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Journal of Function Spaces and Applications
Volume 2012, Article ID 542607, 9 pages
Research Article

On Divergence of Fourier Series by Some Methods of Summability

1Department of Mathematical Analysis, A. Razmadze Mathematical Institute, 1, Aleksidze Street, 0193 Tbilisi, Georgia
2Department of Mathematics, Georgian Technical University, 77, M. Kostava Street, 0175 Tbilisi, Georgia

Received 20 September 2010; Accepted 20 October 2010

Academic Editor: V. M. Kokilashvili

Copyright © 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.


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 Bočkarev’s well-known theorems on divergence of Fourier trigonometric, Walsh, and orthonormal series is established.

1. A New Summability Method of Series

Let𝜆Λ=𝑛(𝑘),𝑛=0,1,2,,𝑘=0,1,2,,𝑛,(1.1) be such triangular matrix which satisfies the following conditions: (1)0𝜆𝑛(𝑘+1)𝜆𝑛(𝑘)1,0𝑘𝑛;(2)𝜆𝑛(0)=1,𝜆𝑛(𝑘)=0,𝑘𝑛+1.(1.2)

By 𝑠𝑛 we denote a partial sum of a series 𝑘=0𝑢𝑘,(1.3) and by 𝜎𝑛 we denote a mean constructed by the Λ matrix, that is, 𝑠𝑛=𝑛𝑘=0𝑢𝑘,𝜎𝑛=𝑛𝑘=0𝜆𝑛(𝑘)𝑢𝑘.(1.4)

Theorem 1.1. Let matrix (1.1) satisfies an inequality lim𝑛𝜆𝑛1(𝑛)>2.(1.5) Then for any series (1.3) which satisfies the following condition: lim𝑛||𝑠𝑛||=+,(1.6) an equality lim𝑛||𝜎𝑛||=+(1.7) holds.

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

Lemma 1.2. For every natural number 𝑛 an inequality ||𝑠𝑛𝜎𝑛||21𝜆𝑛(𝑛)max1𝑘𝑛||𝑠𝑘||(1.8) holds.

Proof of the Lemma. Using Abel transformation and 𝜆𝑛(0)=1 we get 𝑠𝑛𝜎𝑛=𝑛𝑘=0𝑢𝑘𝑛𝑘=0𝜆𝑛(𝑘)𝑢𝑘=𝑛𝑘=1𝑢𝑘𝑛𝑘=1𝜆𝑛(𝑘)𝑢𝑘=𝑛𝑘=11𝜆𝑛𝑢(𝑘)𝑘=𝑛1𝑘=1𝜆𝑛(𝑘+1)𝜆𝑛𝑠(𝑘)𝑘+1𝜆𝑛𝑠(𝑛)𝑛.(1.9) Therefore, ||𝑠𝑛𝜎𝑛||𝑛1𝑘=1||𝜆𝑛(𝑘+1)𝜆𝑛||||𝑠(𝑘)𝑘||+||1𝜆𝑛||||𝑠(𝑛)𝑛||max1𝑘𝑛||𝑠𝑘||𝑛1𝑘=1||𝜆𝑛(𝑘+1)𝜆𝑛||+||(𝑘)1𝜆𝑛||.(𝑛)(1.10) Thus, taking into account (1.1) we immediately get ||𝑠𝑛𝜎𝑛||max1𝑘𝑛||𝑠𝑘||𝑛1𝑘=1𝜆𝑛(𝑘)𝜆𝑛(𝑘+1)+1𝜆𝑛(𝑛)=max1𝑘𝑛||𝑠𝑘||𝜆𝑛(1)𝜆𝑛(𝑛)+1𝜆𝑛(𝑛)max1𝑘𝑛||𝑠𝑘||1𝜆𝑛(𝑛)+1𝜆𝑛(𝑛)=21𝜆𝑛(𝑛)max1𝑘𝑛||𝑠𝑘||.(1.11) So the Lemma is proved.

