q-Summability of multi-dimensional fourier transforms

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Abstract

The summability of Fourier transforms can be generalized for higher dimensions basically in two ways. In this chapter, we study the ℓq-summability of higher dimensional Fourier transforms. As in the literature, we investigate the three cases q = 1, q = 2 and q = ∞. The other type of summability, the so-called rectangular summability, will be investigated in the next chapter. Both types are general summability methods defined by a function θ. We will generalize the results of Sects. 2.5 – 2.9. In the first section, we present the basic definitions of the ℓq-summability. In the next section, we prove the norm convergence of the θ-means. It is shown that the maximal operator of the θ-means is bounded from Hp (ℝd) to Lp(ℝd) for any p > p0, which implies the almost everywhere convergence. In Sect. 5.4, the convergence at Lebesgue points is investigated. Since the proofs are very different for different q’s, therefore each case needs new ideas. Using the result of the ℓ-summability, in the last section we prove the one-dimensional strong summability results presented in Sect. 2.10.

Original languageEnglish
Title of host publicationApplied and Numerical Harmonic Analysis
PublisherSpringer International Publishing
Pages229-382
Number of pages154
Edition9783319568133
DOIs
Publication statusPublished - Jan 1 2017

Publication series

NameApplied and Numerical Harmonic Analysis
Number9783319568133
ISSN (Print)2296-5009
ISSN (Electronic)2296-5017

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ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Weisz, F. (2017). q-Summability of multi-dimensional fourier transforms. In Applied and Numerical Harmonic Analysis (9783319568133 ed., pp. 229-382). (Applied and Numerical Harmonic Analysis; No. 9783319568133). Springer International Publishing. https://doi.org/10.1007/978-3-319-56814-0_5