View of On Fourier series

LB0

Berlin pp 3 -39 f4l Paydon, J. ^{I,'. arrd Wa}

Jormat,iotts, ^D:nke

Rccêvcd, 20. I. 1971,

N. NEGOESCi

Remarl¡.7.2.

If.A:C,e:1, _ll.ll :l.l

and

a1-1, thenthepropo- sitions (ii) and (iii) of theorem 6.1 and

propositions

7.I ^are

^{WáU ána}

plrzDoN's precisions

[4] ^of Worpitzky's

theorem.

(1ä)

g":

_C,

_with e:1,

_llzll

-lxl f ^lyl

^and

llrll: ^{l, is a}

commutative algebra. Theorem 6.1 shows

that, if

llo,ll

<i (n:2,3,...)

(the complex numbers ø,, beTong

of the

sqllare centred

at 0 with the

ver- tíces

at the

points

-¡ 1 4 _and 'f ¹⁾ 4l , tn"r, the

r.alue

of

complex continued

fraction (7.1)

(ø^e

C)

belongs

to _{the set}

defined

by the

inecluation

(7.4) _lo, - ^{xl + l%} -! ^{I <} ]

^tt

^{xl + lyl),}

where z

: _x ! iy

^and

Øt: út]_

iþr.

For

instance,

if ør:2 + ^{3i, then}

(7.4) shows Lhat

z

belongs

to

con-

vex hull of the points

| ^{+ t,,}

²

⁺ +,i,

⁷

+

^gi,,

² +

^8¿.

(iy) út: C, with e: l,

_ll"ll

: rrr"* {lxl,

ly

l} ^{and lllll} :

1,

is

^a comrnntative algebra. Then,

if

llø,,11"

I t":2, 3, ...)

^(n,, r.rlns

a

^sçluare

rvith the sides parallel to the

coordiua'ue axes

and

centred

at 0),

the

valtte of

continucd

tractiorr (7.I)

(a,, e

C)

are

in the set

defined

by

the inccluation

mzrx {l

u.r- xl, _lP, -tl} <; ^max {lrl, ^lyl},

\r'lrere z

: x

_l-

i1

ancl at

- ^i.t

^{-l- il3r.}

MATIIEMATICA

-

RtrVUE D'ANAL,YSE

NUNTÉRIQUE

] : _. ET DE THÉORIE DE L,APPROXIMATION

L'¡ïNALYSB NUIIÉRIQUE BT

L,A

'[I{ÉORIÐ ÐE

n 'APPROXIM.{TION Torne

5,

No2, tr976,

pp"

181-1BB

ON FOURIER SBRIES

^:

G. P. NEVAI

(Bucìapest)

1. I,et us introduce the follou,ing notations.

I,eT

C2^

denote the Banach _space

of

continuous reai-valued. functions

of

period 2æ provided

ivith the

usual

uniform norm and let Li" (l

S

I < ^{oo) bc thc}

^Banach

space

of

real-valued

functions./ of

period

2n lor

r'vhich

l,fl" is

Lebesguc integrable over

,the

^segment

l0, 2nl. I,et

^Jt,

be a real

pararneter

and

let

A, and

_V¡, be operators

oî

C2n(2fi) defined

by the

formulas

(Lrf)(*) :fl*) ^-

^f(x

*

^h)

2t

krf)(*) -f(x)

^{-r f(x} ₂

⁺

^{h) .}

Iret, further, R be a

natttaT

integer

and

let

anÏ; 8).,, (rå) :

,]l_tP, ll(Ln)'',f llr"- ç'r,¡

[1] ^{F'a i} t W y rn aln, A tlt¿oyen¿ of cotouergen,ce for ^lltc non-comn'¿túaliue frøclions. Journal ^of

r\ pproximatioll theor)', ií, 7+ ⁷ G (197 2) .

¡21 Ga¡rrirIf. G, et Gc¡bert,7,,4tgcbrc d,e Battach. UrriversitódeI,iège, Sétninaire d'Analyse ìIath(:rr-LLrtique et cl'Algèbre, pp. 1- 17 (1966 1967)

[3] ^\Vo r piT.zky, Untcrsu,cfunNg¿i!. ü.bei tlic Enttuichcluttg d,er À,Iomodyon,e¡t tnd, illonogert,en

Funhtion tltLrch [(ctt¿nl¡rüch ];yi.e¿lriclt Gltntnasitt.to trtul llealsclt'ulc. Jahresbericht,

l8>0)

' ^C( ) is ^a non-negative, linite ^constant clepending on the parameters lying in the brackets,

be

the ^R-th

^{Cz^(Lå)} modulus

of

smoothness of the function

/. ^Finally,

^let

denote the n-th partial sunr oI the

trigonom.etric

l'ourier

series

of

the function

/.

