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

Letβ€–β€–πœ†Ξ›=𝑛‖‖(π‘˜),𝑛=0,1,2,…,π‘˜=0,1,2,…,𝑛,(1.1) be such triangular matrix which satisfies the following conditions: (1)0β‰€πœ†π‘›(π‘˜+1)β‰€πœ†π‘›(π‘˜)≀1,0β‰€π‘˜β‰€π‘›;(2)πœ†π‘›(0)=1,πœ†π‘›(π‘˜)=0,π‘˜β‰₯𝑛+1.(1.2)

By 𝑠𝑛 we denote a partial sum of a series βˆžξ“π‘˜=0π‘’π‘˜,(1.3) and by πœŽπ‘› we denote a mean constructed by the Ξ› matrix, that is, 𝑠𝑛=π‘›ξ“π‘˜=0π‘’π‘˜,πœŽπ‘›=π‘›ξ“π‘˜=0πœ†π‘›(π‘˜)π‘’π‘˜.(1.4)

Theorem 1.1. Let matrix (1.1) satisfies an inequality limπ‘›β†’βˆžπœ†π‘›1(𝑛)>2.(1.5) Then for any series (1.3) which satisfies the following condition: limπ‘›β†’βˆž||𝑠𝑛||=+∞,(1.6) an equality limπ‘›β†’βˆž||πœŽπ‘›||=+∞(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 ||π‘ π‘›βˆ’πœŽπ‘›||≀21βˆ’πœ†π‘›ξ€Έ(𝑛)β‹…max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||(1.8) holds.

Proof of the Lemma. Using Abel transformation and πœ†π‘›(0)=1 we get π‘ π‘›βˆ’πœŽπ‘›=π‘›ξ“π‘˜=0π‘’π‘˜βˆ’π‘›ξ“π‘˜=0πœ†π‘›(π‘˜)π‘’π‘˜=π‘›ξ“π‘˜=1π‘’π‘˜βˆ’π‘›ξ“π‘˜=1πœ†π‘›(π‘˜)π‘’π‘˜=π‘›ξ“π‘˜=1ξ€·1βˆ’πœ†π‘›ξ€Έπ‘’(π‘˜)π‘˜=π‘›βˆ’1ξ“π‘˜=1ξ€·πœ†π‘›(π‘˜+1)βˆ’πœ†π‘›ξ€Έπ‘ (π‘˜)π‘˜+ξ€·1βˆ’πœ†π‘›ξ€Έπ‘ (𝑛)𝑛.(1.9) Therefore, ||π‘ π‘›βˆ’πœŽπ‘›||β‰€π‘›βˆ’1ξ“π‘˜=1||πœ†π‘›(π‘˜+1)βˆ’πœ†π‘›||β‹…||𝑠(π‘˜)π‘˜||+||1βˆ’πœ†π‘›||β‹…||𝑠(𝑛)𝑛||≀max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||β‹…ξƒ©π‘›βˆ’1ξ“π‘˜=1||πœ†π‘›(π‘˜+1)βˆ’πœ†π‘›||+||(π‘˜)1βˆ’πœ†π‘›||ξƒͺ.(𝑛)(1.10) Thus, taking into account (1.1) we immediately get ||π‘ π‘›βˆ’πœŽπ‘›||≀max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||β‹…ξƒ©π‘›βˆ’1ξ“π‘˜=1ξ€·πœ†π‘›(π‘˜)βˆ’πœ†π‘›ξ€Έ(π‘˜+1)+1βˆ’πœ†π‘›ξƒͺ(𝑛)=max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||β‹…ξ€·πœ†π‘›(1)βˆ’πœ†π‘›(𝑛)+1βˆ’πœ†π‘›ξ€Έ(𝑛)≀max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||β‹…ξ€·1βˆ’πœ†π‘›(𝑛)+1βˆ’πœ†π‘›ξ€Έξ€·(𝑛)=2β‹…1βˆ’πœ†π‘›ξ€Έ(𝑛)β‹…max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||.(1.11) So the Lemma is proved.

