Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 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.

On Divergence of Fourier Series by Some Methods of Summability

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


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 Bočkarevā€™s well-known theorems on divergence of Fourier trigonometric, Walsh, and orthonormal series is established.

1. A New Summability Method of Series

Letā€–ā€–šœ†Ī›=š‘›ā€–ā€–(š‘˜),š‘›=0,1,2,ā€¦,š‘˜=0,1,2,ā€¦,š‘›,(1.1) be such triangular matrix which satisfies the following conditions: (1)0ā‰¤šœ†š‘›(š‘˜+1)ā‰¤šœ†š‘›(š‘˜)ā‰¤1,0ā‰¤š‘˜ā‰¤š‘›;(2)šœ†š‘›(0)=1,šœ†š‘›(š‘˜)=0,š‘˜ā‰„š‘›+1.(1.2)

By š‘ š‘› we denote a partial sum of a series āˆžī“š‘˜=0š‘¢š‘˜,(1.3) and by šœŽš‘› we denote a mean constructed by the Ī› matrix, that is, š‘ š‘›=š‘›ī“š‘˜=0š‘¢š‘˜,šœŽš‘›=š‘›ī“š‘˜=0šœ†š‘›(š‘˜)š‘¢š‘˜.(1.4)

Theorem 1.1. Let matrix (1.1) satisfies an inequality limš‘›ā†’āˆžšœ†š‘›1(š‘›)>2.(1.5) Then for any series (1.3) which satisfies the following condition: limš‘›ā†’āˆž||š‘ š‘›||=+āˆž,(1.6) an equality limš‘›ā†’āˆž||šœŽš‘›||=+āˆž(1.7) holds.

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

Lemma 1.2. For every natural number š‘› an inequality ||š‘ š‘›āˆ’šœŽš‘›||ī€·ā‰¤21āˆ’šœ†š‘›ī€ø(š‘›)ā‹…max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||(1.8) holds.

Proof of the Lemma. Using Abel transformation and šœ†š‘›(0)=1 we get š‘ š‘›āˆ’šœŽš‘›=š‘›ī“š‘˜=0š‘¢š‘˜āˆ’š‘›ī“š‘˜=0šœ†š‘›(š‘˜)š‘¢š‘˜=š‘›ī“š‘˜=1š‘¢š‘˜āˆ’š‘›ī“š‘˜=1šœ†š‘›(š‘˜)š‘¢š‘˜=š‘›ī“š‘˜=1ī€·1āˆ’šœ†š‘›ī€øš‘¢(š‘˜)š‘˜=š‘›āˆ’1ī“š‘˜=1ī€·šœ†š‘›(š‘˜+1)āˆ’šœ†š‘›ī€øš‘ (š‘˜)š‘˜+ī€·1āˆ’šœ†š‘›ī€øš‘ (š‘›)š‘›.(1.9) Therefore, ||š‘ š‘›āˆ’šœŽš‘›||ā‰¤š‘›āˆ’1ī“š‘˜=1||šœ†š‘›(š‘˜+1)āˆ’šœ†š‘›||ā‹…||š‘ (š‘˜)š‘˜||+||1āˆ’šœ†š‘›||ā‹…||š‘ (š‘›)š‘›||ā‰¤max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||ā‹…īƒ©š‘›āˆ’1ī“š‘˜=1||šœ†š‘›(š‘˜+1)āˆ’šœ†š‘›||+||(š‘˜)1āˆ’šœ†š‘›||īƒŖ.(š‘›)(1.10) Thus, taking into account (1.1) we immediately get ||š‘ š‘›āˆ’šœŽš‘›||ā‰¤max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||ā‹…īƒ©š‘›āˆ’1ī“š‘˜=1ī€·šœ†š‘›(š‘˜)āˆ’šœ†š‘›ī€ø(š‘˜+1)+1āˆ’šœ†š‘›īƒŖ(š‘›)=max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||ā‹…ī€·šœ†š‘›(1)āˆ’šœ†š‘›(š‘›)+1āˆ’šœ†š‘›ī€ø(š‘›)ā‰¤max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||ā‹…ī€·1āˆ’šœ†š‘›(š‘›)+1āˆ’šœ†š‘›ī€øī€·(š‘›)=2ā‹…1āˆ’šœ†š‘›ī€ø(š‘›)ā‹…max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||.(1.11) So the Lemma is proved.

