We develop construction techniques in the context of Riesz bases which facilitate the conception of a new family of multi-channel, multi-sampling rate theorems. This family takes advantage of the dilation property of the Fourier transform. In each channel a simple transform is taken which merely involves the addition and subtraction of subsets of the given samples. Each channel is then multipli...

In this paper, we consider the numerical solutions of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two kinds of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection-dispersion equation (RFADE). RFDE is obtained from the standard diffusion equation by replacing the secondo...

This paper presents a formal proof of the Riesz representation theorem in the PVS theorem prover. The Riemann Stieltjes integral was defined in PVS, and the theorem relies on this integral. In order to prove the Riesz representation theorem, it was necessary to prove that continuous functions on a closed interval are Riemann Stieltjes integrable with respect to any function of bounded variation...

this paper is an investigation of $l$-dual frames with respect to a function-valued inner product, the so called $l$-bracket product on $l^{2}(g)$, where g is a locally compact abelian group with a uniform lattice $l$. we show that several well known theorems for dual frames and dual riesz bases in a hilbert space remain valid for $l$-dual frames and $l$-dual riesz bases in $l^{2}(g)$.

The efficient solution of operator equations using wavelets requires that they generate a Riesz basis for the underlying Sobolev space, and that they have cancellation properties of a sufficiently high order. Suitable biorthogonal wavelets were constructed on reference domains as the n-cube, which bases have been used, via a domain decomposition approach, as building blocks to construct biortho...