

Cesàro means of subsequences of partial sums of trigonometric Fourier series

György Gát Institute of Mathematics, University of Debrecen, H-4002 Debrecen, Pf. 400, Hungary

In 1936 Zygmunt Zalcwasser asked with respect to the trigonometric system that how “rare” can a sequence of strictly monotone increasing integers be such that the almost everywhere relation is fulfilled for each integrable function . In this paper, we give an answer to this question. It follows from the main result that this a.e. relation holds for every integrable function and lacunary sequence of natural numbers.

Key words and phrases:
Cesàro means, subsequences of partial sums, trigonometric Fourier series, a.e. convergence, Zalcwasser’s problem.
2010 Mathematics Subject Classification:
Research supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. K111651 and by project EFOP-3.6.1-16-2016-00022 supported by the European Union, co-financed by the European Social Fund.

1. Introduction and the main theorem

In 1936 Zalcwasser [21] asked how “rare” can a sequence of integers be such that

a.e. for every function . In this paper, we give an answer to this question (Theorem 1.1).

It is of main interest in the theory of trigonometric Fourier series that how to reconstruct the function from the partial sums of its Fourier series. It is known from Du Bois-Reymond [8] that the Fourier series of a continuous function can unboundedly diverge at some point.

A. N. Kolmogoroff [12] constructed his famous example of a function such that the partial sums diverge unboundedly almost everywhere. In another paper [13] he constructed an everywhere divergent Fourier series. In particular, it was not clear whether the Fourier series of a continuous function can diverge everywhere. Carleson [6] showed that if , then the partial sums converge to the function almost everywhere. The condition in the Carleson theorem was weakened by Hunt [11] () and recently Antonov [1] who proved that if is in the class , then the partial sums of the Fourier series converge to the function almost everywhere again. It is a fundamental question how to reconstruct a function in from its Fourier series.

Inspired by Fejér’s results, Lebesgue [15] showed that for each integrable function we have the almost everywhere convergence of the Fejér means .

It is also of prior interest what it can be said - with respect to this reconstruction issue - if we have only a subsequence of the partial sums. With respect to the partial sums and the Lebesgue space the bad news is that in 1982 Totik [19] showed that for each subsequence of the sequence of natural numbers there exists an integrable function such that everywhere. Moreover, Konyagin [14] proved that for any increasing sequence of positive integers and any nondecreasing function satisfying the condition , there is a function such that everywhere. That is, a summation method is needed.

In 1936 Zalcwasser [21] asked how “rare” the sequence of integers can be such that

for every function . This problem was completely solved with respect to the trigonometric system for continuous functions and uniform convergence in [16, 20, 3, 7]. That is, if the sequence is convex, then the condition is necessary and sufficient for the uniform convergence for every continuous function.

With respect to convergence almost everywhere and integrable functions the situation is more complicated. In 1936 Zalcwasser [21] proved the a.e. relation for each integrable function . In his paper Salem [16, page 394] writes that this theorem of Zalcwasser is extended to and but there is no citation in [16] about it. Belinsky proved [4] the existence of a sequence such that the relation holds a.e. for every integrable function. In this paper, Belinsky also conjectured that if the sequence is convex, then the condition is necessary and sufficient again. So, that would be the answer to the problem of Zalcwasser [21] in this case (trigonometric system, a.e. convergence and functions). In this paper, - among others - it is proved that this is not the case.

The system of functions () is called the trigonometric system. It is orthogonal over any interval of length , specifically over . Let , that is is an integrable function on . The th Fourier coefficient of is

where is any integer number. The th () partial sum of the Fourier series of is

The th () Fejér or mean of the function is defined in the following way:

It is known that

where the function is known as the th Fejér kernel; we will now find an appropriate expression for it (see e.g. the book of Bary [2]), namely

From this expression one can immediately derive the following properties of the kernel. They will play an essential role later.

Now, we state the main theorem of the paper.

Theorem 1.1.

Let be a function and be a sequence of natural numbers with the property that holds for and for some . Then the almost everywhere relation

holds on .

Corollary 1.2.

Let be a lacunary sequence of natural numbers. Then the almost everywhere relation holds for every .

We remark, that the corresponding version of Corollary 1.2 for the Walsh-Paley system can be found in [10]. On the other hand, no part of the paper [10] could be used here, but the fact that the author of [10] could prove Corollary 1.2 for the Walsh system was an inspiration for this article.

