Abstract We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, to approximate arbitrary functions. We show that the polynomial coefficients in Legendre expansion, thus, the whole series, converge to zero much more rapidly compared to the Taylor expansion of the same order. Furthermore, using numerical analysis with sixth-order polynomial expansion, we demonstrate ...

A numerical method based on Legendre multi-wavelets is applied for solving Lane-Emden equations which form Volterra integro-differential equations. The Lane-Emden equations are converted to Volterra integro-differential equations and then are solved by the Legendre multi-wavelet method. The properties of Legendre multi-wavelets are first presented. The properties of Legendre multi-wavelets are ...

Optimality Theory (henceforth OT) (Prince and Smolensky 1993/2004) is based upon lexicographic optimization. It differs in this respect from Harmonic Grammar (henceforth HG) (Legendre et al. 1990a, Legendre et al. 1990b), which is based upon linear numeric optimization. Differences between the two have been discussed in several places, including (Legendre et al. 2006, Pater et al. 2007a, Prince...

The main result of this paper is a homotopy theoretic classification of Legendre immersions from a compact manifold into a contact manifold. The paper also includes normal form theorems for Legendre submanifolds as well as a multi-jet transversality theorem for Legendre

In this paper, summation formulae for the 2-variable Legendre polynomials in terms of certain multi-variable special polynomials are derived. Several summation formulae for the classical Legendre polynomials are also obtained as applications. Further, Hermite-Legendre polynomials are introduced and summation formulae for these polynomials are also established.

We introduce a new and eecient Chebyshev-Legendre Galerkin method for elliptic problems. The new method is based on a Legendre-Galerkin formulation, but only the Chebyshev-Gauss-Lobatto points are used in the computation. Hence, it enjoys advantages of both the Legendre-Galerkin and Chebyshev-Galerkin methods.

Abstract— A new polynomial method to solve Volterra–Fredholm Integral equations is presented in this work. The method is based upon Shifted Legendre Polynomials. The properties of Shifted Legendre Polynomials and together with Gaussian integration formula are presented and are utilized to reduce the computation of Volterra–Fredholm Integral equations to a system of algebraic equations. Some num...

The Legendre-Stirling numbers were discovered in 2002 as a result of a problem involving the spectral theory of powers of the classical secondorder Legendre differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Legendre expression in Lagrangian symmetric form. Quite remarkably, they share many similar properties with the classical Stirling...

A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree N polynomial in O(N(logN)2/ log logN) operations is derived. The basis of the algorithm is to rewrite a well-known asymptotic formula for Legendre polynomials of large degree as a weighted linear combination of Chebyshev polynomials, which can then be evaluated by using the di...

A new method is proposed for fast and low-complexity computation of exact 3D Legendre moments. The proposed method consists of three main steps. In the first step, the symmetry property is employed where the computational complexity is reduced by 87%. In the second step, exact values of 3D Legendre moments are obtained by mathematically integrating the Legendre polynomials over digital image vo...