On Divergence of Fourier Series by Some Methods of Summability

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

1. A New Summability Method of Series

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

By 𝑠_𝑛 we denote a partial sum of a series ^∞_𝑘=0𝑢_𝑘,(1.3) and by 𝜎_𝑛 we denote a mean constructed by the Λ matrix, that is, 𝑠_𝑛=^𝑛_𝑘=0𝑢_𝑘,𝜎_𝑛=^𝑛_𝑘=0𝜆_𝑛(𝑘)𝑢_𝑘.(1.4)

Theorem 1.1. Let matrix (1.1) satisfies an inequality lim_𝑛→∞𝜆_𝑛(𝑛)>12.(1.5) Then for any series (1.3) which satisfies the following condition: lim_𝑛→∞||𝑠_𝑛||=+∞,(1.6) an equality lim_𝑛→∞||𝜎_𝑛||=+∞(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 ||𝑠_𝑛−𝜎_𝑛||≤21−𝜆_𝑛(𝑛)⋅max_{1≤𝑘≤𝑛}||𝑠_𝑘||(1.8) holds.

Proof of the Lemma. Using Abel transformation and 𝜆_𝑛(0)=1 we get 𝑠_𝑛−𝜎_𝑛=^𝑛_𝑘=0𝑢_𝑘−^𝑛_𝑘=0𝜆_𝑛(𝑘)𝑢_𝑘=^𝑛_𝑘=1𝑢_𝑘−^𝑛_𝑘=1𝜆_𝑛(𝑘)𝑢_𝑘=^𝑛_𝑘=11−𝜆_𝑛(𝑘)𝑢_𝑘=^𝑛−1_𝑘=1𝜆_𝑛(𝑘+1)−𝜆_𝑛(𝑘)𝑠_𝑘+1−𝜆_𝑛(𝑛)𝑠_𝑛.(1.9) Therefore, ||𝑠_𝑛−𝜎_𝑛||≤^𝑛−1_𝑘=1≤max||𝜆_{𝑛1≤𝑘≤𝑛}(𝑘+1)−𝜆_𝑛(𝑘)||⋅||𝑠_𝑘||+||1−𝜆_𝑛(𝑛)||⋅||𝑠_𝑛||||𝑠_𝑘||⋅^𝑛−1_𝑘=1||𝜆_𝑛(𝑘+1)−𝜆_𝑛(𝑘)||+||1−𝜆_𝑛(𝑛)||.(1.10) Thus, taking into account (1.1) we immediately get ||𝑠_𝑛−𝜎_𝑛||≤max_{1≤𝑘≤𝑛}||𝑠_𝑘||⋅^𝑛−1_𝑘=1𝜆_𝑛=max_{1≤𝑘≤𝑛}(𝑘)−𝜆_𝑛(𝑘+1)+1−𝜆_𝑛(𝑛)≤max_{1≤𝑘≤𝑛}||𝑠_𝑘||⋅𝜆_𝑛(1)−𝜆_𝑛(𝑛)+1−𝜆_𝑛(𝑛)=2⋅1||𝑠−_𝑘||𝜆⋅_𝑛1−𝜆_𝑛(𝑛)+1−𝜆_𝑛(𝑛)(𝑛)⋅max_{1≤𝑘≤𝑛}||𝑠_𝑘||.(1.11) So the Lemma is proved.

