In this paper, we consider sampling in a reproducing kernel subspace of L. We introduce a pre-reconstruction operator associated with a sampling scheme and propose a Galerkin reconstruction in general Banach space setting. We show that the proposed Galerkin method provides a quasi-optimal approximation, and the corresponding Galerkin equations could be solved by an iterative approximation-proje...

Under certain conditions, we give an estimate from above on the number of differential equations of order r+1 with prescribed regular singular points, prescribed exponents at singular points, and having a quasi-polynomial flag of solutions. The estimate is given in terms of a suitable weight subspace of the tensor power U(n−) , where n is the number of singular points in C and U(n−) is the enve...

We provide a definition of Multidimensional Chebyshev Systems of order N which is satisfied by the solutions of a wide class of elliptic equations of order 2N . This definition generalizes a very large class of Extended Complete Chebyshev systems in the one-dimensional case. This is the first of a series of papers in this area, which solves the longstanding problem of finding a satisfactory mul...

The mth Chebyshev polynomial of a square matrix A is the monic polynomial that minimizes the matrix 2-norm of p(A) over all monic polynomials p(z) of degree m. This polynomial is uniquely defined if m is less than the degree of the minimal polynomial of A. We study general properties of Chebyshev polynomials of matrices, which in some cases turn out to be generalizations of well known propertie...

Variable degree polynomial (VDP) splines have recently proved themselves as a valuable tool in obtaining shape preserving approximations. However, some usual properties which one would expect of a spline space in order to be useful in geometric modeling, do not follow easily from their definition. This includes total positivity (TP) and variation diminishing, but also constructive algorithms ba...

We review a recently devised Monte Carlo simulation method for the direct study of quasi-stationary properties of stochastic processes with an absorbing state. The method is used to determine the static correlation function and the interparticle gap-length distribution in the critical one-dimensional contact process. We also find evidence for power-law decay of the interparticle distance distri...

The recurrent least squares (RLS) learning approach is proposed for controlling the learning rate in parallel principal subspace analysis (PSA) and in a wide class of principal component analysis (PCA) associated algorithms with a quasi{parallel extraction ability. The purpose is to provide a useful tool for applications where the learning process has to be repeated in an on{line self{adaptive ...

A theorem of Giesy and James states that c0 is nitely representable in James' quasire exive Banach space J2. We extend this theorem to the p th quasi-re exive James space Jp for each p ∈ (1,∞). As an application, we obtain a new closed ideal of operators on Jp, namely the closure of the set of operators that factor through the complemented subspace (`∞ ⊕ `∞ ⊕ · · · ⊕ `∞ ⊕ · · · )`p of Jp. 2010 ...

a computational method for numerical solution of a nonlinear volterra and fredholm integro-differentialequations of fractional order based on chebyshev cardinal functions is introduced. the chebyshev cardinaloperational matrix of fractional derivative is derived and used to transform the main equation to a system ofalgebraic equations. some examples are included to demonstrate the validity and ...