2.Let

_{"f e} C"*

and R be a fixed uatrlrai

integer.

It is a

well known

inequality

due

to H.

I,ebesgue and

I).

_Jackson

that*

Pgrrl)RDNCES

S,(x, .f) :

^u,

ii ør

^{cos /t,x}

-l

^{bu sin hx)}

( i 865)

I l, II.S., 'l-hc conlinued fraction ^ûs a seqltence of linear trans-

llath _Journal, 9, 360-1.t72 (1942).

Facullataa dc n¿alenaticã ;i ^mecanicd

Uniuersil,alea din I açi

R. S. Româniø

782

and therefore

that

is

(n,:7,2, ...). In

1941 s. M.

NrKor,sKrr [B]

^showed

that

C(1)

: ¹

is

the optimal value

of

c(l).

How does

c(R)

behave

for an

arbitrar5. R J

Thi! tuestion

uras open fo.r ?^lo_"g

time

and even

the

estimation

c(R) : :

o(.1) was

not

known

until

1971 when

the author [6]

^and indepenàeátly qf._l¡im

v.,v. Zut< _[11]

solved

this problem. Actually, the

best^ value of 9-(R) . can

be defined easily by the following

wa)¡.

Ilenoting by 1

the

ideutity

operator

let

us remark

that

ArfV¡:1 (An*V¡)À='l

¡f + v,Þ. (;) ^LIYr-o-' : _t

Iret us

choose

now a trigonometric polynomial

H,,

of

degree

at nost

n such

that

llf - H"ll""^:

8,,(f)

rvhere

E"(f)

denotcs

the

measure

of

best approximation

in C", of f

^by

trigonometric polyrror¡ials

of

degree

at most

r¿. Then rve obviously have s,,("f)

- Í:

s,,("f

- ^H,) -

^("f

- ^H,) :

: s,(Af /) - ^{s,(^f H*) +}

+

v,,s,,

(Þ, ffi) Àfyf -'t-r u - ^H,)) - ^u - ^H,).

But

S,(Af

H,,)

:

Lf H,,

: Lf (H, - ^{Í) -F LiÍ}

and

thus

lls,,(A,f

ä,)

_llc2*3

8,,(¡) +

^cìR(-f

;

^k)cr*

Let now

h

: _{n- " .} B)' virtue of

some

result

due

to s.

TDRNSTETN.

[1]

^and

w. RocosrNsr<r

[9]

^{we have for-every g-e Czn}

llVd,(e)ll"r^ I Cllgll"r* (n: ^{r, 2,} '.

').

Therefore

ⁿ

llvss"(D

^(f)A*

vf ---'(Í -

H,,))llc"n< c(2R

- r)E"(n

(n,

: _{t, 2,}

. , .)

t+ h:0 nn

3 ^{oN FoURIER} sERlEs ₁₈₃

Applying now the

Jackson ^theorem

E*(f) : ^{of^*(Í} ,:)",_l

⁽ⁿ

: ^{r, 2,}

^{. . .)}

rve obtain

(1)

s,,(.f)

- l: ^{s,(^:f)} _'; ^{+ ol.*7;ll} _L ^-a

tt tç2nJ

I ^{tn: r, 2, ...,).}

From this

we immediately get

the

following

lls,,(/)

-

^Íllc,*

^{3 ln,tor} " ,*v ;;)",_*, [..Í t:\"",]

^@

:

r,2,. . .)

and. therefore C(R) 5 ¹ . ^Fo,

^every

ⁿ it is

easy construct a'saw-tooth' function _{.f e} ^C

r*

^such

^that

ils"(/) - f|",*= 1 ^to* _"^oVt ;)",_+ ol..(/;

+)""_) and

thus C(R)::,.

3. By the way

rve have

got a very important

expression

(i) for

the deviatiorr

S,,(/) from the

functioÍL

Í

^e

^Cz'. This formula

^has

a lot

^of

applications, -soto"

of which give entirely

new results

for Fourier

series.

Its first

obvious conseçltlence

in the

following TrrEoREM L.

Let f

^{e C2,. Then}

ll/ - s"(/)llc,^;]

^o

if

^and. only

if _for

^euery naturøI iøteger R

,"(S,(.f) ; ^ò)c,,,;*

^0.

