Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 542607 | https://doi.org/10.1155/2012/542607

Shakro Tetunashvili, "On Divergence of Fourier Series by Some Methods of Summability", Journal of Function Spaces, vol. 2012, Article ID 542607, 9 pages, 2012. https://doi.org/10.1155/2012/542607

On Divergence of Fourier Series by Some Methods of Summability

Academic Editor: V. M. Kokilashvili
Received20 Sep 2010
Accepted20 Oct 2010
Published15 Jan 2012

Abstract

A new summability method of series is introduced and studied. The particular cases of this method are, for example, variable-order Cesaro and Riesz methods. Applications to divergence problem of Fourier series are given. An extension of Kolmogorov, Schipp, and Boฤkarevโ€™s well-known theorems on divergence of Fourier trigonometric, Walsh, and orthonormal series is established.

1. A New Summability Method of Series

Letโ€–โ€–๐œ†ฮ›=๐‘›โ€–โ€–(๐‘˜),๐‘›=0,1,2,โ€ฆ,๐‘˜=0,1,2,โ€ฆ,๐‘›,(1.1) be such triangular matrix which satisfies the following conditions: (1)0โ‰ค๐œ†๐‘›(๐‘˜+1)โ‰ค๐œ†๐‘›(๐‘˜)โ‰ค1,0โ‰ค๐‘˜โ‰ค๐‘›;(2)๐œ†๐‘›(0)=1,๐œ†๐‘›(๐‘˜)=0,๐‘˜โ‰ฅ๐‘›+1.(1.2)

By ๐‘ ๐‘› we denote a partial sum of a series โˆž๎“๐‘˜=0๐‘ข๐‘˜,(1.3) and by ๐œŽ๐‘› we denote a mean constructed by the ฮ› matrix, that is, ๐‘ ๐‘›=๐‘›๎“๐‘˜=0๐‘ข๐‘˜,๐œŽ๐‘›=๐‘›๎“๐‘˜=0๐œ†๐‘›(๐‘˜)๐‘ข๐‘˜.(1.4)

Theorem 1.1. Let matrix (1.1) satisfies an inequality lim๐‘›โ†’โˆž๐œ†๐‘›1(๐‘›)>2.(1.5) Then for any series (1.3) which satisfies the following condition: lim๐‘›โ†’โˆž||๐‘ ๐‘›||=+โˆž,(1.6) an equality lim๐‘›โ†’โˆž||๐œŽ๐‘›||=+โˆž(1.7) holds.

Below we prove a Lemma which is used to prove Theorem 1.1.

Lemma 1.2. For every natural number ๐‘› an inequality ||๐‘ ๐‘›โˆ’๐œŽ๐‘›||๎€ทโ‰ค21โˆ’๐œ†๐‘›๎€ธ(๐‘›)โ‹…max1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||(1.8) holds.