Proof of Theorem 1.1. According to the condition of Theorem 1.1 we have lim_𝑛→∞𝜆_𝑛(𝑛)=12+𝛿(1.12) for some 𝛿>0. Note that inequalities 0≤𝜆_𝑛(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 𝜆_𝑛(𝑛)>12+𝛿2.(1.13) So according to the Lemma, for every 𝑛>𝑛₀ an inequality ||𝑠_𝑛−𝜎_𝑛||<2⋅1−12+𝛿2⋅max^{1≤𝑘≤𝑛}||𝑠_𝑘||(1.14) holds true; that is, if 𝑛>𝑛₀, then ||𝑠_𝑛−𝜎_𝑛||<(1−𝛿)⋅max_{1≤𝑘≤𝑛}||𝑠_𝑘||.(1.15) Thus for every 𝑛>𝑛₀ an inequality ‖‖𝑠_𝑛|−|𝜎_𝑛‖‖<(1−𝛿)⋅max_{1≤𝑘≤𝑛}||𝑠_𝑘||(1.16) holds.
So for every 𝑛>𝑛₀ we have||𝜎_𝑛||>||𝑠_𝑛||−(1−𝛿)⋅max_{1≤𝑘≤𝑛}||𝑠_𝑘||.(1.17)
Note that for every natural 𝑛 there exists at least one natural number 1≤𝑞≤𝑛, such that the partial sums of the series (1.3) satisfy the following condition:||𝑠_𝑞||=max_{1≤𝑘≤𝑛}||𝑠_𝑘||.(1.18) We define 𝑝_𝑛 by a formula: 𝑝_𝑛=max𝑞∶1≤𝑞≤𝑛&||𝑠_𝑞||=max_{1≤𝑘≤𝑛}||𝑠_𝑘||.(1.19) So 𝑝_𝑛 is maximal number among the above-mentioned natural 𝑞 numbers. Consequently, 1≤𝑝𝑝_𝑛≤𝑛,||𝑠_𝑝𝑛||=max_{1≤𝑘≤𝑛}||𝑠_𝑘||,(1.20)_𝑛≤𝑝_𝑛+1,||𝑠_𝑝𝑛||≤||𝑠_𝑝𝑛+1||.(1.21) According to the condition of Theorem 1.1, lim_𝑛→∞||𝑠_𝑛||=+∞.(1.22) Therefore, lim_𝑛→∞||𝑠_𝑝𝑛||=+∞,(1.23) that is, lim_𝑛→∞𝑝_𝑛=+∞.(1.24) A consequence of (1.24) is that there exists such natural 𝑛₁ that if 𝑛>𝑛₁ then 𝑝_𝑛>𝑛₀ and since (1.17) holds for every 𝑛>𝑛₀, then (1.17) remains true for every 𝑝_𝑛, where 𝑛>𝑛₁.
So||𝜎_𝑝𝑛||>||𝑠_𝑝𝑛||−(1−𝛿)⋅max_{1≤𝑘≤𝑝𝑛}||𝑠_𝑘||.(1.25) Since 1≤𝑝_𝑛≤𝑛, therefore, max_{1≤𝑘≤𝑝𝑛}||𝑠_𝑘||≤max_{1≤𝑘≤𝑛}||𝑠_𝑘||.(1.26) Note that the last one and (1.25) imply ||𝜎_𝑝𝑛||>||𝑠_𝑝𝑛||−(1−𝛿)⋅max_{1≤𝑘≤𝑛}||𝑠_𝑘||.(1.27) So according to (1.21) we have ||𝜎_𝑝𝑛||>||𝑠_𝑝𝑛||−(1−𝛿)⋅||𝑠_𝑝𝑛||,(1.28) that is, for every 𝑛>𝑛₁ an inequality ||𝜎_𝑝𝑛||>𝛿⋅||𝑠_𝑝𝑛||holds,where0<𝛿≤12.(1.29) Also, (1.23) and (1.29) imply lim_𝑛→∞||𝜎_𝑝𝑛||=+∞.(1.30) So we have finished the proof of Theorem 1.1.

Below we consider some consequences of Theorem 1.1.

Let Λ=‖𝜆_𝑛(𝑘)‖be a triangular matrix, where the sequence {𝛼_𝑛} is from [0,1] and for every 0≤𝑘≤𝑛 number 𝜆_𝑛(𝑘) is defined by the formula: 𝜆_𝑛(𝑘)=𝐴^{𝛼𝑛𝑛−𝑘}𝐴^𝛼𝑛_𝑛,where𝐴^𝛼𝑛_𝑛=𝛼^𝑛+1𝛼_𝑛+2⋯𝛼_𝑛+𝑛𝑛!.(1.31)

If 𝛼_𝑛=𝛼, for every 𝑛≥0 and (1.31) holds true, then the Λ method is Cesaro (𝐶,𝛼) summability method, and if 𝛼_𝑛≡0, then the Λ method coincides with convergence.

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

For (𝐶,{𝛼_𝑛}) we have the following.