The

necessity follows

from the

inecluality

."(S"(,f) ;

^8)c,*

^S lls,(/) -.fll""* f ^."(.f ;

^à)c,,

and the

sufficiency

from

(1).

Let ^us turn io an another application of

^(1).

c.

rr, HARDY ^an'd.

J. E. r,rTTr,EwooD

l2l ^have

^shown-

^that

^if.

f

^e

L{* and <o.(/; 8)r;:

: O(n-"1 where

æþ

) |

^then

f :;f,

^almost everywhere with

f1e I,ipu - - 1

and. therefore

P

I ,t'

lÍ(x) - ^{S,(*, Í)l :Otlog} n' *-"'T)

⁽ⁿ

:1, 2, .'.)

G. P, NÉVAI 2

ll/ - S"(rll"," <

C1n¡ log

na*

^T

f

n

C2r

+

^o

[." v,:)

^czr

R-l

784 ^{G, P, NËVÀI ON FOURIER SERIES}

.(! - ^*)

llog ^(J, - ^x)l

4 ⁵ 185

uniformly

almost everywhere.

this result

can be sharpened. and generali- zed

with

the aid of the relation (1) as follows:

TIIIIoREM

2. Lct

_"f

. Ll*

and

for

^some

fixed. naturøl

^integer

R

let

^*(.f

;

⁸⁾

rî*: O(n-"¡

^(O

< " ^{< R,}

^o.þ

)

¹⁾

Then

uniforru.ly

in x

and

y

^then

the Fourier

series.of

/

convefges_,lulfg11lv ton,ards

î. ^l,"t

^{us say}

^{that the}

^functiou

/

^satisfies

^the

^one-sided

^Dini-Lip-

schitz

condition

if

crnifornr.ly almost eaeryuhere.

We ¡,vould

likc to

crnphasizc

that the

logarithrnic

factor is

absent in

this latter

estimation.

I,et

now

f ^eCrn

^and

^con(/; 8).1_:O(à) (ò--*0) for

^sor11e

fixed

R.

rt

is a ciassical result

that in this

case

S,(/)

converges

uniformly

torvards

f ^lor

ⁿ

+

oo.

With the aid of (1)

one can find.

the exact value of

the speecl

of

convergence

of

S,,(/)

to /.

Nanr.ely, rve have

the

folloiving.

rFrÞìoRrtr'r

3. If f

^e

C2" ancl

u,,*(./i 8)r.,_

: O(ò)

(8

-, 0) _.fo,

^sotne

fixed, natural

^in.tegcr

R

^tltett,

ll/ -

s,,(/)llez_

-

oilrog o,.(/

t :),^t

^.

.*i!' j,)",_l

(n: \,2, ...)

^ønd, th,is estitnate cannot be inr.þroued,.

We shall now

define

the

class

of

piecewise

R

monotonic functions.

The

functio.y

f

^{e C"^}

ls said to ^be

^piecewise

^R (: 1,2, ... but

fixed) monotonic

if

there exists

a partition

p(/) :

-r :

^xo

I ^xt I ... { x*:

ⁿ

of the

segment

l-n,n)

^such

^{that the}

expression

(a{f)(r)

does

not

change

hissignwheneverà ¿0, ^(x, x+

^RS)

C (xr, xr+r)(i:0, 1,..., N-1).

Applying

(1) one can prove

the

following statement.

THDoRETu

4. Let R be ø fixed, natural

^{integer ø.nd,}

^let f

^e

^C2,

^be

þ'ieceuise

R +

1 monotonic.

In

th,is cøse the logarithmic føctor

in-the

^Lebes- gue-Jackson estinta.te cøn be omitted, thøt is

lt/ -

s"(/)il

",n

: r,

þ"[¡, ; )","1

⁽ⁿ

: r, 2,

. . .).

4.

Now we

turn to the

nicest

a cation of

the fo-rmul.a (1). \Me

shall

generalize

th

(more preci- sely Dini-I,ipschitz-Lebesgue)

con\¡er hat

if

f .