Proof of Theorem 1.1. According to the condition of Theorem 1.1 we have lim𝑛𝜆𝑛1(𝑛)=2+𝛿(1.12) for some 𝛿>0. Note that inequalities 0𝜆𝑛(𝑛)1 which hold for every natural 𝑛 imply 1/2+𝛿1, that is, 𝛿1/2.
So, 0<𝛿1/2 holds.
According to (1.12) there exists a natural number 𝑛0 such that for every natural number 𝑛>𝑛0 we have 𝜆𝑛1(𝑛)>2+𝛿2.(1.13) So according to the Lemma, for every 𝑛>𝑛0 an inequality ||𝑠𝑛𝜎𝑛||1<212+𝛿2max1𝑘𝑛||𝑠𝑘||(1.14) holds true; that is, if 𝑛>𝑛0, then ||𝑠𝑛𝜎𝑛||<(1𝛿)max1𝑘𝑛||𝑠𝑘||.(1.15) Thus for every 𝑛>𝑛0 an inequality 𝑠𝑛||𝜎𝑛<(1𝛿)max1𝑘𝑛||𝑠𝑘||(1.16) holds.
So for every 𝑛>𝑛0 we have||𝜎𝑛||>||𝑠𝑛||(1𝛿)max1𝑘𝑛||𝑠𝑘||.(1.17)
Note that for every natural 𝑛 there exists at least one natural number 1𝑞𝑛, such that the partial sums of the series (1.3) satisfy the following condition:||𝑠𝑞||=max1𝑘𝑛||𝑠𝑘||.(1.18) We define 𝑝𝑛 by a formula: 𝑝𝑛||𝑠=max𝑞1𝑞𝑛&𝑞||=max1𝑘𝑛||𝑠𝑘||.(1.19) So 𝑝𝑛 is maximal number among the above-mentioned natural 𝑞 numbers. Consequently, 1𝑝𝑛||𝑠𝑛,𝑝𝑛||=max1𝑘𝑛||𝑠𝑘||,𝑝(1.20)𝑛𝑝𝑛+1,||𝑠𝑝𝑛||||𝑠𝑝𝑛+1||.(1.21) According to the condition of Theorem 1.1, lim𝑛||𝑠𝑛||=+.(1.22) Therefore, lim𝑛||𝑠𝑝𝑛||=+,(1.23) that is, lim𝑛𝑝𝑛=+.(1.24) A consequence of (1.24) is that there exists such natural 𝑛1 that if 𝑛>𝑛1 then 𝑝𝑛>𝑛0 and since (1.17) holds for every 𝑛>𝑛0, then (1.17) remains true for every 𝑝𝑛, where 𝑛>𝑛1.
So||𝜎𝑝𝑛||>||𝑠𝑝𝑛||(1𝛿)max1𝑘𝑝𝑛||𝑠𝑘||.(1.25) Since 1𝑝𝑛𝑛, therefore, max1𝑘𝑝𝑛||𝑠𝑘||max1𝑘𝑛||𝑠𝑘||.(1.26) Note that the last one and (1.25) imply ||𝜎𝑝𝑛||>||𝑠𝑝𝑛||(1𝛿)max1𝑘𝑛||𝑠𝑘||.(1.27) So according to (1.21) we have ||𝜎𝑝𝑛||>||𝑠𝑝𝑛||||𝑠(1𝛿)𝑝𝑛||,(1.28) that is, for every 𝑛>𝑛1 an inequality ||𝜎𝑝𝑛||||𝑠>𝛿𝑝𝑛||1holds,where0<𝛿2.(1.29) Also, (1.23) and (1.29) imply lim𝑛||𝜎𝑝𝑛||=+.(1.30) So we have finished the proof of Theorem 1.1.

Below we consider some consequences of Theorem 1.1.

Let Λ=𝜆𝑛(𝑘)be a triangular matrix, where the sequence {𝛼𝑛} is from [0,1] and for every 0𝑘𝑛 number 𝜆𝑛(𝑘) is defined by the formula: 𝜆𝑛𝐴(𝑘)=𝛼𝑛𝑛𝑘𝐴𝛼𝑛𝑛,where𝐴𝛼𝑛𝑛=𝛼𝑛𝛼+1𝑛𝛼+2𝑛+𝑛𝑛!.(1.31)

If 𝛼𝑛=𝛼, for every 𝑛0 and (1.31) holds true, then the Λ method is Cesaro (𝐶,𝛼) summability method, and if 𝛼𝑛0, then the Λ method coincides with convergence.

We introduce Cesaro summability method with variable orders, denoted by a symbol (𝐶,{𝛼𝑛}), which coincides with Λ summability method defined by (1.31). Means of this method for series (1.3) we denoted by 𝜎𝛼𝑛𝑛.

For (𝐶,{𝛼𝑛}) we have the following.

Theorem 1.3. Let a sequences {𝛼𝑛} be such that for some positive number 𝑚 we have 𝛼𝑛𝑐ln𝑛,where0𝑐<ln2𝑎𝑛𝑑𝑛>𝑚.(1.32) Then for any series (1.3) which satisfies the following condition: lim𝑛||𝑠𝑛||=+,(1.33) an equality lim𝑛||𝜎𝛼𝑛𝑛||=+(1.34) holds.

