ON THE RELATION BETWEEN FOURIER AND LEONT EV COEFFICIENTS WITH RESPECT TO SMIRNOV SPACES
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1 Uraiia Mathatical Joural, Vol. 56, o. 4, 24 O THE RELATIO BETWEE FOURIER AD LEOT EV COEFFICIETS WITH RESPECT TO SMIROV SPACES B. Forstr UDC 57.5 Yu. Ml i show that th Lot coicits κ ( λ i th Dirichlt sris Λ κ λ λ ( λ L ( λ o a uctio E ( D, < <, ar th Fourir coicits o so uctio F L ([, ] a that th irst oulus o cotiuity o F ca b stiat by th irst ouli a aorats i. I th rst ar, w xt his rsults to ouli o arbitrary orr.. Itrouctio Lt D b a clos cox olygo with rtics a,, a, > 2, lt D b its o art, a lt D D\ D b th bouary o D. W assu that th origi blogs to D. As is custoary, w ot by E ( D, < <, th Baach sac o all uctios ( z aalytic i D a satisyig th coitio : su ( z z <. γ Hr, ( γ is a squc o clos rctiiabl Jora cotours γ D that corgs to D. Th sac E ( D is call a Siro sac. Cosir th quasiolyoial L ( z a z, whr C \ { } a a ar th rtics o D,,,. Lt Λ ( λ b its squc o zros. W ca xa uctios E λ ( D with rsct to th aily é ( Λ : ( z ito a sris o colx xotials, aly, th so-call Dirichlt sris whr λ ( z κ( λ, ( L ( λ λ z Ztru Mathati, Tchisch Uirsität Müch, Gray. Publish i Urais yi Matatychyi Zhural, Vol. 56, o. 4, , Aril, 24. Origial articl subitt March 2, /4/ Srigr Scic+Busiss Mia, Ic.
2 O THE RELATIO BETWEE FOURIER AD LEOT EV COEFFICIETS WITH RESPECT TO SMIROV SPACES 629 κ ( λ a λ λ η ( η η. (2 Th ixig o Λ is chos such that ( λ is ocrasig. Th coicits κ( λ ar call Lot coicits. May rsults o ths sris ar u to Lot []. Lwi a Lubarsii show i [2] 2 that, or 2, th aily é ( Λ ors a Risz basis o E ( D, a, hc, sris ( corgs ucoitioally i or. I [3], Sltsii ro that, or arbitrary < <, th Dirichlt sris ( corgs i or sic é ( Λ ors a Schaur basis i E ( D. To stiat th rat o corgc o ths sris, Ml i stui th rlatio btw Lot coicits a Fourir coicits, sic, or th lattr, ay rsults o aroxiatio a rat o corgc o th Fourir sris ar wll ow (s,.g., [4]. H show that, ur crtai coitios, th Lot coicits o E ( D ar th Fourir coicits o so uctio F L ([, 2 π ]. H stiat th rgularity o F with irst ouli o cotiuity. I Sc. 2, w stat his rsults. Extig Ml i s Thor to ouli o soothss o arbitrary orr, w obtai Thor 2 i Sc. 3. Th last sctio cotais th rscti roo. a a 2. Ml i s Rsults I [5] a [6], Ml i cosir th rlatio o th Lot coicits o E ( D to th Fourir coicits o so suit uctio F L ([, 2 π ] or th irst ouli o cotiuity. His irst st was th ructio o th itgral i (2 to a Fourir trasor: La [5]. I. Lt Φ L ( [, 2 π ], < <, a ( >. Dot whr it Φ( t : ( Φ, ( Th Φ o. ( Φ : L ( [, 2 π ], a ξ Φ( ξ ξ, (. Φ cost Φ or so ositi costat ig oly II. Lt E ( ( D, < <. For ix, th Lot coicits ( κ ( λ ( ar th Fourir coicits o so uctio F L ( [, 2 π ], a F L cost E. This rsult was xt i [6] usig th irst ouli o cotiuity. Cosir th aratrizatio z : D [, T ] o D :
