Endpoint Results for the Riesz Transform of the Ornstein–Uhlenbeck Operator

Journal of Fourier Analysis and Applications - In this paper we introduce a new atomic Hardy space $$X^1(\gamma )$$ adapted to the Gauss measure $$\gamma $$ , and prove the boundedness of the first...     

Almost Diagonalization of $$\tau $$ τ -Pseudodifferential Operators with Symbols in Wiener Amalgam and Modulation Spaces

Journal of Fourier Analysis and Applications - In this paper we focus on the almost-diagonalization properties of $$\tau $$ -pseudodifferential operators using techniques from time-frequency...     

Harmonic Functions,Conjugate Harmonic Functions and the Hardy Space $$H^1$$ H 1 in the Rational Dunkl Setting

Journal of Fourier Analysis and Applications - In this work we extend the theory of the classical Hardy space $$H^1$$ to the rational Dunkl setting. Specifically, let $$\Delta $$ be the Dunkl...     

Paley–Wiener Theorems on the Siegel Upper Half-Space

Journal of Fourier Analysis and Applications - In this paper we study spaces of holomorphic functions on the Siegel upper half-space $${\mathcal U}$$ and prove Paley–Wiener type theorems for...     

A Paley–Wiener Type Theorem for Singular Measures on $$(-1/2, 1/2)$$ ( - 1 / 2 , 1 / 2 )

Journal of Fourier Analysis and Applications - For a fixed singular Borel probability measure $$\mu $$ on $$(-1/2, 1/2)$$ , we give several characterizations of when an entire function is the...     

Complementation of the subspace of radial multipliers in the space of Fourier multipliers on $$\mathbb {R}^n$$ R n

Archiv der Mathematik - In this short note, we prove that the subspace of radial multipliers is contractively complemented in the space of Fourier multipliers on the Bochner space $$\mathrm...     

Letter to the Editor: Linear Independence of Time-Frequency Shifts Up To Extreme Dilations

Journal of Fourier Analysis and Applications - Given $$f \in C_0({{\,\mathrm{\mathbb {R}}\,}}^n)$$ and a finite set $$\Lambda \subset {{\,\mathrm{\mathbb {R}}\,}}^{2n}$$ we demonstrate the linear...     

Fast evaluation of trigonometric polynomials from hyperbolic crosses

The discrete Fourier transform in d dimensions with equispaced knots in space and frequency domain can be computed by the fast Fourier transform (FFT) in arithmetic operations. In order to circumvent the ‘curse of dimensionality’ in multivariate approximation, interpolations on sparse grids were introduced. In particular, for frequencies chosen from an hyperbolic cross and spatial knots on a sparse grid fast Fourier transforms that need only arithmetic operations were developed. Recently, the FFT was generalised to nonequispaced spatial knots by the so-called NFFT. In this paper, we propose an algorithm for the fast Fourier transform on hyperbolic cross points for nonequispaced spatial knots in two and three dimensions. We call this algorithm sparse NFFT (SNFFT). Our new algorithm is based on the NFFT and an appropriate partitioning of the hyperbolic cross. Numerical examples confirm our theoretical results.     

$$\pmb {{\mathcal {H}}_{p}}$$Hp-Theory of General Dirichlet Series

Journal of Fourier Analysis and Applications - Inspired by results of Bayart on ordinary Dirichlet series $$\sum a_n n^{-s}$$, the main purpose of this article is to start an $${\mathcal...     

Sums of k-th powers and the Whittaker–Fourier coefficients of automorphic forms

The Ramanujan Journal - In this work, we obtain power-saving bounds for shifted convolution sums involving the Whittaker–Fourier coefficients of automorphic forms and $$r_{s, k}(n)$$ , the...     

Limiting Behavior of Infinite Products Scaled by Pisot Numbers

Journal of Fourier Analysis and Applications - For $$\theta >1$$ , the infinite product $$\Gamma _{\theta }(x)=\prod _{n=0}^{\infty }\cos (\pi \theta ^{-j}x)$$ is the Fourier transform of...     

Improved Bounds on the Dimensions of Sets that Avoid Approximate Arithmetic Progressions

Journal of Fourier Analysis and Applications - In this paper we construct a wavelet basis in $$L^{2}({\mathbb {R}}^{n};\mu )$$ possessing vanishing moments of a fixed order for a general locally...     

Endpoint Estimates for the Maximal Function over Prime Numbers

Journal of Fourier Analysis and Applications - Given an ergodic dynamical system $$(X, \mathcal {B}, \mu , T)$$, we prove that for each function f belonging to the Orlicz space $$L(\log L)^2(\log...     

Graph Products,Fourier Analysis and Spectral Techniques

We consider powers of regular graphs defined by the weak graph product and give a characterization of maximum-size independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In many cases this also characterizes the optimal colorings of these products.We show that the independent sets induced by the base graph are the only maximum-size independent sets. Furthermore we give a qualitative stability statement: any independent set of size close to the maximum is close to some independent set of maximum size.Our approach is based on Fourier analysis on Abelian groups and on Spectral Techniques. To this end we develop some basic lemmas regarding the Fourier transform of functions on generalizing some useful results from the case.     

A Complete Real-Variable Theory of Hardy Spaces on Spaces of Homogeneous Type

Journal of Fourier Analysis and Applications - Let $$(X,d,\mu )$$ be a space of homogeneous type, with the upper dimension $$\omega $$ , in the sense of Coifman and Weiss. Assume that $$\eta $$ is...     

Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl 3,0

First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions Third, we show a set of important properties of the Clifford Fourier transform on Cl_3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl_3,0 multivector functions.     

Generalized Fourier Transforms Associated with the Oscillator Representation

Journal of Fourier Analysis and Applications - By identifying the Fourier transform $$\mathcal {F}$$ with an operator in the oscillator representation of the metaplectic group...     

Wellposedness of NLS in Modulation Spaces

Journal of Fourier Analysis and Applications - Let $${\mathcal {R}}$$ denote the generalized Radon transform, which integrates over a family of N-dimensional smooth submanifolds $${\mathcal...     

Convex functions and Fourier coefficients

Positivity - The aim of this paper is to prove that the cosine Fourier coefficients $$a_{mn}$$ (with $$m,n\ge 1)$$ of a Popoviciu convex function of two variables are nonnegative.     

The Structure of Multiplicative Tilings of the Real Line

Journal of Fourier Analysis and Applications - Suppose $$\Omega , A \subseteq \mathbb {R}\backslash {\left\{ {0}\right\} }$$ are two sets, both of mixed sign, that $$\Omega $$ is Lebesgue...