Proof of the Lemma. Using Abel transformation and ๐œ†๐‘›(0)=1 we get ๐‘ ๐‘›โˆ’๐œŽ๐‘›=๐‘›๎“๐‘˜=0๐‘ข๐‘˜โˆ’๐‘›๎“๐‘˜=0๐œ†๐‘›(๐‘˜)๐‘ข๐‘˜=๐‘›๎“๐‘˜=1๐‘ข๐‘˜โˆ’๐‘›๎“๐‘˜=1๐œ†๐‘›(๐‘˜)๐‘ข๐‘˜=๐‘›๎“๐‘˜=1๎€ท1โˆ’๐œ†๐‘›๎€ธ๐‘ข(๐‘˜)๐‘˜=๐‘›โˆ’1๎“๐‘˜=1๎€ท๐œ†๐‘›(๐‘˜+1)โˆ’๐œ†๐‘›๎€ธ๐‘ (๐‘˜)๐‘˜+๎€ท1โˆ’๐œ†๐‘›๎€ธ๐‘ (๐‘›)๐‘›.(1.9) Therefore, ||๐‘ ๐‘›โˆ’๐œŽ๐‘›||โ‰ค๐‘›โˆ’1๎“๐‘˜=1||๐œ†๐‘›(๐‘˜+1)โˆ’๐œ†๐‘›||โ‹…||๐‘ (๐‘˜)๐‘˜||+||1โˆ’๐œ†๐‘›||โ‹…||๐‘ (๐‘›)๐‘›||โ‰คmax1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||โ‹…๎ƒฉ๐‘›โˆ’1๎“๐‘˜=1||๐œ†๐‘›(๐‘˜+1)โˆ’๐œ†๐‘›||+||(๐‘˜)1โˆ’๐œ†๐‘›||๎ƒช.(๐‘›)(1.10) Thus, taking into account (1.1) we immediately get ||๐‘ ๐‘›โˆ’๐œŽ๐‘›||โ‰คmax1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||โ‹…๎ƒฉ๐‘›โˆ’1๎“๐‘˜=1๎€ท๐œ†๐‘›(๐‘˜)โˆ’๐œ†๐‘›๎€ธ(๐‘˜+1)+1โˆ’๐œ†๐‘›๎ƒช(๐‘›)=max1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||โ‹…๎€ท๐œ†๐‘›(1)โˆ’๐œ†๐‘›(๐‘›)+1โˆ’๐œ†๐‘›๎€ธ(๐‘›)โ‰คmax1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||โ‹…๎€ท1โˆ’๐œ†๐‘›(๐‘›)+1โˆ’๐œ†๐‘›๎€ธ๎€ท(๐‘›)=2โ‹…1โˆ’๐œ†๐‘›๎€ธ(๐‘›)โ‹…max1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||.(1.11) So the Lemma is proved.

Proof of Theorem 1.1. According to the condition of Theorem 1.1 we have lim๐‘›โ†’โˆž๐œ†๐‘›1(๐‘›)=2+๐›ฟ(1.12) for some ๐›ฟ>0. Note that inequalities 0โ‰ค๐œ†๐‘›(๐‘›)โ‰ค1 which hold for every natural ๐‘› imply 1/2+๐›ฟโ‰ค1, that is, ๐›ฟโ‰ค1/2.
So, 0<๐›ฟโ‰ค1/2 holds.
According to (1.12) there exists a natural number ๐‘›0 such that for every natural number ๐‘›>๐‘›0 we have ๐œ†๐‘›1(๐‘›)>2+๐›ฟ2.(1.13) So according to the Lemma, for every ๐‘›>๐‘›0 an inequality ||๐‘ ๐‘›โˆ’๐œŽ๐‘›||๎‚€๎‚€1<2โ‹…1โˆ’2+๐›ฟ2๎‚๎‚โ‹…max1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||(1.14) holds true; that is, if ๐‘›>๐‘›0, then ||๐‘ ๐‘›โˆ’๐œŽ๐‘›||<(1โˆ’๐›ฟ)โ‹…max1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||.(1.15) Thus for every ๐‘›>๐‘›0 an inequality โ€–โ€–๐‘ ๐‘›|โˆ’|๐œŽ๐‘›โ€–โ€–<(1โˆ’๐›ฟ)โ‹…max1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||(1.16) holds.
So for every ๐‘›>๐‘›0 we have||๐œŽ๐‘›||>||๐‘ ๐‘›||โˆ’(1โˆ’๐›ฟ)โ‹…max1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||.(1.17)
Note that for every natural ๐‘› there exists at least one natural number 1โ‰ค๐‘žโ‰ค๐‘›, such that the partial sums of the series (1.3) satisfy the following condition:||๐‘ ๐‘ž||=max1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||.(1.18) We define ๐‘๐‘› by a formula: ๐‘๐‘›๎‚ป||๐‘ =max๐‘žโˆถ1โ‰ค๐‘žโ‰ค๐‘›&๐‘ž||=max1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||๎‚ผ.(1.19) So ๐‘๐‘› is maximal number among the above-mentioned natural ๐‘ž numbers. Consequently, 1โ‰ค๐‘๐‘›||๐‘ โ‰ค๐‘›,๐‘๐‘›||=max1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||,๐‘(1.20)๐‘›โ‰ค๐‘๐‘›+1,||๐‘ ๐‘๐‘›||โ‰ค||๐‘ ๐‘๐‘›+1||.(1.21) According to the condition of Theorem 1.1, lim๐‘›โ†’โˆž||๐‘ ๐‘›||=+โˆž.(1.22) Therefore, lim๐‘›โ†’โˆž||๐‘ ๐‘๐‘›||=+โˆž,(1.23) that is, lim๐‘›โ†’โˆž๐‘๐‘›=+โˆž.(1.24) A consequence of (1.24) is that there exists such natural ๐‘›1 that if ๐‘›>๐‘›1 then ๐‘๐‘›>๐‘›0 and since (1.17) holds for every ๐‘›>๐‘›0, then (1.17) remains true for every ๐‘๐‘›, where ๐‘›>๐‘›1.
So||๐œŽ๐‘๐‘›||>||๐‘ ๐‘๐‘›||โˆ’(1โˆ’๐›ฟ)โ‹…max1โ‰ค๐‘˜โ‰ค๐‘๐‘›||๐‘ ๐‘˜||.(1.25) Since 1โ‰ค๐‘๐‘›โ‰ค๐‘›, therefore, max1โ‰ค๐‘˜โ‰ค๐‘๐‘›||๐‘ ๐‘˜||โ‰คmax1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||.(1.26) Note that the last one and (1.25) imply ||๐œŽ๐‘๐‘›||>||๐‘ ๐‘๐‘›||โˆ’(1โˆ’๐›ฟ)โ‹…max1โ‰ค๐‘˜โ‰ค๐‘›||๐‘ ๐‘˜||.(1.27) So according to (1.21) we have ||๐œŽ๐‘๐‘›||>||๐‘ ๐‘๐‘›||||๐‘ โˆ’(1โˆ’๐›ฟ)โ‹…๐‘๐‘›||,(1.28) that is, for every ๐‘›>๐‘›1 an inequality ||๐œŽ๐‘๐‘›||||๐‘ >๐›ฟโ‹…๐‘๐‘›||1holds,where0<๐›ฟโ‰ค2.(1.29) Also, (1.23) and (1.29) imply lim๐‘›โ†’โˆž||๐œŽ๐‘๐‘›||=+โˆž.(1.30) So we have finished the proof of Theorem 1.1.