Proof of Theorem 1.1. According to the condition of Theorem 1.1 we have limπ‘›β†’βˆžπœ†π‘›1(𝑛)=2+𝛿(1.12) for some 𝛿>0. Note that inequalities 0β‰€πœ†π‘›(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 πœ†π‘›1(𝑛)>2+𝛿2.(1.13) So according to the Lemma, for every 𝑛>𝑛0 an inequality ||π‘ π‘›βˆ’πœŽπ‘›||ξ‚€ξ‚€1<2β‹…1βˆ’2+𝛿2⋅max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||(1.14) holds true; that is, if 𝑛>𝑛0, then ||π‘ π‘›βˆ’πœŽπ‘›||<(1βˆ’π›Ώ)β‹…max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||.(1.15) Thus for every 𝑛>𝑛0 an inequality ‖‖𝑠𝑛|βˆ’|πœŽπ‘›β€–β€–<(1βˆ’π›Ώ)β‹…max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||(1.16) holds.
So for every 𝑛>𝑛0 we have||πœŽπ‘›||>||𝑠𝑛||βˆ’(1βˆ’π›Ώ)β‹…max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||.(1.17)
Note that for every natural 𝑛 there exists at least one natural number 1β‰€π‘žβ‰€π‘›, such that the partial sums of the series (1.3) satisfy the following condition:||π‘ π‘ž||=max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||.(1.18) We define 𝑝𝑛 by a formula: 𝑝𝑛||𝑠=maxπ‘žβˆΆ1β‰€π‘žβ‰€π‘›&π‘ž||=max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||ξ‚Ό.(1.19) So 𝑝𝑛 is maximal number among the above-mentioned natural π‘ž numbers. Consequently, 1≀𝑝𝑛||𝑠≀𝑛,𝑝𝑛||=max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||,𝑝(1.20)𝑛≀𝑝𝑛+1,||𝑠𝑝𝑛||≀||𝑠𝑝𝑛+1||.(1.21) According to the condition of Theorem 1.1, limπ‘›β†’βˆž||𝑠𝑛||=+∞.(1.22) Therefore, limπ‘›β†’βˆž||𝑠𝑝𝑛||=+∞,(1.23) that is, limπ‘›β†’βˆžπ‘π‘›=+∞.(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||πœŽπ‘π‘›||>||𝑠𝑝𝑛||βˆ’(1βˆ’π›Ώ)β‹…max1β‰€π‘˜β‰€π‘π‘›||π‘ π‘˜||.(1.25) Since 1≀𝑝𝑛≀𝑛, therefore, max1β‰€π‘˜β‰€π‘π‘›||π‘ π‘˜||≀max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||.(1.26) Note that the last one and (1.25) imply ||πœŽπ‘π‘›||>||𝑠𝑝𝑛||βˆ’(1βˆ’π›Ώ)β‹…max1β‰€π‘˜β‰€π‘›||π‘ π‘˜||.(1.27) So according to (1.21) we have ||πœŽπ‘π‘›||>||𝑠𝑝𝑛||||π‘ βˆ’(1βˆ’π›Ώ)⋅𝑝𝑛||,(1.28) that is, for every 𝑛>𝑛1 an inequality ||πœŽπ‘π‘›||||𝑠>𝛿⋅𝑝𝑛||1holds,where0<𝛿≀2.(1.29) Also, (1.23) and (1.29) imply limπ‘›β†’βˆž||πœŽπ‘π‘›||=+∞.(1.30) So we have finished the proof of Theorem 1.1.

Below we consider some consequences of Theorem 1.1.

Let Ξ›=β€–πœ†π‘›(π‘˜)β€–be a triangular matrix, where the sequence {𝛼𝑛} is from [0,1] and for every 0β‰€π‘˜β‰€π‘› number πœ†π‘›(π‘˜) is defined by the formula: πœ†π‘›π΄(π‘˜)=π›Όπ‘›π‘›βˆ’π‘˜π΄π›Όπ‘›π‘›,where𝐴𝛼𝑛𝑛=𝛼𝑛𝛼+1𝑛⋯𝛼+2𝑛+𝑛𝑛!.(1.31)

If 𝛼𝑛=𝛼, for every 𝑛β‰₯0 and (1.31) holds true, then the Ξ› method is Cesaro (𝐢,𝛼) summability method, and if 𝛼𝑛≑0, then the Ξ› method coincides with convergence.

We introduce Cesaro summability method with variable orders, denoted by a symbol (𝐢,{𝛼𝑛}), which coincides with Ξ› summability method defined by (1.31). Means of this method for series (1.3) we denoted by πœŽπ›Όπ‘›π‘›.

For (𝐢,{𝛼𝑛}) we have the following.

Theorem 1.3. Let a sequences {𝛼𝑛} be such that for some positive number π‘š we have 𝛼𝑛≀𝑐ln𝑛,where0≀𝑐<ln2π‘Žπ‘›π‘‘π‘›>π‘š.(1.32) Then for any series (1.3) which satisfies the following condition: limπ‘›β†’βˆž||𝑠𝑛||=+∞,(1.33) an equality limπ‘›β†’βˆž||πœŽπ›Όπ‘›π‘›||=+∞(1.34) holds.

