# On Divergence of Fourier Series by Some Methods of Summability

Authors: Shakro Tetunashvili
Journal: Journal of Function Spaces and Applications
 (2012)

---

## 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.


---

## Body

## 1. A New Summability Method of Series

Let(1.1)Λ=‖λn(k)‖,n=0,1,2,…,k=0,1,2,…,n,
be such triangular matrix which satisfies the following conditions:
(1.2)(1)0≤λn(k+1)≤λn(k)≤1,0≤k≤n;(2)λn(0)=1,λn(k)=0,k≥n+1.Bysn we denote a partial sum of a series
(1.3)∑k=0∞uk,
and by σn we denote a mean constructed by the Λ matrix, that is,
(1.4)sn=∑k=0nuk,σn=∑k=0nλn(k)uk.Theorem 1.1.
Let matrix (1.1) satisfies an inequality
(1.5)lim⁡̲n→∞λn(n)>12.
Then for any series (1.3) which satisfies the following condition:
(1.6)lim⁡¯n→∞|sn|=+∞,
an equality
(1.7)lim⁡¯n→∞|σn|=+∞
holds.Below we prove a Lemma which is used to prove Theorem1.1.Lemma 1.2.
For every natural numbern an inequality
(1.8)|sn-σn|≤2(1-λn(n))⋅max⁡1≤k≤n|sk|
holds.Proof of the Lemma.
Using Abel transformation andλn(0)=1 we get
(1.9)sn-σn=∑k=0nuk-∑k=0nλn(k)uk=∑k=1nuk-∑k=1nλn(k)uk=∑k=1n(1-λn(k))uk=∑k=1n-1(λn(k+1)-λn(k))sk+(1-λn(n))sn.
Therefore,
(1.10)|sn-σn|≤∑k=1n-1|λn(k+1)-λn(k)|⋅|sk|+|1-λn(n)|⋅|sn|≤max⁡1≤k≤n|sk|⋅(∑k=1n-1|λn(k+1)-λn(k)|+|1-λn(n)|).
Thus, taking into account (1.1) we immediately get
(1.11)|sn-σn|≤max⁡1≤k≤n|sk|⋅(∑k=1n-1(λn(k)-λn(k+1))+1-λn(n))=max⁡1≤k≤n|sk|⋅(λn(1)-λn(n)+1-λn(n))≤max⁡1≤k≤n|sk|⋅(1-λn(n)+1-λn(n))=2⋅(1-λn(n))⋅max⁡1≤k≤n|sk|.
So the Lemma is proved.Proof of Theorem1.1.
According to the condition of Theorem1.1 we have
(1.12)lim⁡̲n→∞λn(n)=12+δ
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
(1.13)λn(n)>12+δ2.
So according to the Lemma, for every n>n0 an inequality
(1.14)|sn-σn|<2⋅(1-(12+δ2))⋅max⁡1≤k≤n|sk|
holds true; that is, if n>n0, then
(1.15)|sn-σn|<(1-δ)⋅max⁡1≤k≤n|sk|.
Thus for every n>n0 an inequality
(1.16)‖sn|-|σn‖<(1-δ)⋅max⁡1≤k≤n|sk|
holds.
So for everyn>n0 we have(1.17)|σn|>|sn|-(1-δ)⋅max⁡1≤k≤n|sk|.
Note that for every naturaln there exists at least one natural number 1≤q≤n, such that the partial sums of the series (1.3) satisfy the following condition:(1.18)|sq|=max⁡1≤k≤n|sk|.
We define pn by a formula:
(1.19)pn=max⁡{q:1≤q≤n&|sq|=max⁡1≤k≤n|sk|}.
So pn is maximal number among the above-mentioned natural q numbers. Consequently,
(1.20)1≤pn≤n,|spn|=max⁡1≤k≤n|sk|,(1.21)pn≤pn+1,|spn|≤|spn+1|.
According to the condition of Theorem 1.1,
(1.22)lim⁡¯n→∞|sn|=+∞.
Therefore,
(1.23)lim⁡n→∞|spn|=+∞,
that is,
(1.24)lim⁡n→∞pn=+∞.
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(1.25)|σpn|>|spn|-(1-δ)⋅max⁡1≤k≤pn|sk|.
Since 1≤pn≤n, therefore,
(1.26)max⁡1≤k≤pn|sk|≤max⁡1≤k≤n|sk|.
Note that the last one and (1.25) imply
(1.27)|σpn|>|spn|-(1-δ)⋅max⁡1≤k≤n|sk|.
So according to (1.21) we have
(1.28)|σpn|>|spn|-(1-δ)⋅|spn|,
that is, for every n>n1 an inequality
(1.29)|σpn|>δ⋅|spn|holds,where0<δ≤12.
Also, (1.23) and (1.29) imply
(1.30)lim⁡n→∞|σpn|=+∞.
So we have finished the proof of Theorem 1.1.Below we consider some consequences of Theorem1.1.LetΛ=∥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:
(1.31)λn(k)=An-kαnAnαn,whereAnαn=(αn+1)(αn+2)⋯(αn+n)n!.Ifαn=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 σnαn.For(C,{αn}) we have the following.Theorem 1.3.
Let a sequences{αn} be such that for some positive number m we have
(1.32)αn≤cln⁡n,where0≤c<ln⁡2andn>m.
Then for any series (1.3) which satisfies the following condition:
(1.33)lim⁡¯n→∞|sn|=+∞,
an equality
(1.34)lim⁡¯n→∞|σnαn|=+∞
holds.Proof of Theorem1.3.
Note that everyλn(k) satisfies condition (1.1) and condition (1.3). Indeed,
(1.35)λn(k+1)λn(k)=An-k-1αnAn-kαn=n-kαn+n-k≤1
and λn(0)=1, when n≥0.
For everyn≥1 we have(1.36)λn(n)=1Anαn,whereAnαn=(αn+1)(αn+2)⋯(αn+n)n!,
that is,
(1.37)Anαn=(1+αn1)(1+αn2)⋯(1+αnn).
Therefore,
(1.38)ln⁡Anαn=∑k=1nln⁡(1+αkn)<∑k=1nαnk=αn⋅∑k=11k<αn(1+ln⁡n).
Note that the last one and (1.32) imply that
(1.39)c=ln⁡21+γ,forsome0<γ≤1,
and if n>m, we have
(1.40)Anαn<eαn(1+ln⁡n)=eαn⋅eαnln⁡n≤eαn⋅ec=eαn⋅eln⁡⁡(2/(1+γ))=eαn⋅21+γ,
that is,
(1.41)λn(n)=1Anαn>1eαn⋅(12+γ2),whereγ>0.
Note that αn→0 implies the existence of such γ1>0 and natural n2, that if n>n2, then
(1.42)1eαn⋅(12+γ2)>12+γ1,
that is, if n>n2, then
(1.43)λn(n)>12+γ1.
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.Theorem1.3 directly implies the following.Theorem 1.4.
Let{αn} be such sequence that
(1.44){αn}=o(1ln⁡n).
Then for every series (1.3) which satisfies
(1.45)lim⁡¯n→∞|sn|=+∞,
we have
(1.46)lim⁡¯n→∞|σnαn|=+∞.


