Multi-dimensional Fourier Transforms, Lebesgue Points and Strong Summability

Abstract

A general summability method of multi-dimensional Fourier transforms, the so called θ-summability is investigated. Under some conditions on θ we show that the Marcinkiewicz-θ-means of a function f ∈ W (L1, ∞)(R d ) converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so called modified Lebesgue points of f ∈ W (Lp, ∞)(R d ), whenever 1 < p < ∞. As an application we generalize the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya for f ∈ W (L1, ∞)(R) and for strong θ-summability.Mathematics Subject Classification. Primary 42B08; Secondary 42A38, 42A24, 42B25.

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

“…Let us fix a small > 0 and denote the square [0, /2] 3 by /2 . We can prove in the same way as we did in Theorem 4 of [21]…”

“…, ) ∈ R . In [21], we have seen that we may suppose that 1 > 2 > ⋅ ⋅ ⋅ > > 0 and 1 −∑ =2 > 0 and we proved the next lemma. Denote by…”

“…In this section we characterize a wide set of functions for which strong summability holds at each Lebesgue point. For the convergence of ∈ ( , ℓ )(R) (1 < < ∞, 1 ≤ < ∞) at -Lebesgue points we proved the following result in [21]. Note that ( , ℓ )(R) = (R).…”

“…Here ( , ℓ )(R) denotes the Wiener amalgam spaces. Gabisoniya's result was generalized in [21]. The analogous results are given for Fourier series, too.…”

“…Corollary 4 does not hold for = 1 (see Hardy and Littlewood [5]). However, we [21] extended it for = 1, 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 = 1 and for a subspace of ( 1 , ℓ )(R).…”