We start by reviewing some elementary Banach and Hilbert space theory. Two key results here will prove useful in studying the properties of reproducing kernel Hilbert spaces: (a) that a linear operator on a Banach space is continuous if and only if it is bounded, and (b) that all continuous linear functionals on a Banach space arise from the inner product. The latter is often termed Riesz repre...

This is a study of metric spaces, a subject originally initiated by Maurice Fréchet in 1906 and subsequently developed by Hausdorff, Riesz, and ultimately Banach. The original motivation behind the theory idea was to extend the notion of convergence from the real and complex field to more abstract spaces. The idea of the metric was used by Fréchet to induce a topology on the space and later thi...

We study the sample path regularity of the solution of a stochastic wave equation in spatial dimension d = 3. The driving noise is white in time and with a spatially homogeneous covariance defined as a product of a Riesz kernel and a smooth function. We prove that at any fixed time, a.s. the sample paths in the spatial variable belong to certain fractional Sobolev spaces. In addition, for any f...

This article is in continuation of the authors research attempts to derive computational solutions of an unified reaction-diffusion equation of distributed order associated with Caputo derivatives as the time-derivative and Riesz-Feller derivative as space derivative. This article presents computational solutions of distributed order fractional reaction-diffusion equations associated with Riema...

We introduce the space Sω1,ω2 of all C ∞ functions φ such that sup |α|≤m‖e1∂αφ‖∞ and sup |α|≤m‖e2∂αφ̂‖∞ are finite for all k ∈ N0, α ∈ Nn0 , where ω1 and ω2 are two weights satisfying the classical Beurling conditions. Moreover, we give a topological characterization of the space Sω1,ω2 without conditions on the derivatives. For functionals in the dual space S ′ ω1,ω2 , we prove a structure theo...

In this paper the concept of an $Omega$- Almost Boolean ring is introduced and illistrated how a sheaf of algebras can be constructed from an $Omega$- Almost Boolean ring over a locally Boolean space.