We prove L estimates for the maximal Riesz transform in terms of the Riesz transform itself, for 1 < p ≤ ∞. We show that the corresponding weak L1 estimate fails for the maximal Riesz transform, but surprisingly does hold for the maximal Beurling transform.

We give a comprehensive introduction to a general modular frame construction in Hilbert C*-modules and to related linear operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that admit an orthonormal modular Riesz basis. Interrelations and applications t...

In this article we obtain families of frames for the space Bω of functions with band in [−ø, ø] by using the theory of shift-invariant spaces. Our results are based on the Gramian analysis of A. Ron and Z. Shen and a variant, due to Bownik, of their characterization of families of functions whose shifts form frames or Riesz bases. We give necessary and sufficient conditions for the translates o...

We propose Spherical Structured Feature (SSF) maps to approximate shift and rotation invariant kernels as well as b-order arc-cosine kernels (Cho & Saul, 2009). We construct SSF maps based on the point set on d − 1 dimensional sphere Sd−1. We prove that the inner product of SSF maps are unbiased estimates for above kernels if asymptotically uniformly distributed point set on Sd−1 is given. Acco...

In this paper it is investigated how to find a matrix representation of operators on a Hilbert space H with Bessel sequences, frames and Riesz bases. In many applications these sequences are often preferable to orthonormal bases (ONBs). Therefore it is useful to extend the known method of matrix representation by using these sequences instead of ONBs for these application areas. We will give ba...

We present forms of the classical Riesz–Kolmogorov theorem for compactness that are applicable in a wide variety settings. In particular, our theorems apply to classify precompact subsets Lebesgue space $$L^2$$ , Paley–Wiener spaces, weighted Bargmann–Fock and scale Besov–Sobolev spaces holomorphic functions includes Bergman general domains as well Hardy Dirichlet space. criteria characterize c...

In this article, we consider a two-dimensional symmetric space-fractional diffusion equation in which the space fractional derivatives are defined in Riesz potential sense. The well-posed feature is guaranteed by energy inequality. To solve the diffusion equation, a fully discrete form is established by employing Crank-Nicolson technique in time and Galerkin finite element method in space. The ...

We link conditional weak mixing and ergodicity via the tensor product in Riesz spaces. In particular, we characterise of a expectation preserving system by its with itself or other ergodic systems. Along way components order units space terms products units.

Abstract For all admissible values of the numerical parameters sharp sufficient conditions on functional are obtained ensuring boundedness generalized Riesz potential from one general local Morrey-type space to another one, which, for a certain range parameters, coincide with necessary ones.

We study the optimal approximation of the solution of an operator equation A(u) = f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs that are given by an isomorphism A :...