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

1. A New Summability Method of Series

Let‖‖𝜆Λ=𝑛‖‖(𝑘),𝑛=0,1,2,…,𝑘=0,1,2,…,𝑛,(1.1) be such triangular matrix which satisfies the following conditions: (1)0≤𝜆𝑛(𝑘+1)≤𝜆𝑛(𝑘)≤1,0≤𝑘≤𝑛;(2)𝜆𝑛(0)=1,𝜆𝑛(𝑘)=0,𝑘≥𝑛+1.(1.2)

By 𝑠𝑛 we denote a partial sum of a series âˆžî“ğ‘˜=0𝑢𝑘,(1.3) and by ğœŽğ‘› we denote a mean constructed by the Λ matrix, that is, 𝑠𝑛=𝑛𝑘=0𝑢𝑘,ğœŽğ‘›=𝑛𝑘=0𝜆𝑛(𝑘)𝑢𝑘.(1.4)

Theorem 1.1. Let matrix (1.1) satisfies an inequality limğ‘›â†’âˆžğœ†ğ‘›1(𝑛)>2.(1.5) Then for any series (1.3) which satisfies the following condition: limğ‘›â†’âˆž||𝑠𝑛||=+∞,(1.6) an equality limğ‘›â†’âˆž||ğœŽğ‘›||=+∞(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 ||ğ‘ ğ‘›âˆ’ğœŽğ‘›||≤21−𝜆𝑛(𝑛)⋅max1≤𝑘≤𝑛||𝑠𝑘||(1.8) holds.

Proof of the Lemma. Using Abel transformation and 𝜆𝑛(0)=1 we get ğ‘ ğ‘›âˆ’ğœŽğ‘›=𝑛𝑘=0𝑢𝑘−𝑛𝑘=0𝜆𝑛(𝑘)𝑢𝑘=𝑛𝑘=1𝑢𝑘−𝑛𝑘=1𝜆𝑛(𝑘)𝑢𝑘=𝑛𝑘=11−𝜆𝑛𝑢(𝑘)𝑘=𝑛−1𝑘=1𝜆𝑛(𝑘+1)−𝜆𝑛𝑠(𝑘)𝑘+1−𝜆𝑛𝑠(𝑛)𝑛.(1.9) Therefore, ||ğ‘ ğ‘›âˆ’ğœŽğ‘›||≤𝑛−1𝑘=1||𝜆𝑛(𝑘+1)−𝜆𝑛||⋅||𝑠(𝑘)𝑘||+||1−𝜆𝑛||⋅||𝑠(𝑛)𝑛||≤max1≤𝑘≤𝑛||𝑠𝑘||⋅𝑛−1𝑘=1||𝜆𝑛(𝑘+1)−𝜆𝑛||+||(𝑘)1−𝜆𝑛||.(𝑛)(1.10) Thus, taking into account (1.1) we immediately get ||ğ‘ ğ‘›âˆ’ğœŽğ‘›||≤max1≤𝑘≤𝑛||𝑠𝑘||⋅𝑛−1𝑘=1𝜆𝑛(𝑘)−𝜆𝑛(𝑘+1)+1−𝜆𝑛(𝑛)=max1≤𝑘≤𝑛||𝑠𝑘||⋅𝜆𝑛(1)−𝜆𝑛(𝑛)+1−𝜆𝑛(𝑛)≤max1≤𝑘≤𝑛||𝑠𝑘||⋅1−𝜆𝑛(𝑛)+1−𝜆𝑛(𝑛)=2⋅1−𝜆𝑛(𝑛)⋅max1≤𝑘≤𝑛||𝑠𝑘||.(1.11) So the Lemma is proved.

