This article introduces a numerical method based on an M(n+ 1) set of general, hybrid orthonormal Bernstein functions coupled with Block-Pulse Functions(HOBB) on the interval [0,1] for approximating solutions of a Coupled System of linear and non linear Volterra integral and Integro-Differential equations. This method reduces a Coupled System of Volterra integral and IntegroDifferential equatio...

We consider rank-$1$ lattices for integration and reconstruction of functions with series expansion supported on a finite index set. explore the connection between periodic Fourier space non-periodic cosine Chebyshev space, via tent transform then transform, to transfer known results from setting into new insights settings. Fast discrete can be applied phase. To reduce size auxiliary set in ass...

In the first part of this paper we construct an algorithm for implementing the discrete wavelet transform by means of matrices in SO2(R) for orthonormal compactly supported wavelets and matrices in SLm(R),m ≥ 2, for compactly supported biorthogonal wavelets. We show that in 1 dimension the total operation count using this algorithm can be reduced to about 50% of the conventional convolution and...

A novel whiteness test of residuals is proposed, which makes use generalized bases orthonormal transfer functions. It can be viewed as a robustified version the classical in sense that it reduces risk type II errors, by introducing frequency weighting assessment flatness residual power spectrum density. This weighting, depends on basis poles, employed for validation reduced order models, when d...

We construct an explicit orthonormal basis of piecewise i+1Fi hypergeometric polynomials for the Alpert multiresolution analysis. The Fourier transform of each basis function is written in terms of 2F3 hypergeometric functions. Moreover, the entries in the matrix equation connecting the wavelets with the scaling functions are shown to be balanced 4F3 hypergeometric functions evaluated at 1, whi...

We use the concept of multiresolution analysis and orthonormal basis of wavelet functions to construct an estimator for an unknown density in a given Sobolev’s space. We show consistency with the rate of convergence equal to the optimal minimax rate of convergence. We also discuss some of the features of this new method of estimation on a numerical example.