Strong Summability of Fourier Transforms at Lebesgue Points and Wiener Amalgam Spaces

Abstract

We characterize the set of functions for which strong summability holds at each Lebesgue point. More exactly, if is in the Wiener amalgam space and is almost everywhere locally bounded, or , then strong -summability holds at each Lebesgue point of . The analogous results are given for Fourier series, too.

1. Introduction

It was proved by Lebesgue [1] that the Fejér means [2] of the trigonometric Fourier series of an integrable function converge almost everywhere to the function; that is, for almost every , where denotes the torus and the th partial sum of the Fourier series of the one-dimensional function . The set of convergence is characterized as the Lebesgue points of .

Hardy and Littlewood [3] considered the so-called strong summability and verified that the strong means tend to 0 at each Lebesgue point of , as , whenever () (for Fourier transforms see Giang and Móricz [4]). This result does not hold for (see Hardy and Littlewood [5]). However, the strong means tend to almost everywhere for all . This is because of Marcinkiewicz [6] for and Zygmund [7] for all (see also Bary [8]). Later Gabisoniya [9] characterized the set of convergence as the so-called Gabisoniya points. Strong summability with lacunary partial sums and Lebesgue points are investigated by Belinsky et al. [10–13].

In a general method of summation, the so-called -summation method, which is generated by a single function and which includes the well-known Fejér, Riesz, Weierstrass, and Abel summability methods, is studied intensively in the literature (see, e.g., Butzer and Nessel [14], Trigub and Belinsky [15–17], Liflyand [18], and Weisz [19, 20]). In this paper we generalize some of the above-mentioned results for strong -summability of Fourier transforms and for Wiener amalgam spaces. We characterize the set of functions for which strong summability holds at each Lebesgue point. More exactly, we will show that at each Lebesgue point of () when is locally bounded at , where . Moreover, the convergence holds at each Lebesgue point of if (). Here denotes the Wiener amalgam spaces. Gabisoniya’s result was generalized in [21]. The analogous results are given for Fourier series, too.

2. Wiener Amalgam Spaces and Lebesgue Points

Let us fix , . For a set let be its Cartesian product taken with itself -times. We briefly write instead of the space equipped with the norm with the usual modification for , where is the Lebesgue measure. () denotes the space of measurable functions for which is locally integrable. We say that is locally bounded at if there exists a neighborhood of such that is bounded on this neighborhood.

Now we generalize the spaces. A measurable function belongs to the Wiener amalgam space () if with the obvious modification for . It is easy to see that and the following continuous embeddings hold true: . Thus

A point is called a -Lebesgue point (or a Lebesgue point of order ) of if It was proved by Feichtinger and Weisz [22, 23] that almost every point is a -Lebesgue point of (). In context of Lebesgue points of -functions we call also for the earlier papers of Belinsky et al. [12, 13].

In this paper the constants and may vary from line to line and the constants are depending only on .

3. The Kernel Functions

The Fourier transform of is given by where . Suppose first that for some . The Fourier inversion formula motivates the definition of the Dirichlet integral () introduced by where the Dirichlet kernel is defined by Obviously, .

It is easy to see that, with the help of the integral in (11), the definition of can be extended to all with . Note that , where . It is known (see, e.g., Grafakos [24] or [20]) that, for , ,

This convergence does not hold for . However, using a summability method, we can generalize these results. We may take a general summability method, the so-called -summation defined by a function . This summation contains all well-known summability methods, such as the Marcinkiewicz-Fejér, Riesz, Weierstrass, Abel, Picard, and Bessel summations.

Suppose that is continuous on ; the support of is for some and is differentiable on . Suppose further that where , denotes the maximum, and denotes the minimum.

For the -means of a function () are defined by It is easy to see that Note that this formula is well defined for all . Here denotes the -kernel. Thus for all with . Note that for the Fejér means (i.e., for we get the usual definition In Feichtinger and Weisz [22, 23] we have proved that, under conditions (14) and (25) with , for all Lebesgue points of . In this paper, we investigate the problem of the strong summability, that is, whether the convergence holds for Lebesgue points and some . Usually is increasing; then we can take the absolute value of in the integral.

To this end we have to introduce some -dimensional definitions. In the -dimensional case we define the Dirichlet kernel by and the so called Marcinkiewicz--kernel by where . In [21], we have seen that we may suppose that and and we proved the next lemma. Denote by

Lemma 1. Let for some . Then If in addition and , where , then

The next lemma is due to the author [25].

Lemma 2. If (14) and (25) are satisfied for some and , then ().