Proof of Theorem 1.1. According to the condition of Theorem 1.1 we have limš‘›ā†’āˆžšœ†š‘›1(š‘›)=2+š›æ(1.12) for some š›æ>0. Note that inequalities 0ā‰¤šœ†š‘›(š‘›)ā‰¤1 which hold for every natural š‘› imply 1/2+š›æā‰¤1, that is, š›æā‰¤1/2.
So, 0<š›æā‰¤1/2 holds.
According to (1.12) there exists a natural number š‘›0 such that for every natural number š‘›>š‘›0 we have šœ†š‘›1(š‘›)>2+š›æ2.(1.13) So according to the Lemma, for every š‘›>š‘›0 an inequality ||š‘ š‘›āˆ’šœŽš‘›||ī‚€ī‚€1<2ā‹…1āˆ’2+š›æ2ī‚ī‚ā‹…max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||(1.14) holds true; that is, if š‘›>š‘›0, then ||š‘ š‘›āˆ’šœŽš‘›||<(1āˆ’š›æ)ā‹…max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||.(1.15) Thus for every š‘›>š‘›0 an inequality ā€–ā€–š‘ š‘›|āˆ’|šœŽš‘›ā€–ā€–<(1āˆ’š›æ)ā‹…max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||(1.16) holds.
So for every š‘›>š‘›0 we have||šœŽš‘›||>||š‘ š‘›||āˆ’(1āˆ’š›æ)ā‹…max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||.(1.17)
Note that for every natural š‘› there exists at least one natural number 1ā‰¤š‘žā‰¤š‘›, such that the partial sums of the series (1.3) satisfy the following condition:||š‘ š‘ž||=max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||.(1.18) We define š‘š‘› by a formula: š‘š‘›ī‚»||š‘ =maxš‘žāˆ¶1ā‰¤š‘žā‰¤š‘›&š‘ž||=max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||ī‚¼.(1.19) So š‘š‘› is maximal number among the above-mentioned natural š‘ž numbers. Consequently, 1ā‰¤š‘š‘›||š‘ ā‰¤š‘›,š‘š‘›||=max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||,š‘(1.20)š‘›ā‰¤š‘š‘›+1,||š‘ š‘š‘›||ā‰¤||š‘ š‘š‘›+1||.(1.21) According to the condition of Theorem 1.1, limš‘›ā†’āˆž||š‘ š‘›||=+āˆž.(1.22) Therefore, limš‘›ā†’āˆž||š‘ š‘š‘›||=+āˆž,(1.23) that is, limš‘›ā†’āˆžš‘š‘›=+āˆž.(1.24) A consequence of (1.24) is that there exists such natural š‘›1 that if š‘›>š‘›1 then š‘š‘›>š‘›0 and since (1.17) holds for every š‘›>š‘›0, then (1.17) remains true for every š‘š‘›, where š‘›>š‘›1.
So||šœŽš‘š‘›||>||š‘ š‘š‘›||āˆ’(1āˆ’š›æ)ā‹…max1ā‰¤š‘˜ā‰¤š‘š‘›||š‘ š‘˜||.(1.25) Since 1ā‰¤š‘š‘›ā‰¤š‘›, therefore, max1ā‰¤š‘˜ā‰¤š‘š‘›||š‘ š‘˜||ā‰¤max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||.(1.26) Note that the last one and (1.25) imply ||šœŽš‘š‘›||>||š‘ š‘š‘›||āˆ’(1āˆ’š›æ)ā‹…max1ā‰¤š‘˜ā‰¤š‘›||š‘ š‘˜||.(1.27) So according to (1.21) we have ||šœŽš‘š‘›||>||š‘ š‘š‘›||||š‘ āˆ’(1āˆ’š›æ)ā‹…š‘š‘›||,(1.28) that is, for every š‘›>š‘›1 an inequality ||šœŽš‘š‘›||||š‘ >š›æā‹…š‘š‘›||1holds,where0<š›æā‰¤2.(1.29) Also, (1.23) and (1.29) imply limš‘›ā†’āˆž||šœŽš‘š‘›||=+āˆž.(1.30) So we have finished the proof of Theorem 1.1.