3 63 B. FORSTER z ( u a + a a + + a a ( u T or T u T,,,, whr + T :, T a a a + T : T : a a. For E ( D a < h <, lt h a+ a δ (, h : a + θ θ / + a h + a + a θ θ /. Th uctio δ (, h is cotiuous, oicrasig, a aishig as h +. Thor [6]. Lt E ( D, < <, a lt b ix. Th th Lot coicits ( κ ( λ, (, o ar th Fourir coicits o so uctio F L ([, 2 π ]. Furthror, ω ( F, h cost ( ω ( z, h + δ (, h. Th roo ca b uc as a scial cas o Sc Ml i us his rsults i [6] to ro irct aroxiatio thors or th irst ouli. As w will s i Sc. 3, Thor ca also b ro or ouli o arbitrary orr. 3. Extsio to Mouli o Arbitrary Orr To xt Thor, w ha to i ouli o soothss o orr or uctios E ( D. This ca b o by usig th bst aroxiatio by algbraic olyoials. Lt E ( D a lt I D b a arc. For, th quatio E (, I i P L ( I P is th algbraic bst aroxiatio o th arc I. Hr, th iiu is ta or all algbraic olyoials P o gr at ost. Th oulus o orr is i as ollows:
4 O THE RELATIO BETWEE FOURIER AD LEOT EV COEFFICIETS WITH RESPECT TO SMIROV SPACES 63 Diitio. Lt E ( D, < <. For h >, cosir all artitios D I, whr h / 2 I h. Th th trical oulus o soothss o th uctio is i as ollows: ω (, h : ω D, (, h : su i P P L ( I su E(, I. Hr, th suru is ta or all such artitios. O ca show that ths ouli ar quialt to usual ouli o soothss i o iit itrals [7]. W ca orulat Thor or th th ouli. Thor 2. Lt E ( D, < <, a lt b ix. Th th Lot coicits ( κ ( λ, (, ar th Fourir coicits o so uctio F L ([, 2 π ]: κ ( ( λ iθ F( θ θ π : c( F. 2 Th th oulus o F ca b stiat as ollows: ω ( F, h cost ( ω(, h + δ(, h, (3 whr δ (, h : h a+ a a θ θ / + a h a + a θ θ /. Th uctio δ (, h is cotiuous a oicrasig or < h < 2 π / h a satisis th rlatio li h + δ (, h. This rsult abls us to trasor th Lot coicits (2 i th Dirichlt sris ( ito th Fourir coicits o crtai uctios F. Sic Thor 2 rois ioratio o th rgularity o F, classical Brsti thors ca b ali to th corrsoig Fourir sris. This ca b us to ro w rsults o th rat o aroxiatio o th Dirichlt sris (. Th tr δ (, h caot b oitt ro th thor, as th ollowig xal shows: Lt 2 a ( z. Suos that L (. Th ω (, h 2, whras δ (, h 2 O( h or h a all. For th Lot coicits, w ha
5 632 B. FORSTER κ ( ( λ ( λ O as. W ow ro La that ( F κ ( λ i L 2 ([, 2 π ]. ( Th Brsti thor [4] yils ω ( F, h 2 O h gis S ( sic th aroxiatio with th artial sris ( ( F κ ( λ ( i F S( F 2 i ( > > ( λ > > ( 2 2 / ( λ O as. Thus, th tr δ (, h 2 is cssary i (3 (s also [6]. 4. Proo o Thor Prliiaris. Lt us irst ta a closr loo at th quasiolyoial L ( z a z, whr C \ { } a a,,,, ar th rtics o D. Lt Λ ( λ b th squc o its zros. Th tir uctio L has th ollowig rortis [] (Cha., Sc. 2: ( I. For suicit larg C, th zros λ whr ( o L such that λ > C ar o th or λ ( ( ( λ δ +, ( λ i a+ + q a iβ, ( a δ a. Hr, < a cost,,,, >, a a + : a. Th aratrs b a q ar i by th orula q a a iβ ( + +,