4. Strong Summability of Fourier Transforms

In this section we characterize a wide set of functions for which strong summability holds at each Lebesgue point. For the convergence of () at -Lebesgue points we proved the following result in [21]. Note that .

Theorem 3. Suppose that (14) and (25) hold for some and . Let for some and . If is a -Lebesgue point of for all , then

If all () are equal, then we obtain the following.

Corollary 4. Suppose that (14) and (25) hold for some even and . Let for some and . If is a -Lebesgue point of , then

Obviously, the convergence holds almost everywhere. Corollary 4 does not hold for (see Hardy and Littlewood [5]). However, we [21] extended it for , but for much more specialized points than the Lebesgue points, for the so-called Gabisoniya points, which were introduced in [9]. In the next theorem we generalize Theorem 3 and Corollary 4 for and for a subspace of .

Theorem 5. Suppose that (14) and (25) hold for some and . Let for some . If is a Lebesgue point of and is locally bounded at for all , then

Proof. It is easy to see that Since we obtain For simplicity we will prove the rest of the theorem for , only. It can be proved for higher dimensions similarly. As we mentioned earlier, we may suppose that and . Let us fix a small and denote the square by . We can prove in the same way as we did in Theorem of [21] that if is small enough and is large enough. The estimation of this integral on the set of that proof does not work now, because there we used a modified maximal function which is not necessarily bounded in our case. So we have to show here that To this end let us introduce the sets for a given small and large . Then we have to estimate the integral in (35) for , . First of all observe that for and . On the set we have and so . Hence Since is locally bounded at , we get by (37) that as . On we have and so Similarly, On the set we have if is large enough and so . Then The second term is less than if is large enough and the first term is less than if is small enough. The set can be handled in the same way.
On we have . Thus and . Then On we have where is chosen such that . Hence On , (44) implies that Observe that contradicts .
Consider the set . If , then and if , then . Observe that . Hence Since , we can integrate in to obtain as in (43). Similarly, holds as well on . If , then and so as before. If , then If , then the integral on the set can be estimated in the same way. If , then because and on this set. Writing instead of in the sets (), we obtain the definition of the sets . On we have by (50) that Moreover, contradicts , more exactly, .
On and , thus . Similar to (48), on . Then which is small enough if is large and is small enough. On we use the estimation to obtain as just before.
On again. If , then and if , then . This case can be handled in the same way as the set .
On we get which implies that where . Similarly, Moreover, on Since , the proof of the theorem is complete.

Corollary 6. Suppose that (14) and (25) hold for some even and . Let for some . If is a Lebesgue point of and is locally bounded at , then

If is almost everywhere locally bounded, then the corollary holds almost everywhere. Unfortunately, it is not true that an integrable function is almost everywhere locally bounded (see [21]). The strong summability can be extended to all exponents in the usual way (see [21]).

Corollary 7. Suppose that (14) holds for all , is nonincreasing, and . Under the same conditions as in Corollaries 4 or 6, respectively, one gets that

Note that under the same conditions we get for the Fejér summation that

5. Strong Summability of Fourier Series

In this section we formulate the above results for Fourier series. For an integrable function the th Fourier coefficient is defined by where denotes the torus. For and the th partial sum is introduced by where is the Dirichlet kernel. Let be a two-parameter sequence of real numbers satisfying where The -means of are defined by where the -kernel is given by Hence

Basically, we will consider two types of examples. The first type of examples is the Cesàro summation, which is generated by a sequence .

Example 8 (Cesàro summation). For , , let (see Zygmund [26, p. 77]). Let for some . Since , the Cesàro operators can be given by If , we get back the Fejér means.

The other type is generated by a function (see all other examples in this paper). Let be continuous on and Suppose that Note that for the Fejér means we get the usual definition All examples of this paper satisfy (26) and (27).

Theorem 9. Suppose that (26), (27), and (69) are satisfied for some and . Let for some . If is a -Lebesgue point of for all , then Moreover, if is even, then

Theorem 10. Suppose that (26), (27), and (69) are satisfied for some and . If , is a Lebesgue point of , and is locally bounded at for all , then (80) and (81) hold.

Corollary 11. Suppose that , , and (69) is satisfied for all . Under the same conditions as in Theorems 9 or 10, respectively, one gets that

Note that under the same conditions we get for the Fejér summation that

Finally we note that the Weierstrass, Abel, Picard, Bessel, Fejér, de La Vallée-Poussin, Rogosinski, and Riesz summations can be considered as special cases of -summation (see Weisz [21]).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank the referee for reading the paper carefully and for calling his attention to some papers of the references. This research was supported by the Hungarian Scientific Research Funds (OTKA) no. K115804.