Below we consider some consequences of Theorem 1.1.

Let Ī›=ā€–šœ†š‘›(š‘˜)ā€–be a triangular matrix, where the sequence {š›¼š‘›} is from [0,1] and for every 0ā‰¤š‘˜ā‰¤š‘› number šœ†š‘›(š‘˜) is defined by the formula: šœ†š‘›š“(š‘˜)=š›¼š‘›š‘›āˆ’š‘˜š“š›¼š‘›š‘›,whereš“š›¼š‘›š‘›=ī€·š›¼š‘›š›¼+1ī€øī€·š‘›ī€øā‹Æī€·š›¼+2š‘›ī€ø+š‘›š‘›!.(1.31)

If š›¼š‘›=š›¼, for every š‘›ā‰„0 and (1.31) holds true, then the Ī› method is Cesaro (š¶,š›¼) summability method, and if š›¼š‘›ā‰”0, then the Ī› method coincides with convergence.

We introduce Cesaro summability method with variable orders, denoted by a symbol (š¶,{š›¼š‘›}), which coincides with Ī› summability method defined by (1.31). Means of this method for series (1.3) we denoted by šœŽš›¼š‘›š‘›.

For (š¶,{š›¼š‘›}) we have the following.

Theorem 1.3. Let a sequences {š›¼š‘›} be such that for some positive number š‘š we have š›¼š‘›ā‰¤š‘lnš‘›,where0ā‰¤š‘<ln2š‘Žš‘›š‘‘š‘›>š‘š.(1.32) Then for any series (1.3) which satisfies the following condition: limš‘›ā†’āˆž||š‘ š‘›||=+āˆž,(1.33) an equality limš‘›ā†’āˆž||šœŽš›¼š‘›š‘›||=+āˆž(1.34) holds.

Proof of Theorem 1.3. Note that every šœ†š‘›(š‘˜) satisfies condition (1.1) and condition (1.3). Indeed, šœ†š‘›(š‘˜+1)šœ†š‘›=š“(š‘˜)š›¼š‘›š‘›āˆ’š‘˜āˆ’1š“š›¼š‘›š‘›āˆ’š‘˜=š‘›āˆ’š‘˜š›¼š‘›+š‘›āˆ’š‘˜ā‰¤1(1.35) and šœ†š‘›(0)=1, when š‘›ā‰„0.
For every š‘›ā‰„1 we havešœ†š‘›1(š‘›)=š“š›¼š‘›š‘›,whereš“š›¼š‘›š‘›=ī€·š›¼š‘›š›¼+1ī€øī€·š‘›ī€øā‹Æī€·š›¼+2š‘›ī€ø+š‘›,š‘›!(1.36) that is, š“š›¼š‘›š‘›=ī‚€š›¼1+š‘›1š›¼ī‚ī‚€1+š‘›2ī‚ā‹Æī‚€š›¼1+š‘›š‘›ī‚.(1.37) Therefore, lnš“š›¼š‘›š‘›=š‘›ī“š‘˜=1ī‚€š›¼ln1+š‘˜š‘›ī‚<š‘›ī“š‘˜=1š›¼š‘›š‘˜=š›¼š‘›ā‹…ī“š‘˜=11š‘˜<š›¼š‘›(1+lnš‘›).(1.38) Note that the last one and (1.32) imply that 2š‘=ln1+š›¾,forsome0<š›¾ā‰¤1,(1.39) and if š‘›>š‘š, we have š“š›¼š‘›š‘›<š‘’š›¼š‘›(1+lnš‘›)=š‘’š›¼š‘›ā‹…š‘’š›¼š‘›lnš‘›ā‰¤š‘’š›¼š‘›ā‹…š‘’š‘=š‘’š›¼š‘›ā‹…š‘’ln(2/(1+š›¾))=š‘’š›¼š‘›ā‹…2,1+š›¾(1.40) that is, šœ†š‘›1(š‘›)=š“š›¼š‘›š‘›>1š‘’š›¼š‘›ā‹…ī‚€12+š›¾2ī‚,whereš›¾>0.(1.41) Note that š›¼š‘›ā†’0 implies the existence of such š›¾1>0 and natural š‘›2, that if š‘›>š‘›2, then 1š‘’š›¼š‘›ā‹…ī‚€12+š›¾2ī‚>12+š›¾1,(1.42) that is, if š‘›>š‘›2, then šœ†š‘›1(š‘›)>2+š›¾1.(1.43) A consequence of (1.43) is that if (1.32) holds, then the Ī› 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 {š›¼š‘›} be such sequence that ī€½š›¼š‘›ī€¾ī‚€1=š‘œī‚.lnš‘›(1.44) Then for every series (1.3) which satisfies limš‘›ā†’āˆž||š‘ š‘›||=+āˆž,(1.45) we have limš‘›ā†’āˆž||šœŽš›¼š‘›š‘›||=+āˆž.(1.46)

