The Strong Approximation of Functions by Fourier-Vilenkin Series in Uniform and H¨older Metrics

T.V.Iofina and S.S.Volosivets

Saratov State University，Faculty ofM athematics and M echanics，Astrakhanskaya str.，83，410012，Saratov，Russia

The Strong Approximation of Functions by Fourier-Vilenkin Series in Uniform and H¨older Metrics

T.V.Iofina and S.S.Volosivets∗

Saratov State University，Faculty ofM athematics and M echanics，Astrakhanskaya str.，83，410012，Saratov，Russia

Received 15M ay 2012；A ccep ted（in revised version）15 January 2015

.We will study the strong approximation by Fou rier-Vilenkin series using m atrices with Some generalm onotone condition.The strong Vallee-Poussin，which Meansof Fourier-Vilenkin seriesare also investigated.

Vilenkin system s，strong approximation，generalized m onotonicity.

AM SSub ject Classifications：40F05，42C10，43A 55，43A 75

1 In troduction

For x=k/ml，0＜k＜ml，k，l∈N，we take the expansion with a finite num ber of xn/= 0.Let G（P）be the Abel group of sequences x=（x1，x2，···），xn∈Z（pn），with add ition x⊕y=z=（z1，z2，···），where zn∈Z（pn）and zn=xn+yn（m od pn），n∈N.We define m aps g：［0，1）→G（P）andλ：G（P）→［0，1）by form u las g（x）=（x1，x2，···），where x is in the form（1.1）andwhere x∈G（P）.Then for x，y∈［0，1），we can in troduce x⊕y：=λ（g（x）⊕g（y）），if z=g（x）⊕g（y）does not satisfy equality zi=pi-1 for all i≥i0.In a sim ilar w ay，we introduce x⊖y and for all x，y∈［0，1）generalized d istanceρ（x，y）=λ（g（x）⊖g（y））.Every k∈Z+=｛0，1，2，···｝can be exp ressed uniquely in the form of

For a given x∈［0，1）with expansion（1.1）and k∈Z+with expansion（1.2），we setThe systemis called am u ltip licative or Vilenkin system.It isorthonorm aland com p lete in L［0，1）and we have

for a.e.y，w henever x∈［0，1）is fixed（see［8，Section 1.5］）.

The Fou rier-Vilenkin coefficientsand partial Fou rier-Vilenkin sum s for f∈L1［0，1）are defined by

Let us introduce a Modu lus of continuityω∗（f，δ）p=sup0＜h＜δ‖f（x⊖h）-f（x）‖pin Lp［0，1），1≤p＜∞.If Pn=｛f∈L1［0，1）：ˆf（k）=0，k≥n｝，then En（f）p=inf｛‖f-tn‖p，tn∈Pn｝，1≤p≤∞.Letω（δ）be a function ofModu lus of con tinuity type（ω（δ）∈Ω），i.e.，ω（δ） is continuous and increasing on［0，1）andω（0）=0.Then the spaceconsists of f∈Lp［0，1）（1≤p＜∞）or f∈C∗［0，1）（p=∞）such thatω∗（f，δ）p≤Cω（δ），where C depends only on f.Denote bythe subspace ofconsioting of all functions f such that limh→0ω∗（f，h）p/ω（h）=0.The spaces［0，1）and，1≤p≤∞，with the norm）are Banach ones.Inwe can consider

Using matrix A， we can define a summation method by formula

In the case of trigonom etric system and m onotoneby k sequence｛ank｝∞n，k=0，the estim ates of‖f-Tn（f）‖∞wereobtained by P.Chand ra［4］in term sofModu lusof continuity.Later L.Leind ler［10］generalized these Results to the cases

and

Here C doesn't depend on m，n.For Vilenkin systemthe estim ates of‖f-are obtained in［9］.Fu rtherwe shall consider

The estim ates of‖Rn（f，r）‖∞form onotone by k sequence

with add itional restrictions on their oscillations were proved by T.Xie and X.Sun in［19］.For m atrices satisfying（1.4）and（1.5），sim ilar Results are established by B.Szal［16］.In［17］，Some estim ates close to onesof P.Chand ra［3］and L.Leind ler［8］are obtained.

In the present paper，we study the rate of‖Rn（f，r）‖p，1＜p≤∞，where am atrix A satisfiesone of the follow ing cond itions：

or

In both cases K does not depend on n，m.The class GM of real non-negative sequencessatisfying inequalitym∈N，was introduced by S.Tikhonov［18］.In particu lar，in［18］it isestablished that GM con tains the classof quasi m onotone sequences QM（with p roperty ann-τ↓0 for Someτ≥0 and n∈N）.Further，we assum e thatω（t）∈ΩsatisfiesΔ2-cond ition，i.e.，ω（t）≤Cω（t/2），t∈［0，1）.