6 O THE RELATIO BETWEE FOURIER AD LEOT EV COEFFICIETS WITH RESPECT TO SMIROV SPACES 633 ( whr + :. Hc, th zros λ ar sil. Th st o zros Λ ca b rrst i th or ( Λ { λ},, { λ } (,( +,. II. Thr ar ositi costats A a c such that, or all ( a ξ [ a, a ], w ha ( ( λ ξa λ ( ( ξa A c. Hr, [ a, a ] ots th li btw th rtics a a a i th colx la. For silicity, w assu that all zros o L ar sil. W us rortis I a II to trat th zros o L a to stiat th colx xotials i th Dirichlt sris (. I aitio, w th ollowig rsult o ultilirs: Thor 3 (J. Marciiwicz, [8], Thor 4.4. Lt ( a C b so sris such that a < M a a a M or all a so suit ositi costat M. Lt c i L ( [, 2 π ], < <. Th thr xists a uctio h L ( [, 2 π ] with h i ca a h CM (, whr th costat C ( > s oly o. W ow ha all as or th roo o Thor Proo. Th xistc o a uctio F L ( [, 2 π ] with th iicat rortis is show i assrtio II o La. Thus, w ust ha to xai th rgularity o F. Usig coitios I a II o Sc. 4., w ca writ
7 634 B. FORSTER κ ( ( λ a ( z z a λ ( ( za a ( z z a ( λ ( za + O( c + a + a a a a + i β a a + + q θ i θ θ θ +, + a a a a i β a a q i a a π π θ π θ θ θ a a a O( c. Th irst tr is obiously th th Fourir coicit o so uctio with oulus o orr ω D, (, h. Usig assrtio I o La or th sco tr, w ust ha to aalys th rgularity o Φ with rsct to th rgularity o so uctio Φ L ([, 2 π ] sic a R i a + a a >. Th th rquir assrtio ollows ro th iquality ω ( a a h ω (, h. [, ], + Lt h >, ( >, R, a Φ L ([, 2 π ]. W will show that th sris o coicits ( ( Φ ( is th sris o Fourir coicits o so uctio Φ L ([, 2 π ] with ω ( Φ, h cost ( ω ( Φ, h + δ ( Φ, h, whr ( Φ, h : δ h / / u u + Φ( u u Φ( h. Lt ϕ L ([, ]. Th A : ( ϕ( u u u 2 u π ( ( ϕ( u u u + ( ( ( ( u ϕ u u ϕ u u ( π+ ( ( u u ϕ u u ϕ u u 2
8 O THE RELATIO BETWEE FOURIER AD LEOT EV COEFFICIETS WITH RESPECT TO SMIROV SPACES 635 ( + ( ( u u ϕ u u ϕ u u π 2 2 ( π ( u u ( ϕ u. With th otatio o assrtio I o La, w ha ϕˇ ( t ( i ( ( ϕ ( t ( ( ϕ ( ( i it ( ξ i it ϕξ ( ξ( ( u ( ϕ ( + ( ( ( u u A ( ( i ( ( ( ϕ ( ( it i ( it + ( ( ( u ϕ u u ( i ( ( ( ( it ( ( ϕµ + it ( ( u ( ϕ u uµ it, (4 whr µ i ( ( a µ ( ( ( µ. W will show that µ a µ ar ultilirs i L ([, ]. Thus, th Fourir sris wight with ( µ ( a ( µ ( both corg i L ([, ]. First, cosir µ. W ha µ µ + + ix ( ( x x (5 x ( ( a, urthror,
9 636 B. FORSTER x ix ( ( x ( ( x ix ( x ( Lt ε > b ix a lt < i { (, }. W slit th wight Fourir sris i two arts. First, lt < ε. For x +, w ha. x ix ( ( x ( ( x ix ( x ( ix ( x ( x ix x. (6 W istigat how this tr bhas as bcaus, or > γ with so γ >, th tr is bou i th oai < ε or cotiuity rasos. For, it is asily s that x ix x i( sig( ( x 2 ( x ix ix x. isig( Th sco tr corgs to i. Sic x + a < ε, w ca stiat th whol tr by so costat it o a x. Hc, µ cost or < ε a so 2 costat it o. By iuctio a Eq. (6, w gt x ix ( ( x cost, ( ( whr th costat os ot o x a. Furthror, µ < cost or so costat it o. Usig Eq. (5, w uc µ + µ cost + x cost, a, thus, + < ε/ µ µ cost. For Φ L ([, ] a Φ as i La, by usig Thor 3, w coclu that c( Φ µ i < ε/ cost c( < ε/ Φ i cost Φ, (7 sic ( ( Φ Z is th squc o Fourir coicits o Φ. c
10 O THE RELATIO BETWEE FOURIER AD LEOT EV COEFFICIETS WITH RESPECT TO SMIROV SPACES 637 Sco, lt ε. W i µ : ( a uc µ µ + ( + ( ( ( + ( ( ( ( ( + ( ( R( ( ( ( + ( ( R( ( ( + ( ( R( C (, ( + ( ( C(,, ε R( or ositi costats C (, a C (,, ε. Thus, sic < i { (, }, w ha µ + µ ε/ µ µ l [ ε/ ]+ l+ [ ε/ ]+ l ([ ε ]+ l R C (,, / ( ε l C (,, ε C (,, ε [ / ]R ( R ε ( ( R C (,, εr (.