Proof of Theorem 1.1. According to the condition of Theorem 1.1 we have limğ‘›â†’âˆžğœ†ğ‘›1(𝑛)=2+𝛿(1.12) for some 𝛿>0. Note that inequalities 0≤𝜆𝑛(𝑛)≤1 which hold for every natural 𝑛 imply 1/2+𝛿≤1, that is, 𝛿≤1/2.
So, 0<𝛿≤1/2 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 ||ğ‘ ğ‘›âˆ’ğœŽğ‘›||1<2⋅1−2+𝛿2⋅max1≤𝑘≤𝑛||𝑠𝑘||(1.14) holds true; that is, if 𝑛>𝑛0, then ||ğ‘ ğ‘›âˆ’ğœŽğ‘›||<(1−𝛿)⋅max1≤𝑘≤𝑛||𝑠𝑘||.(1.15) Thus for every 𝑛>𝑛0 an inequality ‖‖𝑠𝑛|−|ğœŽğ‘›â€–â€–<(1−𝛿)⋅max1≤𝑘≤𝑛||𝑠𝑘||(1.16) holds.
So for every 𝑛>𝑛0 we have||ğœŽğ‘›||>||𝑠𝑛||−(1−𝛿)⋅max1≤𝑘≤𝑛||𝑠𝑘||.(1.17)
Note that for every natural 𝑛 there exists at least one natural number 1â‰¤ğ‘žâ‰¤ğ‘›, such that the partial sums of the series (1.3) satisfy the following condition:||ğ‘ ğ‘ž||=max1≤𝑘≤𝑛||𝑠𝑘||.(1.18) We define 𝑝𝑛 by a formula: 𝑝𝑛||𝑠=maxğ‘žâˆ¶1â‰¤ğ‘žâ‰¤ğ‘›&ğ‘ž||=max1≤𝑘≤𝑛||𝑠𝑘||.(1.19) So 𝑝𝑛 is maximal number among the above-mentioned natural ğ‘ž numbers. Consequently, 1≤𝑝𝑛||𝑠≤𝑛,𝑝𝑛||=max1≤𝑘≤𝑛||𝑠𝑘||,𝑝(1.20)𝑛≤𝑝𝑛+1,||𝑠𝑝𝑛||≤||𝑠𝑝𝑛+1||.(1.21) According to the condition of Theorem 1.1, limğ‘›â†’âˆž||𝑠𝑛||=+∞.(1.22) Therefore, limğ‘›â†’âˆž||𝑠𝑝𝑛||=+∞,(1.23) that is, limğ‘›â†’âˆžğ‘ğ‘›=+∞.(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||ğœŽğ‘ğ‘›||>||𝑠𝑝𝑛||−(1−𝛿)⋅max1≤𝑘≤𝑝𝑛||𝑠𝑘||.(1.25) Since 1≤𝑝𝑛≤𝑛, therefore, max1≤𝑘≤𝑝𝑛||𝑠𝑘||≤max1≤𝑘≤𝑛||𝑠𝑘||.(1.26) Note that the last one and (1.25) imply ||ğœŽğ‘ğ‘›||>||𝑠𝑝𝑛||−(1−𝛿)⋅max1≤𝑘≤𝑛||𝑠𝑘||.(1.27) So according to (1.21) we have ||ğœŽğ‘ğ‘›||>||𝑠𝑝𝑛||||𝑠−(1−𝛿)⋅𝑝𝑛||,(1.28) that is, for every 𝑛>𝑛1 an inequality ||ğœŽğ‘ğ‘›||||𝑠>𝛿⋅𝑝𝑛||1holds,where0<𝛿≤2.(1.29) Also, (1.23) and (1.29) imply limğ‘›â†’âˆž||ğœŽğ‘ğ‘›||=+∞.(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 0≤𝑘≤𝑛 number 𝜆𝑛(𝑘) is defined by the formula: 𝜆𝑛𝐴(𝑘)=𝛼𝑛𝑛−𝑘𝐴𝛼𝑛𝑛,where𝐴𝛼𝑛𝑛=𝛼𝑛𝛼+1𝑛⋯𝛼+2𝑛+𝑛𝑛!.(1.31)

If 𝛼𝑛=𝛼, for every 𝑛≥0 and (1.31) holds true, then the Λ method is Cesaro (𝐶,𝛼) summability method, and if 𝛼𝑛≡0, 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 ğœŽğ›¼ğ‘›ğ‘›.

For (𝐶,{𝛼𝑛}) we have the following.