Proof of Theorem 1.3. Note that every 𝜆𝑛(𝑘) satisfies condition (1.1) and condition (1.3). Indeed, 𝜆𝑛(𝑘+1)𝜆𝑛=𝐴(𝑘)𝛼𝑛𝑛𝑘1𝐴𝛼𝑛𝑛𝑘=𝑛𝑘𝛼𝑛+𝑛𝑘1(1.35) and 𝜆𝑛(0)=1, when 𝑛0.
For every 𝑛1 we have𝜆𝑛1(𝑛)=𝐴𝛼𝑛𝑛,where𝐴𝛼𝑛𝑛=𝛼𝑛𝛼+1𝑛𝛼+2𝑛+𝑛,𝑛!(1.36) that is, 𝐴𝛼𝑛𝑛=𝛼1+𝑛1𝛼1+𝑛2𝛼1+𝑛𝑛.(1.37) Therefore, ln𝐴𝛼𝑛𝑛=𝑛𝑘=1𝛼ln1+𝑘𝑛<𝑛𝑘=1𝛼𝑛𝑘=𝛼𝑛𝑘=11𝑘<𝛼𝑛(1+ln𝑛).(1.38) Note that the last one and (1.32) imply that 2𝑐=ln1+𝛾,forsome0<𝛾1,(1.39) and if 𝑛>𝑚, we have 𝐴𝛼𝑛𝑛<𝑒𝛼𝑛(1+ln𝑛)=𝑒𝛼𝑛𝑒𝛼𝑛ln𝑛𝑒𝛼𝑛𝑒𝑐=𝑒𝛼𝑛𝑒ln(2/(1+𝛾))=𝑒𝛼𝑛2,1+𝛾(1.40) that is, 𝜆𝑛1(𝑛)=𝐴𝛼𝑛𝑛>1𝑒𝛼𝑛12+𝛾2,where𝛾>0.(1.41) Note that 𝛼𝑛0 implies the existence of such 𝛾1>0 and natural 𝑛2, that if 𝑛>𝑛2, then 1𝑒𝛼𝑛12+𝛾2>12+𝛾1,(1.42) that is, if 𝑛>𝑛2, then 𝜆𝑛1(𝑛)>2+𝛾1.(1.43) A consequence of (1.43) is that if (1.32) holds, then the Λ 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 {𝛼𝑛} be such sequence that 𝛼𝑛1=𝑜.ln𝑛(1.44) Then for every series (1.3) which satisfies lim𝑛||𝑠𝑛||=+,(1.45) we have lim𝑛||𝜎𝛼𝑛𝑛||=+.(1.46)

2. On Divergence of Fourier Series

It is well known the following.

Theorem A (Kolmogorov [1]). There exists such summable function 𝑓 that Fourier trigonometric series of 𝑓𝑎02+𝑘=1𝑎𝑘cos𝑘𝑥+𝑏𝑘sin𝑘𝑥(2.1) unboundedly diverges everywhere.

Let 𝑊={𝑤𝑛(𝑡)}𝑛=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 𝑔 that Fourier-Walsh series of 𝑔𝑛=1𝑎𝑛𝑤𝑛(t)(2.2) unboundedly diverges everywhere.

Let Φ={𝜑𝑛(𝑡)} be orthonormal functions system defined on [0,1], such that ||𝜑𝑛||[](𝑡)𝑀,𝑡0,1,𝑛=1,2,(2.3)

Then below-mentioned theorem holds.

Theorem C (Bŏ𝐜karev [4]). For every orthonormal system Φ which satisfies (2.3), there exists such summable function defined on [0,1] that its Fourier series constructed by Φ system 𝑛=1𝑎𝑛𝜑𝑛(𝑡)(2.4) unboundedly diverges in any point of some set 𝐸[0,1] with positive measure.

Denote by 𝜎𝛼𝑛𝑛(𝑥;𝑓), 𝜎𝛼𝑛𝑛(𝑡,𝑔,𝑊), and 𝜎𝛼𝑛𝑛(𝑡,,Φ) means of series (2.1), (2.2), and (2.4), respectively.

Theorem 1.3 implies that if {𝛼𝑛} satisfies (1.32), then Theorems A, B, and C hold for (𝐶,{𝛼𝑛}) summability method.

Namely, the following Theorems hold true.

Theorem 2.1. Let a sequence {𝛼𝑛} satisfies (1.32). Then there exists such summable function 𝑓, that sequence {𝜎𝛼𝑛𝑛(𝑥;𝑓)} unboundedly diverges everywhere.

Theorem 2.2. Let a sequence {𝛼𝑛} satisfies (1.32). Then there exists such summable function 𝑔 that sequence {𝜎𝛼𝑛𝑛(𝑡,𝑔,𝑊)} unboundedly diverges everywhere.

Theorem 2.3. If orthonormal system Φ satisfies (2.3) and a sequence {𝛼n} satisfies (1.32), then there exists such summable function , defined on [0,1], that sequence {𝜎𝛼𝑛𝑛(𝑡;;Φ)} unboundedly diverges at every point of some set 𝐸[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 if𝛼𝑛1=𝑜.ln𝑛(2.5)

Remark 2.4. If every number 𝜆𝑛(𝑘) will be replaced by (1𝑘/(𝑛+1))𝛼𝑛 in (1.31), then we get a summability method defined by Λ=𝜆𝑛(𝑘) matrix, which we call Riesz summability method with variable orders and denote it by symbol (𝑅,{𝛼𝑛}).

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 (𝑅,{𝛼𝑛}) method, where {𝛼𝑛} satisfies (1.32).


This paper is supported by the Grant GNSF/STO9_23_3–100.


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