Theorem 1.3. Let a sequences {𝛼_𝑛} be such that for some positive number 𝑚 we have 𝛼_𝑛≤𝑐ln𝑛,where0≤𝑐<ln2𝑎𝑛𝑑𝑛>𝑚.(1.32) Then for any series (1.3) which satisfies the following condition: lim_𝑛→∞||𝑠_𝑛||=+∞,(1.33) an equality lim_𝑛→∞||𝜎^𝛼𝑛_𝑛||=+∞(1.34) holds.

Proof of Theorem 1.3. Note that every 𝜆_𝑛(𝑘) satisfies condition (1.1) and condition (1.3). Indeed, 𝜆_𝑛(𝑘+1)𝜆_𝑛(𝑘)=𝐴𝐴^{𝛼𝑛𝑛−𝑘−1𝛼𝑛}_𝑛−𝑘=𝑛−𝑘𝛼_𝑛+𝑛−𝑘≤1(1.35) and 𝜆_𝑛(0)=1, when 𝑛≥0.
For every 𝑛≥1 we have𝜆_𝑛(𝑛)=1𝐴^𝛼𝑛_𝑛,where𝐴^𝛼𝑛_𝑛=𝛼^𝑛+1𝛼_𝑛+2⋯𝛼_𝑛+𝑛𝑛!,(1.36) that is, 𝐴^𝛼𝑛_𝑛=1+𝛼^𝑛11+𝛼2^𝑛⋯1+𝛼𝑛^𝑛.(1.37) Therefore, ln𝐴^𝛼𝑛_𝑛=^𝑛_𝑘=1ln1+𝛼^𝑘𝑛<^𝑛_𝑘=1𝛼_𝑛𝑘=𝛼^𝑛⋅_𝑘=11𝑘<𝛼^𝑛(1+ln𝑛).(1.38) Note that the last one and (1.32) imply that 𝑐=ln21+𝛾,forsome0<𝛾≤1,(1.39) and if 𝑛>𝑚, we have 𝐴^𝛼𝑛_𝑛<𝑒^{𝛼𝑛(1+ln𝑛)}=𝑒^𝛼𝑛⋅𝑒^{𝛼𝑛ln𝑛}≤𝑒^𝛼𝑛⋅𝑒^𝑐=𝑒^𝛼𝑛⋅𝑒^{ln(2/(1+𝛾))}=𝑒^𝛼𝑛⋅21+𝛾,(1.40) that is, 𝜆_𝑛(𝑛)=1𝐴^𝛼𝑛_𝑛>1𝑒^𝛼𝑛⋅12+𝛾2,where𝛾>0.(1.41) Note that 𝛼_𝑛→0 implies the existence of such 𝛾₁>0 and natural 𝑛₂, that if 𝑛>𝑛₂, then 1𝑒^𝛼𝑛⋅12+𝛾2>12+𝛾¹,(1.42) that is, if 𝑛>𝑛₂, then 𝜆_𝑛(𝑛)>12+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 {𝛼_𝑛} be such sequence that 𝛼_𝑛=𝑜ln1𝑛.(1.44) Then for every series (1.3) which satisfies lim_𝑛→∞||𝑠_𝑛||=+∞,(1.45) we have lim_𝑛→∞||𝜎^𝛼𝑛_𝑛||=+∞.(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=1𝑎_𝑘cos𝑘𝑥+𝑏_𝑘sin𝑘𝑥(2.1) unboundedly diverges everywhere.

Let 𝑊={𝑤_𝑛(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 𝑔^∞_𝑛=1𝑎_𝑛𝑤_𝑛(t)(2.2) unboundedly diverges everywhere.

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

Then below-mentioned theorem holds.

Theorem C (Bo𝐜̆karev [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 ^∞_𝑛=1𝑎_𝑛𝜑_𝑛(𝑡)(2.4) unboundedly diverges in any point of some set 𝐸⊂[0,1] with positive measure.

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

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

Namely, the following Theorems hold true.

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

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

Theorem 2.3. If orthonormal system Φ satisfies (2.3) and a sequence {𝛼_n} satisfies (1.32), then there exists such summable function ℎ, defined on [0,1], that sequence {𝜎^𝛼𝑛_𝑛(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 if𝛼_𝑛=𝑜1ln𝑛.(2.5)

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

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,{𝛼_𝑛}) method, where {𝛼_𝑛} satisfies (1.32).

Acknowledgment

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