2. On Divergence of Fourier Series

It is well known the following.

Theorem A (Kolmogorov [1]). There exists such summable function š‘“ that Fourier trigonometric series of š‘“š‘Ž02+āˆžī“š‘˜=1š‘Žš‘˜cosš‘˜š‘„+š‘š‘˜sinš‘˜š‘„(2.1) unboundedly diverges everywhere.

Let š‘Š={š‘¤š‘›(š‘”)}āˆžš‘›=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 š‘” that Fourier-Walsh series of š‘”āˆžī“š‘›=1š‘Žš‘›š‘¤š‘›(t)(2.2) unboundedly diverges everywhere.

Let Ī¦={šœ‘š‘›(š‘”)} be orthonormal functions system defined on [0,1], such that ||šœ‘š‘›||[](š‘”)ā‰¤š‘€,š‘”āˆˆ0,1,š‘›=1,2,ā€¦(2.3)

Then below-mentioned theorem holds.

Theorem C (BoĢ†šœkarev [4]). For every orthonormal system Ī¦ which satisfies (2.3), there exists such summable function ā„Ž defined on [0,1] that its Fourier series constructed by Ī¦ system āˆžī“š‘›=1š‘Žš‘›šœ‘š‘›(š‘”)(2.4) unboundedly diverges in any point of some set šøāŠ‚[0,1] with positive measure.

Denote by šœŽš›¼š‘›š‘›(š‘„;š‘“), šœŽš›¼š‘›š‘›(š‘”,š‘”,š‘Š), and šœŽš›¼š‘›š‘›(š‘”,ā„Ž,Ī¦) means of series (2.1), (2.2), and (2.4), respectively.

Theorem 1.3 implies that if {š›¼š‘›} satisfies (1.32), then Theorems A, B, and C hold for (š¶,{š›¼š‘›}) summability method.

Namely, the following Theorems hold true.

Theorem 2.1. Let a sequence {š›¼š‘›} satisfies (1.32). Then there exists such summable function š‘“, that sequence {šœŽš›¼š‘›š‘›(š‘„;š‘“)} unboundedly diverges everywhere.

Theorem 2.2. Let a sequence {š›¼š‘›} satisfies (1.32). Then there exists such summable function š‘” that sequence {šœŽš›¼š‘›š‘›(š‘”,š‘”,š‘Š)} unboundedly diverges everywhere.

Theorem 2.3. If orthonormal system Ī¦ satisfies (2.3) and a sequence {š›¼n} satisfies (1.32), then there exists such summable function ā„Ž, defined on [0,1], that sequence {šœŽš›¼š‘›š‘›(š‘”;ā„Ž;Ī¦)} unboundedly diverges at every point of some set šøāŠ‚[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 ifš›¼š‘›ī‚€1=š‘œī‚.lnš‘›(2.5)

Remark 2.4. If every number šœ†š‘›(š‘˜) will be replaced by (1āˆ’š‘˜/(š‘›+1))š›¼š‘› in (1.31), then we get a summability method defined by Ī›=ā€–šœ†š‘›(š‘˜)ā€– matrix, which we call Riesz summability method with variable orders and denote it by symbol (š‘…,{š›¼š‘›}).

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 (š‘…,{š›¼š‘›}) method, where {š›¼š‘›} satisfies (1.32).


This paper is supported by the Grant GNSF/STO9_23_3ā€“100.


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Copyright Ā© 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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