Proof of Theorem 1.3. Note that every πœ†π‘›(π‘˜) satisfies condition (1.1) and condition (1.3). Indeed, πœ†π‘›(π‘˜+1)πœ†π‘›=𝐴(π‘˜)π›Όπ‘›π‘›βˆ’π‘˜βˆ’1π΄π›Όπ‘›π‘›βˆ’π‘˜=π‘›βˆ’π‘˜π›Όπ‘›+π‘›βˆ’π‘˜β‰€1(1.35) and πœ†π‘›(0)=1, when 𝑛β‰₯0.
For every 𝑛β‰₯1 we haveπœ†π‘›1(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=π‘›ξ“π‘˜=1𝛼ln1+π‘˜π‘›ξ‚<π‘›ξ“π‘˜=1π›Όπ‘›π‘˜=π›Όπ‘›β‹…ξ“π‘˜=11π‘˜<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>π‘š, we have 𝐴𝛼𝑛𝑛<𝑒𝛼𝑛(1+ln𝑛)=𝑒𝛼𝑛⋅𝑒𝛼𝑛ln𝑛≀𝑒𝛼𝑛⋅𝑒𝑐=𝑒𝛼𝑛⋅𝑒ln(2/(1+𝛾))=𝑒𝛼𝑛⋅2,1+𝛾(1.40) that is, πœ†π‘›1(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 πœ†π‘›1(𝑛)>2+𝛾1.(1.43) A consequence of (1.43) is that if (1.32) holds, then the Ξ› 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=π‘œξ‚.ln𝑛(1.44) Then for every series (1.3) which satisfies limπ‘›β†’βˆž||𝑠𝑛||=+∞,(1.45) we have limπ‘›β†’βˆž||πœŽπ›Όπ‘›π‘›||=+∞.(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 π‘“π‘Ž02+βˆžξ“π‘˜=1π‘Žπ‘˜cosπ‘˜π‘₯+π‘π‘˜sinπ‘˜π‘₯(2.1) unboundedly diverges everywhere.

Let π‘Š={𝑀𝑛(𝑑)}βˆžπ‘›=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 π‘”βˆžξ“π‘›=1π‘Žπ‘›π‘€π‘›(t)(2.2) unboundedly diverges everywhere.

Let Ξ¦={πœ‘π‘›(𝑑)} be orthonormal functions system defined on [0,1], such that ||πœ‘π‘›||[](𝑑)≀𝑀,π‘‘βˆˆ0,1,𝑛=1,2,…(2.3)

Then below-mentioned theorem holds.

Theorem C (BoΜ†πœkarev [4]). For every orthonormal system Ξ¦ which satisfies (2.3), there exists such summable function β„Ž defined on [0,1] that its Fourier series constructed by Ξ¦ system βˆžξ“π‘›=1π‘Žπ‘›πœ‘π‘›(𝑑)(2.4) unboundedly diverges in any point of some set πΈβŠ‚[0,1] with positive measure.

Denote by πœŽπ›Όπ‘›π‘›(π‘₯;𝑓), πœŽπ›Όπ‘›π‘›(𝑑,𝑔,π‘Š), and πœŽπ›Όπ‘›π‘›(𝑑,β„Ž,Ξ¦) 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 (𝐢,{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 {πœŽπ›Όπ‘›π‘›(π‘₯;𝑓)} unboundedly diverges everywhere.

Theorem 2.2. Let a sequence {𝛼𝑛} satisfies (1.32). Then there exists such summable function 𝑔 that sequence {πœŽπ›Όπ‘›π‘›(𝑑,𝑔,π‘Š)} unboundedly diverges everywhere.

Theorem 2.3. If orthonormal system Ξ¦ satisfies (2.3) and a sequence {𝛼n} satisfies (1.32), then there exists such summable function β„Ž, defined on [0,1], that sequence {πœŽπ›Όπ‘›π‘›(𝑑;β„Ž;Ξ¦)} unboundedly diverges at every point of some set πΈβŠ‚[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=π‘œξ‚.ln𝑛(2.5)

Remark 2.4. If every number πœ†π‘›(π‘˜) will be replaced by (1βˆ’π‘˜/(𝑛+1))𝛼𝑛 in (1.31), then we get a summability method defined by Ξ›=β€–πœ†π‘›(π‘˜)β€– 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.