We consider the problem of estimating a function ( ) on the unit disk ( ): 2 + 2 1 , given a discrete and noisy data recorded on a regular square grid. An estimate of ( ) based on a class of orthogonal and complete functions over the unit disk is proposed. This class of functions has a distinctive property of being invariant to rotation of axes about the origin of coordinates yielding therefore...

This work is concerned with the development of inverse-type inequalities for piecewise polynomial functions and, in particular, functions belonging to hp-finite element spaces. The cases of positive and negative Sobolev norms are considered for both continuous and discontinuous finite element functions.The inequalities are explicit both in the local polynomial degree and the local mesh size.The...

Exact and approximate analytical solutions of linear and nonlinear singular two-point boundary value problems (BVPs) are obtained for the first time by the Legendre operational matrix of differentiation. Different from other numerical techniques, shifted Legendre polynomials and their properties are employed for deriving a general procedure for forming this matrix. The accuracy of the technique...

This paper presents a computational technique based on the Tau method and Legendre polynomials for the solution of a class of time-fractional telegraph equations. An appropriate representation of the solution via the Legendre operational matrix of fractional derivative is used to reduces its numerical treatment to the solution of a set of linear algebraic equations. The fractional derivatives a...

The main aim of this paper is the development of suitable bases (replacing the power basis x (n ∈ N≥0) which enable the direct series representation of orthogonal polynomial systems on non-uniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows to write solutions of arbitrary divided-difference equations in terms...

We introduce a new join operation on colored simplicial complexes that preserves the Cohen-Macaulay property. An example of this operation puts the connection between the central Delannoy numbers and Legendre polynomials in a wider context.

Apéry’s proof [13] of the irrationality of ζ(3) is now over 25 years old, and it is perhaps surprising that his methods have not yielded any significant new results (although further progress has been made on the irrationality of zeta values [1, 14]). Shortly after the initial proof, Beukers produced two elegant reinterpretations of Apéry’s arguments; the first using iterated integrals and Lege...

An efficient technique for the analysis of scattering by uniaxial anisotropic objects is presented. The technique is based on the method of higher order MoM of the surface integral equations. This higher order MoM solution uses the higher order hierarchical basis functions which are based on the modified Legendre polynomials. Numerical results are given to demonstrate that the higher order hier...

The finite Fourier transform of a family of orthogonal polynomials is the usual transform of these polynomials extended by 0 outside their natural domain of orthogonality. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families. 2010 Mathematics subject classification: primary 33C45; secondary 44A38, 33C47, 33C10, 42C10.

It is often the case that the exact moments of a statistic of the continuous type can be explicitly determined, while its density function either does not lend itself to numerical evaluation or proves to be mathematically intractable. The density approximants discussed in this article are based on the first n exact moments of the corresponding distributions. A unified semiparametric approach to...