We study the orthogonal polynomial expansion on sparse grids for a function of d variables in a weighted L space. A fast algorithm is developed to compute the orthogonal polynomial expansion by combining the fast cosine transform, a fast transform from the Chebyshev orthogonal polynomial basis to the orthogonal polynomial basis for the weighted L space, and a fast algorithm of computing hierarc...

This paper presents a novel ECG signal measuring approach using compressive sensing method. The signal representing sparsity in any orthogonal basis can be well recovered using minimize L1 norm optimization, while satisfying the RIP condition for the measurement matrix  and orthogonal basis . First, based on this theorem, an analysis for evaluating the sparsity of ECG signal in orthogonal bas...

Shift-orthogonal wavelets are a new type of multiresolution wavelet bases that are orthogonal with respect to translation (or shifts) within one level but not with respect to dilations across scales. In this paper, we characterize these wavelets and investigate their main properties by considering two general construction methods. In the first approach, we start by specifying the analysis and s...

The purpose of this presentation is to describe a recent family of basis functions—the fractional B-splines—which appear to be intimately connected to fractional calculus. Among other properties, we show that they are the convolution kernels that link the discrete (finite differences) and continuous (derivatives) fractional differentiation operators. We also provide simple closed forms for the ...

A fast algorithm is developed to compute orthogonal polynomial expansions on sparse grids for a function of d variables in a weighted L space. The proposed algorithm combines the fast cosine transform, a fast transform from the Chebyshev orthogonal polynomial basis to the orthogonal polynomial basis for the weighted L space and a fast algorithm of computing hierarchically structured basis funct...

We present a simple, explicit orthogonal basis of eigenvectors for the Johnson and Kneser graphs, based on Young’s orthogonal representation of the symmetric group. Our basis can also be viewed as an orthogonal basis for the vector space of all functions over a slice of the Boolean hypercube (a set of the form {(x1, . . . , xn) ∈ {0, 1}n : ∑ i xi = k}), which refines the eigenspaces of the John...