Below we consider some consequences of Theorem 1.1.

Let ฮ›=โ€–๐œ†๐‘›(๐‘˜)โ€–be a triangular matrix, where the sequence {๐›ผ๐‘›} is from [0,1] and for every 0โ‰ค๐‘˜โ‰ค๐‘› number ๐œ†๐‘›(๐‘˜) is defined by the formula: ๐œ†๐‘›๐ด(๐‘˜)=๐›ผ๐‘›๐‘›โˆ’๐‘˜๐ด๐›ผ๐‘›๐‘›,where๐ด๐›ผ๐‘›๐‘›=๎€ท๐›ผ๐‘›๐›ผ+1๎€ธ๎€ท๐‘›๎€ธโ‹ฏ๎€ท๐›ผ+2๐‘›๎€ธ+๐‘›๐‘›!.(1.31)

If ๐›ผ๐‘›=๐›ผ, for every ๐‘›โ‰ฅ0 and (1.31) holds true, then the ฮ› method is Cesaro (๐ถ,๐›ผ) summability method, and if ๐›ผ๐‘›โ‰ก0, then the ฮ› method coincides with convergence.

We introduce Cesaro summability method with variable orders, denoted by a symbol (๐ถ,{๐›ผ๐‘›}), which coincides with ฮ› summability method defined by (1.31). Means of this method for series (1.3) we denoted by ๐œŽ๐›ผ๐‘›๐‘›.

For (๐ถ,{๐›ผ๐‘›}) we have the following.

Theorem 1.3. Let a sequences {๐›ผ๐‘›} be such that for some positive number ๐‘š we have ๐›ผ๐‘›โ‰ค๐‘ln๐‘›,where0โ‰ค๐‘<ln2๐‘Ž๐‘›๐‘‘๐‘›>๐‘š.(1.32) Then for any series (1.3) which satisfies the following condition: lim๐‘›โ†’โˆž||๐‘ ๐‘›||=+โˆž,(1.33) an equality lim๐‘›โ†’โˆž||๐œŽ๐›ผ๐‘›๐‘›||=+โˆž(1.34) holds.

