For a Lebesgue integrable complex-valued function $f$ defined on ${ℝ}^{+}:=[0,\infty )$ let $\widehat{f}$ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that $\widehat{f}\left(y\right)\to 0$ as $y\to \infty$. But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of $L^1\left({ℝ}^{+}\right)$ there is a definite rate at which the Walsh-Fourier transform tends to zero. We...

We propose a definition of Riesz transforms associated to the Ornstein-Uhlenbeck operator based on the Dunkl Laplacian. In the case related to the group ℤ ₂ it is proved that the Riesz transform is bounded on the corresponding $L^p$ spaces, 1 < p < ∞.

The aim of this paper is to prove new uncertainty principles for integral operators with bounded kernel for which there is a Plancherel Theorem. The first of these results is an extension of Faris’s local uncertainty principle which states that if a nonzero function $f\in L²({ℝ}^d,\mu )$ is highly localized near a single point then (f) cannot be concentrated in a set of finite measure. The second result extends the Benedicks-Amrein-Berthier uncertainty principle and states that a nonzero function $f\in L²({ℝ}^d,\mu )$ and its integral...

For any 0 < ϵ < 1, p ≥ 1 and each function $f\in L^p[0,1]$ one can find a function $g\in L^{\infty }[0,1)$ with mesx ∈ [0,1): g ≠ f < ϵ such that its greedy algorithm with respect to the Walsh system converges uniformly on [0,1) and the sequence $|c_k\left(g\right)|:k\in spec\left(g\right)$ is decreasing, where $c_k\left(g\right)$ is the sequence of Fourier coefficients of g with respect to the Walsh system.