^Cr,

and.

rf@)

_fu)t: _{r (}

*;, ) ^{u. _yl *o)}

rvhere e(ð)

= 0 and

^e(8)

*0 for I -' f0.

rnEoREM

5. If

_{.f e} ^C2, satisfies the one-sid'ed,

Dini-Liþschitz

^cond'ition

th,en the

Fourier

sãries of

f

^conuerges

^{uniformly touød;} f

^ønd. llLoleouer ue Itaae the

follouing

^{estimøte :}

(2)

f(x) - ^Í(v)

^>-

-

ll/ - S"(rllc,n: ^o

where

*

_denotes

_the

convolution and,

lÍ@) -

^s,,(x,

^.f)l : o

^þ

þr

"[;)

log

(x<u\

log

n (tr,

:

7,

2,

)

Í

:r+=d'+"G)1 ^{(n: r,2, .,.)}

of

^course,

^this

^statement

is nofe

genefal

than the

Dini-I,ipschitz theorem,

but it can be

easily checked thaL

it is

more .general

than

the

Jordan-Ílirichlet

convergencg

test as

rvell.

This fact is

interesling-

by iiself

^since

it is well klown that the Dini-I,ipschítz

^and

the

_Jordan-

Dirichlet

convergence theorems

are not

comparable.

To

^prove

this

theorem

lgt us apply

(1.) with.

n : l',Pv virtue

^of

.it

we

have'to

estimate

only S,(4"/) since/is

continuous. We have

by

the

Dirichlet

formula

r,,(o; ¡¡ : (L:Í)*D,,

is the n-th Dirichlet

kernel. llence

S,,14

",f)

:

^*Dn

fr

+

7l

ÎE

,,

I86

and

by virtue of

⁽²⁾ rve have

: l(^"/)

^'*

lD,,| +o["(-1)]

^,rl ^s ilu-f@))*ur*D,t-tD,t)l +ol'(;)]

tr"r =

s +lr - ^f@l,'

^11"

^{D* +} D,l*'[' _(;)]

where

T*D,(t) : O-(,

7 ^{oN FoURIER}

SERIES ¹⁸⁷

Lct

f

e

Cz- ÀLip a

⁽⁰

<

^o

<

^{1) and' Iet}

^f ^{be o{}

^rnono-

toni

^s

lc-: ^c(f)

^{suctt' t'kat}

tnìfi'ni¡åi' itij i

^c"(*)

^is

mon'oton'íc

in ^{In this}

^{cflsc ruc ka'ae}

ll/ -

^{S,(flll czn}

: O(n-o)

⁽ⁿ

: l' 2'

' ')

'I'his

follows immcdiately from a'heorem

5' Ulrfortunatelv' for

^a

:

¹

it is not truc. At ii';ä;"1ti-"' t'o'i]":;; ;; "'" o¡t"i" ^iiom

^{(4) the}

following ^theorem'

Tr{EoREM

7. Let

_.f

t Cr,O ^l"ip |

øtød' tet

f

^be

^of

^cotlyex

^{tyþe tkat}

^is

1C:C(f)

^suclt'

tn.l'Åt ¡"iltto'' ¡i*¡ * ^{Cx' is}

conaex' Then

ue

haue

il/ - ^s,(flt: ol+)

⁽ⁿ

-- r' 2' "

^')

Let ^now Í t Lt*' It

^is

T. E.

LrrrDr,wooD [3] ^that

^ior

"

e

Ll*

such

that

a

^classical

^result of

^{G' H' HARDY ancL}

a fixed poínl x

^{one can}

^{find a}

^function

tf(xi

^s)

-/(ø)':'(d-) ^0-.ol

and. Sn(ø,

/)

^diverges

^{lor n} n co'-At the

^same

time' however' s'

^rzuMr

and c. suNoucur Liì ná"""ptoved that if f

^e

Ll*

^anð'

lÍu) - ^fþ)l :

"(t"r|' - utl ^þ'

^z

ⁿ

^x)

then

S,(ø,

f)--*Í(x) ^{for n} *

^co'

The Izumi-So"oo"ttl''

theorem also

has a

one-sicled analogue:

TIfEoI{ÊlfB.LetÍ.LL^and'letxbetl'Lebesgueþointoftkefunc-

G. P. NÉVÀI

^

.lrì

ls,,(a,/)l s l(a^,f)xlD,l

^I

+,Ë

I 1tl

[1

+1xlD,,l]:

.

From

this

we obtain Therefore

ls,(x,

î

(3) lS,,(x,

L*f _)l :

O

[!",,t; t)rz,,T-!-* *"u* * _"(;)]

since

it

can

be

easily computed

that

lo "(, - ^{z) + o,Q)l} : ^o (-r,

_*"ul

Th.eorern

5

follows irnmediately

from

^(3).