Proof of Theorem 1.3. Note that every ๐œ†๐‘›(๐‘˜) satisfies condition (1.1) and condition (1.3). Indeed, ๐œ†๐‘›(๐‘˜+1)๐œ†๐‘›=๐ด(๐‘˜)๐›ผ๐‘›๐‘›โˆ’๐‘˜โˆ’1๐ด๐›ผ๐‘›๐‘›โˆ’๐‘˜=๐‘›โˆ’๐‘˜๐›ผ๐‘›+๐‘›โˆ’๐‘˜โ‰ค1(1.35) and ๐œ†๐‘›(0)=1, when ๐‘›โ‰ฅ0.
For every ๐‘›โ‰ฅ1 we have๐œ†๐‘›1(๐‘›)=๐ด๐›ผ๐‘›๐‘›,where๐ด๐›ผ๐‘›๐‘›=๎€ท๐›ผ๐‘›๐›ผ+1๎€ธ๎€ท๐‘›๎€ธโ‹ฏ๎€ท๐›ผ+2๐‘›๎€ธ+๐‘›,๐‘›!(1.36) that is, ๐ด๐›ผ๐‘›๐‘›=๎‚€๐›ผ1+๐‘›1๐›ผ๎‚๎‚€1+๐‘›2๎‚โ‹ฏ๎‚€๐›ผ1+๐‘›๐‘›๎‚.(1.37) Therefore, ln๐ด๐›ผ๐‘›๐‘›=๐‘›๎“๐‘˜=1๎‚€๐›ผln1+๐‘˜๐‘›๎‚<๐‘›๎“๐‘˜=1๐›ผ๐‘›๐‘˜=๐›ผ๐‘›โ‹…๎“๐‘˜=11๐‘˜<๐›ผ๐‘›(1+ln๐‘›).(1.38) Note that the last one and (1.32) imply that 2๐‘=ln1+๐›พ,forsome0<๐›พโ‰ค1,(1.39) and if ๐‘›>๐‘š, we have ๐ด๐›ผ๐‘›๐‘›<๐‘’๐›ผ๐‘›(1+ln๐‘›)=๐‘’๐›ผ๐‘›โ‹…๐‘’๐›ผ๐‘›ln๐‘›โ‰ค๐‘’๐›ผ๐‘›โ‹…๐‘’๐‘=๐‘’๐›ผ๐‘›โ‹…๐‘’ln(2/(1+๐›พ))=๐‘’๐›ผ๐‘›โ‹…2,1+๐›พ(1.40) that is, ๐œ†๐‘›1(๐‘›)=๐ด๐›ผ๐‘›๐‘›>1๐‘’๐›ผ๐‘›โ‹…๎‚€12+๐›พ2๎‚,where๐›พ>0.(1.41) Note that ๐›ผ๐‘›โ†’0 implies the existence of such ๐›พ1>0 and natural ๐‘›2, that if ๐‘›>๐‘›2, then 1๐‘’๐›ผ๐‘›โ‹…๎‚€12+๐›พ2๎‚>12+๐›พ1,(1.42) that is, if ๐‘›>๐‘›2, then ๐œ†๐‘›1(๐‘›)>2+๐›พ1.(1.43) A consequence of (1.43) is that if (1.32) holds, then the ฮ› matrix satisfies conditions of Theorem 1.1. This completes the proof of Theorem 1.3.

Theorem 1.3 directly implies the following.

Theorem 1.4. Let {๐›ผ๐‘›} be such sequence that ๎€ฝ๐›ผ๐‘›๎€พ๎‚€1=๐‘œ๎‚.ln๐‘›(1.44) Then for every series (1.3) which satisfies lim๐‘›โ†’โˆž||๐‘ ๐‘›||=+โˆž,(1.45) we have lim๐‘›โ†’โˆž||๐œŽ๐›ผ๐‘›๐‘›||=+โˆž.(1.46)

2. On Divergence of Fourier Series

It is well known the following.

