Journal of Function Spaces

1. A New Summability Method of Series

Let be such triangular matrix which satisfies the following conditions:

By we denote a partial sum of a series and by we denote a mean constructed by the matrix, that is,

Theorem 1.1. Let matrix (1.1) satisfies an inequality Then for any series (1.3) which satisfies the following condition: an equality holds.

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

Lemma 1.2. For every natural number an inequality holds.

Proof of the Lemma. Using Abel transformation and we get Therefore, Thus, taking into account (1.1) we immediately get So the Lemma is proved.

Proof of Theorem 1.1. According to the condition of Theorem 1.1 we have for some . Note that inequalities which hold for every natural imply , that is, .
So, holds.
According to (1.12) there exists a natural number such that for every natural number we have So according to the Lemma, for every an inequality holds true; that is, if , then Thus for every an inequality holds.
So for every we have
Note that for every natural there exists at least one natural number , such that the partial sums of the series (1.3) satisfy the following condition: We define by a formula: So is maximal number among the above-mentioned natural numbers. Consequently, According to the condition of Theorem 1.1, Therefore, that is, A consequence of (1.24) is that there exists such natural that if then and since (1.17) holds for every , then (1.17) remains true for every , where .
So Since , therefore, Note that the last one and (1.25) imply So according to (1.21) we have that is, for every an inequality Also, (1.23) and (1.29) imply So we have finished the proof of Theorem 1.1.

Below we consider some consequences of Theorem 1.1.

Let be a triangular matrix, where the sequence is from and for every number is defined by the formula:

If , for every and (1.31) holds true, then the method is Cesaro summability method, and if , then the method coincides with convergence.

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

For we have the following.

Theorem 1.3. Let a sequences be such that for some positive number we have Then for any series (1.3) which satisfies the following condition: an equality holds.

Proof of Theorem 1.3. Note that every satisfies condition (1.1) and condition (1.3). Indeed, and , when .
For every we have that is, Therefore, Note that the last one and (1.32) imply that and if , we have that is, Note that implies the existence of such and natural , that if , then that is, if , then A consequence of (1.43) is that if (1.32) holds, then the 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 be such sequence that Then for every series (1.3) which satisfies we have

2. On Divergence of Fourier Series

It is well known the following.

Theorem A (Kolmogorov [1]). There exists such summable function that Fourier trigonometric series of unboundedly diverges everywhere.

Let 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 that Fourier-Walsh series of unboundedly diverges everywhere.

Let be orthonormal functions system defined on , such that

Then below-mentioned theorem holds.

Theorem C (Bokarev [4]). For every orthonormal system which satisfies (2.3), there exists such summable function defined on that its Fourier series constructed by system unboundedly diverges in any point of some set with positive measure.

Denote by , , and means of series (2.1), (2.2), and (2.4), respectively.

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

Namely, the following Theorems hold true.

Theorem 2.1. Let a sequence satisfies (1.32). Then there exists such summable function , that sequence unboundedly diverges everywhere.

Theorem 2.2. Let a sequence satisfies (1.32). Then there exists such summable function that sequence unboundedly diverges everywhere.

Theorem 2.3. If orthonormal system satisfies (2.3) and a sequence satisfies (1.32), then there exists such summable function , defined on , that sequence unboundedly diverges at every point of some set 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 if

Remark 2.4. If every number will be replaced by in (1.31), then we get a summability method defined by matrix, which we call Riesz summability method with variable orders and denote it by symbol .

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 method, where satisfies (1.32).

Acknowledgment

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