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...

We prove that a biorthogonal wavelet basis yields an unconditional basis in all spaces $L^p\left({ℝ}^d\right)$ with 1 < p < ∞, provided the biorthogonal wavelet set functions satisfy weak decay conditions. The biorthogonal wavelet set is associated with an arbitrary dilation matrix in any dimension.

By a general Franklin system corresponding to a dense sequence = (tₙ, n ≥ 0) of points in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots , that is, the nth function of the system has knots t₀, ..., tₙ. The main result of this paper is that each general Franklin system is an unconditional basis in $L^p[0,1]$, 1 < p < ∞.

We give a simple geometric characterization of knot sequences for which the corresponding orthonormal spline system of arbitrary order k is an unconditional basis in the atomic Hardy space H¹[0,1].

We prove that given any natural number k and any dense point sequence (tₙ), the corresponding orthonormal spline system is an unconditional basis in reflexive $L^p$.

Exact conditions for α, β, a, b > −1 and 1 ≤ p ≤ ∞ are determined under which the inclusion property $$L_{w^{(a,b)}}^p[-1,1]$$ ⊂ $$L_{w^{(\alpha ,\beta )}}^1[-1,1]$$ is valid. It is shown that the conditions characterize the inclusion property. The paper concludes with some results, in which the inclusion property can be detected in relation with estimates of Jacobi differential operators and with Muckenhoupt’s transplantation theorems and multiplier theorems for Jacobi series.

We establish conditions on the partial moduli of continuity which guarantee uniform convergence of the N-dimensional Walsh-Fourier series of functions f from the class $C_W\left(I^N\right)\cap {\bigcap }_{i=1}^{N}B{V}_{i,p\left(n\right)}$, where p(n)↑ ∞ as n → ∞.

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.