Some Resultsare devoted to the strong Fejer and de la Valle-PoussinMeans（Lemmas 2.7，2.8，Theorem 3.5，Corollaries 3.1，3.2）.

2 Auxiliary propositions

Lemma 2.1.For f∈Lp［0，1），1＜p＜∞，wehave‖Sn（f）‖p≤C‖f‖p，n∈N，where C does not depend on fand n.Asa corollary，weobtain inequality

Let g=（g1，g2，···，gj，···），where gjarem easu rable on［0，1）functions.Letus define

The p roof of Lemma 2.2 is sim ilar to the case r=2，stud ied by S.Frid li［5］.

holds.Forq=∞，wealso have

In an im p licit form，inequality（2.1）is proved in［3］for bounded sequencesM.Avd ispahic and M.Pep ic［2］obtained itsanalog in am ore generalcase.

The follow ing Lemma is due to A.V.Efim ov（see［8，Section 10.5］）.

Lemma 2.4.Let f∈Lp［0，1），1≤p＜∞，or f∈C∗［0，1）.Then

Lemma 2.5.Ifω（t）∈Ωsatisfies theΔ2-condition，then fromit follows that En（f）p≤ Cω（1/n），n∈N.

Thus，Lemma 2.5 is proved.

Lemma 2.6.（i）Let amatrix A satisfies conditions（1.3）and（1.6）.Then an，i≤（K+1）an，mfor m≤i≤2m≤n，where K is the constant from（1.6）.

（ii）Letamatrix A satisfies conditions（1.3）and（1.7）.Then an，i≤（K+1）an，mfor［m/2］≤i≤m，where K is the constant from（1.7）.

Proof.Part（i）m ay be found in［18］.In order to establish（ii），we find for［m/2］≤i＜m that

w hence（K+1）an，m≥an，i.In the case i=m，the statem ent（ii）is eviden t.Thus，Lemma 2.6 is proved.

The trigonom etric coun terpartof Lemma 2.7 is due to L.Leind ler［11］.

Lemma 2.7.Let f∈C∗［0，1），1≤r＜∞.Then

whereM doesnotdepend on n∈N and f.

Proof.Letus consider i∈N such that n∈［mi-1，mi）.Then

We have

Using（2.1）and（2.3），we find that

So，Lemma 2.7 is proved.The inequality（2.5）of Lemma 2.8 in the case m=［n/2］is stated withou t p roof by S. Frid li and F.Schipp［6］for Some general system s.In［6］also one can find the idea of app lication of（2.1）to problem sof strong approximation（see also［7］）.

and

where M（ν）doesnot depend on n，m∈N and f.

Proof.By（2.2）we have w hence（2.4）follows in virtue of inequalityνn≤m.

The inequality（2.5）is derived from（2.4）by substitu tion f-tn-minstead of f，where tn-m∈Pn-mand‖f-tn-m‖∞=En-m（f）∞.Herewe use the equality Sk（tn-m）=tn-mfor k≥n-m and M inkow skiinequality in lras follows：

So，Lemma 2.8 is proved.

Rem ark 2.1.The counterpartsof（2.4）and（2.5）for‖·‖pand p≥r are easily follows from Lemma 2.1 and Lemma 2.2（see the p roof of Theorem 3.2）.

The follow ing Lemma is an analog of Leind ler-M eir-Totik theorem［12］.

Lemma 2.9.Letω，µ∈Ωbe such thatλ（t）=ω（t）/µ（t）is increasing on（0，1）.Then for an operator An（f）=Kn∗f，Kn∈L1［0，1），andthe inequality

holds.

The p roof of Lemma 2.9 is sim ilar to one of Theorem 8 in［9］.

Lemma 2.10.Letω，µ∈Ωbe such thatλ（t）=ω（t）/µ（t）is increasing on（0，1）.Ifωsatisfies Δ2-condition and，n∈N.

Thus，E2n（f）p，µ≤2C4λ（n-1）.Using m onotonicity of best approximations andΔ2-cond ition，we get the inequality of Lemma.

Rem ark 2.2.The cond ition of increasing ofω（t）/µ（t）in troduced by J.Prestin and S. Pr¨ossdorf［13］is suitable for Someapp lications，for example，the theory ofm u ltip licators o f Lipschitz classes（see［1］）.

3 M ain Results

Theorem 3.1.Letamatrix A satisfies conditions（1.3）and（1.7），f∈C∗［0，1），r≥1.Then Proof.Let n∈N and j=j（n）∈Z+be defined by inequality 2j≤n＜2j+1，i.e.，j=［log2n］. Then we have

Using Abel's transform（summ ation by parts），（1.7）and Lemma 2.6，we obtain

Accord ing to（2.5），

It is clear that（1.7）impliesSince［（n+1）/2］≤（n+1）/2≤ 2j，using of Abel's transform and（2.5）gives

From（3.1）and（3.2），the statem entof theorem follows.

