Riesz potentials of fractal measures μ in metric spaces and their inverses are introduced . They define self–adjoint operators in the Hilbert space L2(μ) and the former are shown to be compact. In the Euclidean case the corresponding spectral asymptotics are derived with Besov space methods. The inverses of the Riesz potentials are fractal pseudodifferential operators. For the order two operato...

We investigate the relation between the Riesz and the Báez-Duarte criterion for the Riemann Hypothesis. In particular we present the relation between the function R(x) appearing in the Riesz criterion and the sequence ck appearing in the Báez-Duarte formulation. It is shown that R(x) can be expressed by ck, and, vice versa, the sequence ck can be obtained from the values of R(x) at integer argu...

In this paper we use the lifting scheme to construct biorthogonal spline wavelet bases on regularly refined non-uniform grids. The wavelets have at least one vanishing moment and on each resolution level they form an L2 Riesz basis. Furthermore we are interested in determining the exact range of Sobolev exponents for which the complete multilevel system forms a Riesz basis. Hereto we need to ex...

We study a.e. convergence on L, and Lorentz spaces L, p > 2d d−1 , for variants of Riesz means at the critical index d( 1 2 − 1 p )− 1 2 . We derive more general results for (quasi-)radial Fourier multipliers and associated maximal functions, acting on L spaces with power weights, and their interpolation spaces. We also include a characterization of boundedness of such multiplier transformation...

We introduce a natural definition of Riesz measures and Wishart laws associated to an Ω-positive (virtual) quadratic map, where Ω ⊂ Rn is a regular open convex cone. We give a general formula for moments of the Wishart laws. Moreover, if the quadratic map has an equivariance property under the action of a linear group acting on the cone Ω transitively, then the associated Riesz measure and Wish...

Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on Rn or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p > 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(L)−1/2 on the Lp space. As an application, for 1 < p < 3+ ε, we establish the Lp boundedness...

We introduce a new notion for the deformation of Gabor systems. Such deformations are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. With this new notion we prove a strong deformation result for Gabor frames and Gabor Riesz sequences that covers the known perturbation and deformation results. Our proof of the deformation theore...

Using the existence of solutions for equilibrium equations with a Neumann type boundary condition as developed by Shi and Liao (J. Inequal. Appl. 2015:363, 2015), we obtain the Riesz integral representation for continuous linear maps associated with additive set-valued maps with values in the set of all closed bounded convex non-empty subsets of any Banach space, which are generalizations of in...

In his studies of mixing conditions on Markov chains, Rosenblatt [32; 33, Chap. 71 used the Riesz convexity (interpolation) theorem to compare different measures of dependence between two given families of random variables on a probability space. Rosenblatt [34] also suggested that by using other results in operator theory, one might be able to obtain more information about the relationships be...

The Dual-Phase-Lagging (DPL) equation is formulated as an abstract differential equation. In the absence of a heat source term the DPL equation with homogeneous boundary conditions generates a contraction semigroup. The exact expression of the semigroup is achieved. It is proved that the associated eigenfunctions form a Riesz basis. The stability of semigroup is proved. Moreover, it is also sho...