problem (OCP) for systems governed by Volterra integro-differential (VID) equation is considered. The method is developed by means of the Legendre wavelet approximation and collocation method. The properties of Legendre wavelet together with Gaussian integration method are utilized to reduce the problem to the solution of nonlinear programming one. Some numerical examples are given to confirm t...

We construct wavelets on general n-dimensional domains or manifolds via a domain decomposition technique, resulting in so-called composite wavelets. With this construction, wavelets with supports that extend to more than one patch are only continuous over the patch interfaces. Normally, this limited smoothness restricts the possibility for matrix compression, and with that the application of th...

The purpose of this note is to highlight some of the unique properties of spline wavelets. These wavelets can be classified in four categories: othogonal (Battle-Lemarié), semi-orthogonal (e.g., B-spline), shift-orthogonal, and biorthogonal (Cohen-DaubechiesFeauveau). Unlike most other wavelet bases, splines have explicit formulae in both the time and frequency domain, which greatly facilitates...

The two-dimensional (2D) and three-dimensional (3D) orthogonal moments are useful tools for 2D and 3D object recognition and image analysis. However, the problem of computation of orthogonal moments has not been well solved because there exist few algorithms that can efficiently reduce the computational complexity. As is well known, the calculation of 2D and 3D orthogonal moments by a straightf...

We present a new unified kernel regression framework on manifolds. Starting with a symmetric positive definite kernel, we formulate a new bivariate kernel regression framework that is related to heat diffusion, kernel smoothing and recently popular diffusion wavelets. Various properties and performance of the proposed kernel regression framework are demonstrated. The method is subsequently appl...

We describe a method for the numerical evaluation of normalized versions of the associated Legendre functions P−μ ν and Q −μ ν of degrees 0 ≤ ν ≤ 1, 000, 000 and orders −ν ≤ μ ≤ ν on the interval (−1, 1). Our algorithm, which runs in time independent of ν and μ, is based on the fact that while the associated Legendre functions themselves are extremely expensive to represent via polynomial expan...

Legendre expansion of the ν ¯ ν ⇀ ↽ e + e − kernel: Influence of high order terms. Abstract We calculate the Legendre expansion of the rate of the process ν + ¯ ν ↔ e + + e − up to 3 rd order extending previous results of other authors which only consider the 0 th and 1 st order terms. Using different closure relations for the moment equations of the radiative transfer equation we discuss the p...