11 638 B. FORSTER Obiously, µ C ε or so ositi costat C ε ig oly o ε. Hc, or Φ L ([, ] with Φ as i La, by usig Thor 3 w gt c( Φ i µ cost ε i c( Φ ε cost Φ. (8 Dot Φ ε( t : c( Φ µ it, ε i.., c ( Φ ε c ( Φµ. Hc, c( Φ i µ c( Φ µ ( ε ε i i c ( ( ε Φ ε i i ε i ε c ( Φ Φ ε ( + Φ ε cost Φ, (9 whr w ha us (8. Usig (7 a (9, w ca uc c( Φ i µ c( Φ i µ + c( Φ i µ cost ( ( < ε ε Φ. Hc, µ is a ultilir i L ([, ]. Th sa ca b show or µ. W ha µ + µ ( + ( µ ( µ + R( µ µ ( + + R( µ µ ( + + a ( +, cost µ µ + + µ + µ + + ( + R( or so ositi costat a >. Thus, µ is a ultilir i L ([, ], too. W ha
12 O THE RELATIO BETWEE FOURIER AD LEOT EV COEFFICIETS WITH RESPECT TO SMIROV SPACES 639 c( Φ i µ cost ( Φ. For < h, usig rlatios (4 a La w gt Φ ( i Φ µ + ( u ( Φ( u uµ ( i i cost ( Φ + cost ( ( ( u Φ( u u i cost Φ u + cost ( ( χ ( [, ]( u Φ u u ( i cost + ( ( Φ ( χ [ ] Φ, ( i cost + Φ ( χ [, ] Φ ( cost ω( Φ, h + Φ( u u / cost ( ω ( Φ, h + δ ( Φ, h, whr χ [ ab, ] ots th charactristic uctio o th itral [ a, b ]. Passig to th suru, w gt ω ( Φ, h su < < h Φ cost ( ω ( Φ, h + δ ( Φ, h, a th assrtio is ro. Th author wats to xrss hr gratitu to V. V. Arisii a R. Lassr or ay aluabl cots o th subct. This wor was suort by th Dutsch Forschugsgischat through th grauat rogra Agwat Algorithisch Mathati, Tchisch Uirsität Müch.
13 64 B. FORSTER REFERECES. A. F. Lot, Exotial Sris [i Russia], aua, Moscow ( B. Ja. Lwi a Ju. I. Lubarsii, Itrolatio by as o scial classs o tir uctios a rlat xasios i sris o xotials, Math. USSR Iz., 9, ( A. M. Sltsii, Bass o xotial uctios i th sac E o cox olygos, Math. USSR Iz., 3(2, ( I. Achisr, Thory o Aroxiatio, Dor, w Yor ( Yu. I. Ml i, So rortis o xotial sris rrstig uctios big rgular i cox olygos, i: So Probls i th Thory o Aroxiatio o Fuctios [i Russia], Ki (985, Yu. I. Ml i, Dirct thors or th aroxiatio o uctios rgular i cox olygos by xotial olyoials i th itgral tric, Ur. Mat. Zh., 4, ( Yu. A. Bruyi, Picwis olyoial aroxiatio, big thor a ratioal aroxiatio, Lct. ots Math., 556, ( A. Zygu, Trigootric Sris, Cabrig Ui. Prss (988.
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