Theorem 3.2.Letamatrix A satisfiesconditions（1.3）and（1.7），f∈Lp［0，1），1＜p＜∞，p≥r≥1. Then

Proof.App lying Lemma 2.2，we have

Theorem 3.3.Letamatrix A satisfies conditions（1.3）and（1.6），f∈C∗［0，1），r≥1.Then

Proof.We shall use again j=j（n）with p roperty 2j≤n＜2j+1，i.e.，j=［log2n］.App lying Abel's transform，we obtain

By（1.6），Lemma 2.6 and（2.5），we have

Since

by Lemma 2.6，we find that

whence the inequality of theorem follows.

Sim ilarly to Theorem 3.2，one can prove

Theorem 3.4.Ifamatrix A satisfies conditions（1.3）and（1.6），f∈Lp［0，1），1＜p＜∞，p≥r≥1，then（3.3）holds.

Theorem s3.3 and 3.4 im p ly

Corollary 3.1.Let f∈Lp［0，1），1＜p＜∞，1≤r≤p，or f∈C∗［0，1）（p=∞），1≤r＜∞.Then

In particu lar，for r=1 and f∈Lip∗（α，p）（i.e.，ω∗（f，h）p=O（hα））we obtain

Rem ark 3.1.It is well know n that for，the equality limn→∞‖f-σn（f）‖p，ω=0 holds（see［9］forω（h）=hα）.In particu lar，forwe have limn→∞En（f）p，ω=0.

Theorem 3.5 givesan analog of the estim ate（2.5）for H¨olderm etric.

Proof.By M inkow ski inequality and commu tativity of translation and convolu tion，we have

Hence，in virtue of（2.4）and Rem ark 2.1，it follows that

where in the case p=∞，the constant C1isequal to M（ν）from Lemma2.8.Let tn-m∈Pn-mbe such that‖f-tn-m‖p，ω=En-m（f）p，ω.Using equality Sk（tn-m）=tn-mfor k≥n-m，we obtain sim ilarly to（2.6）

On the other hand，by（3.4）and（3.5）（we use notationΔhf=f（·⊖h）-f（·））

Com bining estim ates（3.5）and（3.6），we finish the p roofof theorem.

Corollary 3.2.Let 1＜p≤∞，ω，µ∈Ω，whereω（t）satisfiesΔ2-cond ition，whileλ（t）= ω（t）/µ（t）is increasing on（0，1）and limt→0λ（t）=0.If，and num bers n，m∈N are such thatνn≤m≤n，ν∈（0，1），then‖Vn，m（f，r）‖p，µ≤Cλ（（n-m）-1），（n-m）∈N. Proof.In virtue of Theorem 3.5，‖Vn，m（f，r）‖p，µ≤C1（ν）En-m（f）p，µ，while by Lemma 2.10，we have En-m（f）p，µ≤C2λ（1/（n-m））.Substituting the second inequality into first one，we prove the theorem.

Follow ing the idea of Szal［16］，we assum e in tw o last theorem s that there existsα∈（0，1），such thatωα（t）/µ（t）is increasing on（0，1）.We also require thatω，µ∈Ωandω satisfiesΔ2-cond ition.

Theorem 3.6.Letamatrix A satisfies conditions（1.3）and（1.7）Then

Proof.In virtue of Theorem 3.1 and M inkow skiinequality

By Lemmas 2.4 and 2.5，the estim ates

hold.On theother hand，Ek（Δhf）∞≤‖Δhf‖∞≤ω（h），k∈N，and as Corollary，

Theorem 3.7.Let amatrix A satisfies conditions（1.3）and（1.6），f∈Lp［0，1），1＜p＜∞，or f∈C∗［0，1）（for p=∞），p≥r≥1.If f，then

The p roofof Theorem 3.7 is similar to the one of Theorem 3.6，and uses Theorem s 3.3 and 3.4 instead of Theorem 3.1.

Rem ark 3.2.The con terparts of Theorem s 3.1 and 3.7，proved in［16］，con tain the term ln2nan，ninstead of nan，nin the present paper（by authors op inion，it ism ore correctly to w rite 1+ln+nan，n）.Such estim atesm ay have a better order of decreasing（for example，ifan，n=1，an，k=0，1≤k＜n）.Itwillbe interesting to refine Theorem s3.1 and 3.7 in a sim ilar m anner and to study‖Rn（f，r）‖pin the case of p=1.

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∗Correspond ing author.Emailaddresses：Iof inaT@mai l.ru（T.V.Iofina），VolosivetsSS@mai l.ru（S.S.Volosivets）