We say some preliminary words about the main ideas of the proof of Theorem 1.1. Let be the th de la Vallée-Poussin mean of the integrable function . Instead of the means of we can investigate means of , since for the de la Vallée-Poussin means we have the a.e. convergence . We define some sequence of sets and the whole sections and are dedicated to prove the “orthogonality” lemma (Lemma 6.1). That is,

where constant depends only on and it is uniform in and . To be honest, “orthogonality” lemma (Lemma 6.1) is proved for lacunary sequences (with quotient greater than ), but this fact will be got around in the proof of the main theorem (Theorem 1.1). In the “replacement” lemma (Lemma 6.2) we basically prove that the sequences

are a.e. equiconvergent. This will lead to the a.e. convergence . Roughly speaking, the proof of the main theorem is based on the “orthogonality” and “replacement” lemmas. In the sequel, we start the process on the way to the “orthogonality” lemma.

2. A Decomposition Lemma

The dyadic subintervals of are defined in the following way.

The elements of are said to be dyadic intervals. If , then there exists a unique such that , and consequently . Each has disjoint elements ().

The following Calderon-Zygmund type decomposition lemma can be found for instance in [18, page 17] or [17, page 90] (more precisely, in a slightly different way) or in [9] (with an elementary proof). This will play a prominent role in the proof of the main theorem of this paper.

Lemma 2.1.

Let , and . Then there exists a sequence of integrable functions such that

are disjoint dyadic intervals depending only on (and ),

. Moreover, ,

and for the union

of the disjoint dyadic intervals () we have .

Also using the notation with respect to Lemma 2.1 we define . That is, is the set of dyadic intervals, whose union is the set . Moreover, for any dyadic interval , if and only if and for every dyadic interval . For any positive number let , . Besides, remark that (), ( is the characteristic function of the complement of ).

Let be the dyadic interval of measure with . For any dyadic interval and integer let be the th neighbour of (the addition is done modulo , that is a circle represents ). For instance and for , are the right and left adjacent (modulo ) dyadic intervals of with the same measure. We also use the notation .

For any let be the tripled of . That is, is an interval with the same center as and with the tripled measure of . Similarly, we define

and so on. That is, we can define . Moreover, use the notation for any odd integer

We need a preliminary lemma:

Lemma 2.2.

Let be an odd natural number, be a dyadic interval. Then we have



Without loss of generality we can suppose that . If , then and . The sets are disjoint and consequently

Now, turn our attention to . Suppose that we take the sum for pairs satisfying and take fixed .

The intersection of dyadic intervals can be different from if and only if . By the first inequality of Lemma 2.2 proved already we have

Taking into account that and the fact that the case can be discussed in the same way the proof of Lemma 2.2 is complete. ∎

3. The integral on

For and set a version of Hilbert transforms of at for (that is, ) as:


The transform is of type for any and it is also of weak type as it can be seen in the following way. Let . Then . Then we have

Since the maximal (“ordinary”) Hilbert transform ([5, Chapter 3]) and the integral mean value operator are of type for any and they are also of weak type , then so is the “newly defined” Hilbert transform.

The first lemma needed is:

Lemma 3.1.

Let be odd integers, , with and . Then the inequality

holds, where the constant can depend only on and (it is uniform in ) and is the characteristic function of the complement of .


Without loss of generality, in order not to write too many conjugate signs, we suppose that is a real function. By applying the Calderon-Zygmund decomposition lemma (Lemma 2.1) for we have , where . Since the Hilbert transform is of type and , then we have

That is, instead of we have to investigate only . Since the elements of are disjoint dyadic intervals, therefore

and then

The sum over pairs will be divided into the following parts: ()

First, we investigate . For we have

as it comes from Lemma 2.1. Similarly,


Let , where . Then, there exists an such that , where . We give an upper bound for the nonnegative real number .


because for and we have and consequently . Remark that means for a . That is, if is not in interval , then it is shifted by .

Moreover, if , then by which is equivalent with (and of course ) we also have , that is, and consequently by the fact that

The same can be said about because . This yields


Henceforth it is easy to give a bound for :

Thus, by Lemma 2.2 and Lemma 2.1 we have

Next, turn our attention to . Since , then as in the investigation of we have

again. Meanwhile, as in the investigation of for any () it yields


( gives .) Recall that and fix a .

Let be a subset of . We call the dyadic interval maximal with respect to if and there is no interval such that . This will be abbreviated as maximal. If , then we simply use the term maximal intervals. Thus, recalling that two dyadic intervals are disjoint or one of them contains the other, one has

Since by Lemma 2.2

in the estimation of we have

Since the maximal dyadic intervals are disjoint, then

Two dyadic intervals are disjoint or one of them is contained in the other. Is it possible that ? If yes, then gives a contradiction since . That is, either or . On the other hand, if , then gives the same contradiction. That is, either or . The same situation concerns which means that is either or .

In the first case () there is nothing to prove. In the second case () we have , because by Lemma 2.1 . Then, denoting by the center of the interval we have



Besides, as above we have . Thus,


Since is the center of , we have (