Theorem 1.3. Let a sequences {𝛼𝑛} be such that for some positive number 𝑚 we have 𝛼𝑛≤𝑐ln𝑛,where0≤𝑐<ln2ğ‘Žğ‘›ğ‘‘ğ‘›>𝑚.(1.32) Then for any series (1.3) which satisfies the following condition: limğ‘›â†’âˆž||𝑠𝑛||=+∞,(1.33) an equality limğ‘›â†’âˆž||ğœŽğ›¼ğ‘›ğ‘›||=+∞(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)𝛼𝑛𝑛−𝑘−1𝐴𝛼𝑛𝑛−𝑘=𝑛−𝑘𝛼𝑛+𝑛−𝑘≤1(1.35) and 𝜆𝑛(0)=1, when 𝑛≥0.
For every 𝑛≥1 we have𝜆𝑛1(𝑛)=𝐴𝛼𝑛𝑛,where𝐴𝛼𝑛𝑛=𝛼𝑛𝛼+1𝑛⋯𝛼+2𝑛+𝑛,𝑛!(1.36) that is, 𝐴𝛼𝑛𝑛=𝛼1+𝑛1𝛼1+𝑛2⋯𝛼1+𝑛𝑛.(1.37) Therefore, ln𝐴𝛼𝑛𝑛=𝑛𝑘=1𝛼ln1+𝑘𝑛<𝑛𝑘=1𝛼𝑛𝑘=𝛼𝑛⋅𝑘=11𝑘<𝛼𝑛(1+ln𝑛).(1.38) Note that the last one and (1.32) imply that 2𝑐=ln1+𝛾,forsome0<𝛾≤1,(1.39) and if 𝑛>𝑚, we have 𝐴𝛼𝑛𝑛<𝑒𝛼𝑛(1+ln𝑛)=𝑒𝛼𝑛⋅𝑒𝛼𝑛ln𝑛≤𝑒𝛼𝑛⋅𝑒𝑐=𝑒𝛼𝑛⋅𝑒ln(2/(1+𝛾))=𝑒𝛼𝑛⋅2,1+𝛾(1.40) that is, 𝜆𝑛1(𝑛)=𝐴𝛼𝑛𝑛>1𝑒𝛼𝑛⋅12+𝛾2,where𝛾>0.(1.41) Note that 𝛼𝑛→0 implies the existence of such 𝛾1>0 and natural 𝑛2, that if 𝑛>𝑛2, then 1𝑒𝛼𝑛⋅12+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 𝛼𝑛1=𝑜.ln𝑛(1.44) Then for every series (1.3) which satisfies limğ‘›â†’âˆž||𝑠𝑛||=+∞,(1.45) we have limğ‘›â†’âˆž||ğœŽğ›¼ğ‘›ğ‘›||=+∞.(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 ğ‘“ğ‘Ž02+âˆžî“ğ‘˜=1ğ‘Žğ‘˜cos𝑘𝑥+𝑏𝑘sin𝑘𝑥(2.1) unboundedly diverges everywhere.

Let 𝑊={𝑤𝑛(𝑡)}âˆžğ‘›=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 ğ‘”âˆžî“ğ‘›=1ğ‘Žğ‘›ğ‘¤ğ‘›(t)(2.2) unboundedly diverges everywhere.

Let Φ={𝜑𝑛(𝑡)} be orthonormal functions system defined on [0,1], such that ||𝜑𝑛||[](𝑡)≤𝑀,𝑡∈0,1,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 ℎ defined on [0,1] that its Fourier series constructed by Φ system âˆžî“ğ‘›=1ğ‘Žğ‘›ğœ‘ğ‘›(𝑡)(2.4) unboundedly diverges in any point of some set 𝐸⊂[0,1] with positive measure.

Denote by ğœŽğ›¼ğ‘›ğ‘›(𝑥;𝑓), ğœŽğ›¼ğ‘›ğ‘›(𝑡,𝑔,𝑊), and ğœŽğ›¼ğ‘›ğ‘›(𝑡,ℎ,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 {ğœŽğ›¼ğ‘›ğ‘›(𝑥;𝑓)} unboundedly diverges everywhere.

Theorem 2.2. Let a sequence {𝛼𝑛} satisfies (1.32). Then there exists such summable function 𝑔 that sequence {ğœŽğ›¼ğ‘›ğ‘›(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 ℎ, defined on [0,1], that sequence {ğœŽğ›¼ğ‘›ğ‘›(𝑡;ℎ;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 if𝛼𝑛1=𝑜.ln𝑛(2.5)

Remark 2.4. If every number 𝜆𝑛(𝑘) will be replaced by (1−𝑘/(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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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.

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