Summability of Multi-Dimensional Trigonometric Fourier Series

By Ferenc Weisz. 2012

67 citations recorded by the source corpus.

We consider the summability of oneand multi-dimensional trigonometric Fourier series. The Fejér and Riesz summability methods are investigated in detail. Different types of summation and convergence are considered. We will prove that the maximal operator of the summability means is bounded from the Hardy space Hp to Lp, for all p > p0, where p0 depends on the summability method and the dimension. For p = 1, we obtain a weak type inequality by interpolation, which ensures the almost everywhere convergence of the summability means. Similar results are formulated for the more general θ-summability and for Fourier transforms. MSC: Primary 42B08, 42A38, Secondary 42B30.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/593d720ba0c2fc1b0b9f2ecc7ea37c3af5fa4af3

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In Applied Mathematics

1. Johannes Jakob Meyer
2. Maria Schuld
3. Ryan Sweke

In Statistics and Probability

1. Johannes Jakob Meyer
2. Maria Schuld
3. Ryan Sweke

Citations in the corpus

• Effect of data encoding on the expressive power of variational quantum-machine-learning models (2020, methodology): “To begin with, we note that we can approximate any given g ∈ L 2 ([0 , 2 π ] N ), up to an arbitrarily small error in L 2 norm, by using a truncated Fourier series [43].”