## 2. On Divergence of Fourier Series

It is well known the following.Theorem A (Kolmogorov [1]).
There exists such summable functionf that Fourier trigonometric series of  f(2.1)a02+∑k=1∞akcos⁡kx+bksin⁡⁡kx
unboundedly diverges everywhere.LetW={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 functiong that Fourier-Walsh series of  g(2.2)∑n=1∞anwn(t)
unboundedly diverges everywhere.LetΦ={φn(t)} be orthonormal functions system defined on [0,1], such that
(2.3)|φn(t)|≤M,t∈[0,1],n=1,2,…Then below-mentioned theorem holds.Theorem C (Boc̆karev [4]).
For every orthonormal systemΦ which satisfies (2.3), there exists such summable function  h defined on  [0,1] that its Fourier series constructed by  Φ system
(2.4)∑n=1∞anφn(t)
unboundedly diverges in any point of some set E⊂[0,1] with positive measure.Denote byσnαn(x;f), σnαn(t,g,W), and σnαn(t,h,Φ) means of series (2.1), (2.2), and (2.4), respectively.Theorem1.3 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 {σnαn(x;f)} unboundedly diverges everywhere.Theorem 2.2.
Let a sequence{αn} satisfies (1.32). Then there exists such summable function g that sequence {σnαn(t,g,W)} unboundedly diverges everywhere.Theorem 2.3.
If orthonormal systemΦ satisfies (2.3) and a sequence {αn} satisfies (1.32), then there exists such summable function h, defined on [0,1], that sequence {σnαn(t;h;Φ)} unboundedly diverges at every point of some set E⊂[0,1] with positive measure.It is obvious that a consequence of Theorem1.4 is that Theorems 2.1, 2.2, and 2.3 hold true if(2.5)αn=o(1ln⁡n).Remark 2.4.
If every numberλn(k)will be replaced by(1-k/(n+1))αnin (1.31), then we get a summability method defined byΛ=∥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 Theorems2.1, 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).



---

*Source: 542607-2012-01-15.xml*