Theorem A (Kolmogorov [1]). There exists such summable function ๐‘“ that Fourier trigonometric series of ๐‘“๐‘Ž02+โˆž๎“๐‘˜=1๐‘Ž๐‘˜cos๐‘˜๐‘ฅ+๐‘๐‘˜sin๐‘˜๐‘ฅ(2.1) unboundedly diverges everywhere.

Let ๐‘Š={๐‘ค๐‘›(๐‘ก)}โˆž๐‘›=1 be the Walsh system. Below we formulate Theorem B which is analogous of Theorem A and holds for Fourier-Walsh series.

Theorem B (Schipp [2, 3]). There exists such summable function ๐‘” that Fourier-Walsh series of ๐‘”โˆž๎“๐‘›=1๐‘Ž๐‘›๐‘ค๐‘›(t)(2.2) unboundedly diverges everywhere.

Let ฮฆ={๐œ‘๐‘›(๐‘ก)} be orthonormal functions system defined on [0,1], such that ||๐œ‘๐‘›||[](๐‘ก)โ‰ค๐‘€,๐‘กโˆˆ0,1,๐‘›=1,2,โ€ฆ(2.3)

Then below-mentioned theorem holds.

Theorem C (Boฬ†๐œkarev [4]). For every orthonormal system ฮฆ which satisfies (2.3), there exists such summable function โ„Ž defined on [0,1] that its Fourier series constructed by ฮฆ system โˆž๎“๐‘›=1๐‘Ž๐‘›๐œ‘๐‘›(๐‘ก)(2.4) unboundedly diverges in any point of some set ๐ธโŠ‚[0,1] with positive measure.

Denote by ๐œŽ๐›ผ๐‘›๐‘›(๐‘ฅ;๐‘“), ๐œŽ๐›ผ๐‘›๐‘›(๐‘ก,๐‘”,๐‘Š), and ๐œŽ๐›ผ๐‘›๐‘›(๐‘ก,โ„Ž,ฮฆ) means of series (2.1), (2.2), and (2.4), respectively.

Theorem 1.3 implies that if {๐›ผ๐‘›} satisfies (1.32), then Theorems A, B, and C hold for (๐ถ,{๐›ผ๐‘›}) summability method.

Namely, the following Theorems hold true.

Theorem 2.1. Let a sequence {๐›ผ๐‘›} satisfies (1.32). Then there exists such summable function ๐‘“, that sequence {๐œŽ๐›ผ๐‘›๐‘›(๐‘ฅ;๐‘“)} unboundedly diverges everywhere.

Theorem 2.2. Let a sequence {๐›ผ๐‘›} satisfies (1.32). Then there exists such summable function ๐‘” that sequence {๐œŽ๐›ผ๐‘›๐‘›(๐‘ก,๐‘”,๐‘Š)} unboundedly diverges everywhere.

Theorem 2.3. If orthonormal system ฮฆ satisfies (2.3) and a sequence {๐›ผn} satisfies (1.32), then there exists such summable function โ„Ž, defined on [0,1], that sequence {๐œŽ๐›ผ๐‘›๐‘›(๐‘ก;โ„Ž;ฮฆ)} unboundedly diverges at every point of some set ๐ธโŠ‚[0,1] with positive measure.

It is obvious that a consequence of Theorem 1.4 is that Theorems 2.1, 2.2, and 2.3 hold true if๐›ผ๐‘›๎‚€1=๐‘œ๎‚.ln๐‘›(2.5)

Remark 2.4. If every number ๐œ†๐‘›(๐‘˜) will be replaced by (1โˆ’๐‘˜/(๐‘›+1))๐›ผ๐‘› in (1.31), then we get a summability method defined by ฮ›=โ€–๐œ†๐‘›(๐‘˜)โ€– matrix, which we call Riesz summability method with variable orders and denote it by symbol (๐‘…,{๐›ผ๐‘›}).

It can be proved analogously that Theorems 2.1, 2.2, and 2.3 remain true for Riesz summability method with variable orders, that is, for (๐‘…,{๐›ผ๐‘›}) method, where {๐›ผ๐‘›} satisfies (1.32).

Acknowledgment

This paper is supported by the Grant GNSF/STO9_23_3โ€“100.

References

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Copyright ยฉ 2012 Shakro Tetunashvili. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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