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 meanstend 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. [1013].

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 [1517], 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 thatat 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 normwith 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 () ifwith 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 ifIt 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 bywhere . Suppose first that for some . The Fourier inversion formulamotivates the definition of the Dirichlet integral () introduced bywhere the Dirichlet kernel is defined byObviously, .

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 thatwhere , denotes the maximum, and denotes the minimum.

For the -means of a function () are defined byIt is easy to see thatNote that this formula is well defined for all . Heredenotes the -kernel. Thusfor all with . Note that for the Fejér means (i.e., for we get the usual definitionIn 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 convergenceholds 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 byand the so called Marcinkiewicz--kernel bywhere . In [21], we have seen that we may suppose that and and we proved the next lemma. Denote by

Lemma 1. Letfor some . ThenIf 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 thatSincewe obtainFor 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] thatif 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 thatTo this end let us introduce the setsfor 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 . HenceSince is locally bounded at , we get by (37) thatas . On we haveand soSimilarly,On the set we have if is large enough and so . ThenThe 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 . ThenOn we havewhere is chosen such that . HenceOn , (44) implies thatObserve that contradicts .
Consider the set . If , then and if , then . Observe that . HenceSince , we can integrate in to obtainas in (43). Similarly, holds as well on . If , thenand soas before. If , thenIf , 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) thatMoreover, contradicts , more exactly, .
On and , thus . Similar to (48),on . Thenwhich is small enough if is large and is small enough. On we use the estimationto obtainas just before.
On again. If , then and if , then . This case can be handled in the same way as the set .
On we getwhich implies thatwhere . Similarly,Moreover, on