

q:;'(ü,,)

: _çfþ-, (tr):

0

for, k:0,2,tu _

^.L

ql _ft,,) * o:

^q2k0(tþ).

(4.6) a,nd (4.7)

2rL-1

,nt

- ^---

l[T-ç,

,IL ^2-')(ù/

v:Lr2

sina ^r¿t

* å

sino rr,l)

]-r

t- ;

^{sinz r¿ú}

^{+ '+}

^{sina r¿l}

- ^-f,

^{sinu n'l}

_]

ff

wo clenote

by

(4.S)

br,,

(t): ^9f

^(r)

then, from (4.6)

and

(4.?),

^.rve _have, ^¡"(ú)

(4.9)

_{b'o,, Qr¡}

:

^g

a,nd

hi',t(to)

: ₈

g;J(¿,,)

- ^Aí,;(tr) bi'2(tt\:Ort¡*j.

therefore lrom

(4.9), (4.8)

and

(4.?), we obtajn

(4.1-0)

_Lçto,r(t) b^,,r(t)f',-,0

:.f'(n¡),k : _{07 n} -l (4.1L)

_Lsr,^(t)

br,r(t)1í:,¡ : _o,

rt

{ j

and

lgu.z$)lt,,,r(t)l!,!,r:.T(nn)

_{

8

qi,(úo)

-

^6;

(ür)}

I

1,' @r)

_

_{1,, (ø*),}

To

complete

the

proof, u'e shall require

(4.L2) f.

il ^f

^- -

^{n,^ ls-v quz@)}

: o(V=-*)'-" , ^v

^--=

^r,!

(4.13) _{,É l* -} {ttl4-vlqi."@)¡ : ^g(VI - n:¡ì'-"

- u .= ^',

¡lo' \ " ) ,v'-=1,2,3

(* *""

1

7l

)

o 5

+ ['

1

t;,

14,t I<. B. SRTVASfÁ,VA, R. B. SAJ<DN¿'

I

I ON INTER,POLATION OPËRATON.S

- ^TV 145

and and

(4.L+) ¿f ,

q,,lu-',1qtr',,(n)l

:o (l/=")' ,v:1,z.

\4re skotch proofs

oT

(a.12)-(4.14).

FromL¡ernma

1, (3.1),

(4.L2)

can

oasily

be

deduced.

For

(4.13)

to establish

'we

ortce

nrore

tako

^(3.3)

into

consideration ancl 'lvrite

Ð

^1.,

-

erl-l'r'-v

q'l¡fu) : It *

^Iz

¡i:0

|

¡li'ft) I (

56 n¿

)XfjilÌ 9i*ily.dernonsrlare

^l,hc

proof of

(a.1a).

rne

sÍùrne methorr applies

to

^get,

flre

'ersion bf

theorem

3

rot, nt

:

2 IìJIITDIìIìNCIìS

where

(4.15) r,

'-=

;-

_srn 1

¿

'i-

_{"o*

_f -

^cos ¿-)'t-"

I

8ç[(¿)

e;(r) -

^{Aå,,(¿) ef (¿)}

Tj^.1,::J: U.te approrimotion ol _{functions bu}

,2

^(1967), 1ôJ-I22 (Rrrssrnrr.¡.

rcIt proccss, .Ann. Uui\'. Sci. Bu<iapest, l0

ot:Ìmatiott, Canad. ìInth. l3ull., Êl (192U), pp.

tt.,,jrll

lntopolotíon ttperntors __I _{(A proof of}

ns), L.,an:rlyse numcriquc

"t ^f"

Ur.ùi"i"i"i_

lllA

pr.ool oI f.inran,s theorcnr for dilfcrcn_

_ ltl11}

^{prool oI} l.cìyakor.skii_Gopengauz,s 0, z, 1981, pp.24?_262.

ouslcii_Gopenguz lheorcnt througlt irtlcrpolalion, 279.

on lht

_opprot:imstion of luncliotts by algebraic b2 _ _2Gl-r.

^3,(¿)

t¡+ j

â,nd

(4.16) t, : #

^(cos

^{Í -}

^cos

^¿,)i

^"

I

8e; ^(¿) _et

0\-

^;,,

(¿) ,pl ^(¿)

^3,,

(¿)

tr'or

the

esl,imation

of

(4.15), ^'we haYc,

from

^(3.4)

.'"#:E,l

**i {*t", *io! _lt -

^t,,,

}'-'* {-* j

^(ü

- Í,,)}8-'?"]lsi{-ar

^I

l:lccciled _4.III.1g84 27-u n

sin f

f (sin f )4-"

z5r

_{lsin? nf I}

r

¹

|

^1z

^{n¡o-" ,Êo}

^{l,u* lo}

" ' (zn¡a-'"

2n-1 lsin? ^{rr,f I}

_I)e paLt ntc nI o f M atlrcntttics,

^Unioe-rsity ol' ftar es Salcrunt,

P. O. lloa JîTtip, I)or es Saloánt TanzenÌa:

I )epat lmetú of Il,I uil:intaties atd Aslronont¡¡,

- LttcA'nou¡ (J niucrsity.

Lucknota* p2AOOf 1U,n¡

Imlia

k=,(t

I

h+j ^vâ

þ+i

. n /sinf

/ 5tð--\) ,

\Z

-

_sin¿

\

^2n,

'-'x j'

: o(V=o)'",

^u

: L, z,

^:i

Ilere

we have rnad,e use of

(4.17)

lçí(¿)l *

,';l

ancl

[(^i,,(¿)l ^<

^8m

Similarly,

owing

to

leí,'(¿)

l . +: