A scheme for constructing orthogonal systems of bivariate polynomials in the Bernstein–Bézier form over triangular domains is formulated. The orthogonal basis functions have a hierarchical ordering by degree, facilitating computation of least-squares approximations of increasing degree (with permanence of coefficients) until the approximation error is subdued below a prescribed tolerance. The o...

We present some comments to the paper "Orthogonal basis with a conicoid first mode for shape specification of optical surfaces: comment."

two latin squares of order $n$ are orthogonal if in their superposition, each of the$n^{2}$ ordered pairs of symbols occurs exactly once. colbourn, zhang and zhu, in a seriesof papers, determined the integers $r$ for which there exist a pair of latin squares oforder $n$ having exactly $r$ different ordered pairs in their superposition. dukes andhowell defined the same problem for latin squares ...

Suppose we have observations yi = si+zi, i = 1; :::; n, where (si) is signal and (zi) is i.i.d. Gaussian white noise. Suppose we have available a library L of orthogonal bases, such as the Wavelet Packet bases or the Cosine Packet bases of Coifman and Meyer. We wish to select, adaptively based on the noisy data (yi), a basis in which best to recover the signal (\de-noising"). Let Mn be the tota...

The order of a wavelet transform is typically given by the number of vanishing moments of the analysis wavelet. The Strang-Fix conditions imply that the error for an orthogonal wavelet approximation at scale a i = − 2 globally decays as a , where L is the order of the transform. This is why, for a given number of scales, higher order wavelet transforms usually result in better signal approximat...

The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function Wγ(x) = x γ1 1 · · ·x γd d (1− |x|)d+1 when all γi > −1 and they are eigenfunctions of a second order partial differential operator Lγ . The singular cases that some, or all, γ1, . . . , γd+1 are −1 are studied in this paper. Firstly a complete basis of polynomials that are eigenfunctions of Lγ ...

This paper introduces a new set of orthogonal moment Functions-franklin Moments (FMs). The kernel functions of FMs is Franklin set, which is a class of complete orthogonal spline function set of degree 1. The implementation of FMs does not involve any numerical approximation and has a rather low computation complexity, since the basis set has the advantages of lower order. These properties make...

In this paper, a triangular spectral element method using orthogonal rational functions as basis is presented for problems on complex domains.

One-way wave equations conveniently describe wave propagation in media with discontinuous and/or rapid variations in one direction, but with smooth and slow variations in the complementary transverse directions. In the past, reciprocity theorems have been developed in terms of one-way wave fields. The boundaries of the integration volumes and the variations of the medium parameters must adhere ...

Abstract Orthogonal polynomials in two variables on cubic curves are considered. For an integral with respect to appropriate weight function defined a curve, explicit basis of orthogonal is constructed terms families one variable. We show that these can be used approximate functions and square root singularities, demonstrate their usage for solving differential equations singular solutions.