Narrow Results

In this paper, we introduce a generalization of the $q$-Taylor expansion theorems. We expand a function in a neighborhood of two points instead of one in three different theorems. The first is a $q$-analog of the Lidstone theorem where the two points are 0 and 1 and we expand the function in $q$-analogs of Lidstone polynomials which are in fact $q$-Bernoulli polynomials as in the classical case. The definitions of these $q$-Bernoulli polynomials and numbers are introduced. We also introduce $q$-analogs of Euler polynomials and numbers. On the other two expansion theorems, we expand an analytic function around arbitrary points $z_1$ and $z_2$ either in terms of the polynomials $\{1,{\prod }_{k=0}^{n-1}(z-z_1{q}^k)(z-z_2{q}^k),n\in \mathbb{N}\}$ or in terms of the polynomials $\{1,{\prod }_{k=0}^{n-1}(z_1-z{q}^k)(z_2-z{q}^k),n\in \mathbb{N}\}$. As an application, we introduce a new series expansion for the basic hypergeometric series ${}_2{\varphi }_1$.

In this paper, we obtain a plane wave decomposition for the delta distribution in superspace, provided that the superdimension is not odd and negative. This decomposition allows for explicit inversion formulas for the super Radon transform in these cases. Moreover, we prove a more general Radon inversion formula valid for all possible integer values of the superdimension. The proof of this result comes along with the study of fractional powers of the super Laplacian, their fundamental solutions, and the plane wave decompositions of super Riesz kernels.

This is a continuation of the study of adaptive Fourier decomposition (AFD).¹⁵ Under a mild condition not in terms of smoothness, a convergence rate is provided. We prove that the selection of the parameters corresponding to Fourier series in the average sense is optimal. We also present the transformation matrices between the adaptive rational orthogonal system and the related sequence of the shifted Cauchy kernels and their derivatives.

A sequence of special functions in Hardy space are constructed from Cauchy kernel on unit disk 𝔻. Applying projection operator of the sequence of functions leads to an analytic sampling approximation to f, any given function in . That is, f can be approximated by its analytic samples in 𝔻^s. Under a mild condition, f is approximated exponentially by its analytic samples. By the analytic sampling approximation, a signal in can be approximately decomposed into components of positive instantaneous frequency. Using circular Hilbert transform, we apply the approximation scheme in to L^s(𝕋²) such that a signal in L^s(𝕋²) can be approximated by its analytic samples on ℂ^s. A numerical experiment is carried out to illustrate our results.

The main goal of this paper is to extend A.Connes's result on Fredholm index to whole Hilbert space. A Fredholm module will be constructed by using singular integral operators with the Cauchy kernel and the relationship between the Fredholm index of the singular integral operators and K-group was set up.