The condition (2)

can

be

generalized suppose

that for

sorne

R(: 2, 3,

^. _{' .)}

^¡/

^òJ

^> -

I ^'(ò)iog 8l

ancl ^r,ve can

obtain by similar

arguments

that

(4) lt/

^-_

^s,(/)l c,n: ol.^-, _(t

t

:)",,r .[;)]

⁽ⁿ

: r, 2,

^.

..).

'I'he results

mentioned.

in this part

have also some applications

to

some well known results. R. sar,EM arrcl a. zYGMUND

f10] have proved

^the following

10,

tken S*(*-' _'f)

'f(x\ ^for n*

^æ'

localization

of the rniin ^relation

⁽¹⁾

of

Theorem ^5.

ä":ltr;*;oned

^here were Publis-

by the following

waY.

We

can

tÐ>0)

tion

f. If

Í(v) -Í(z) ^> -

llog(e-1)l^{'(z - !)}

(y

< ^z; Y, z -'x)

1BB G, P, NÊVÄI _o

() MATITEMATICA

_

REVUE D,ANALYSE NUM.É'RIQUE

ET DE THÉORIE DE L'APPROXIMATION Ìì,B1TPRÞNCES

c sotntttalion _des sêy.ies h,igonométriquas. Contptes

A conucrgett.ce critct,iott Jor Fottyier series, NIatln.

Sonte neu coltuergellce criteria for ^Fouriet, series,

Journ. B, (19S1),29g_30S. ^tc Fotl'vier attølysis (XI,\¡III), Tohoku l\'Iath.

[5] Névai, G' P', A note ott a G. I' Natanson's tlteorctn. (in Russian), .A.cta ],ra¡r Acac1.

iøterþolation _and, Fourier vrnts (irr Russian), j4uergence test (in Russian), Acta ì{ath. Acacl.

of lhe tenn. ^,in. case of a.þþroxi_

Doklad R sZ, (1941),-386:_389.

trigono Rcihett, nrath. Aruralelr int.ation, by _þa,rti.al s Luu,s of I;otn,ie r se ri ¿s.

14 22.

,.2r.þeriodic funclion by I.inear rnctltuls oJ

ih Uceblrylr Zirvctlcrrii, Ácr. Mnt. t 1f O;.:¡,

L'ANALYSB

NUMERIQUE

ET LA

THEORIE

DE

L'APPROXIMATION

Tomc 5,

No

2,

1976,

pp. 189-192

COIUPLÉMBNTS REGAIìDANT I,A RAPIDITÉ DE

CONVERGBNCIi DES SERIBS

pâr

ANDREI NEY (Cluj-Napoca)

Matheutati,cal Institule of ^'the Hungarian A cad.cmy of Sii.ences

Préliminaires. I,'accélération de

la

convergence d'une série

ayant

pour

but l'amélioration de l'approxirnation

c1e sa somtne est basée

sur

deux concepts :

1- la définition du fait ^qu'une

^série conr¡erge pll1s

vite

que l'autre, et

2" la

procédttre

d'appliquer une tratrsformation au

termes d'tLne série convergente,

de telle

façon qtt'elle reste crlcore convergente

et

con-

serve la

somme

de la série

origirrelle,

-

^{d.e plus, 1a}

^suite

^{des sommes}

partielles de

la

transformée converge

plus \¡ite à

cette sorlrue que la

suite

des sommes

partielles de la

série originelle,

D'après eRTNGSHETM,

l4l, ^sí

^Ðu,,

eL 2uu sol1t deux séries

nt1-

nrériques convergentes

dont

les rcstes

du Íang n

soient

notés par

t,, respectivement p,,,

alors la série

Xu,, convergc

- ^Ilar définition -

^plus

Received 20. Vf. 19?4.

vite

^que

la

série

2u,

sr

on a lim & :

9.

¡t+æ /tt

On

démontre facilernent,

que

potlr

deux

séries convergentes

et

^à

terrnes positifs, Ðu,,

et

2a,,, I'égaTitn

tffi;": 0

entraine l;égalité

:*7,:0.

On est tenté, parfois, d'étudier

setl1eme11t

le rapport

des termes correspondants

en vue da la

comparaison des rapidités de convergence de deirx séries

-

^{1nême atltres}

^qu'à

^termes

^positifs, tl l, t3] ^{Or i1}

^est

su,ffisant

de

considérer 1'exelrplc classiqtte

I

des

sérirs > \+et _{> -L--}

pour

constater qüe malgré

la

décroissance Plus

vite du

terme général de

la

deuxième série

- ^{à zéro} -

^que

^celui

^de

la prernière, la

d'euxième série diverge, tandis que

la

première converge'

¿,*i