A general theory is provided that allows to write multi-variate Fourier transforms or convolutions of radial functions as very simple univariate operations. As a byproduct, an interesting group of operators fI g 2IR with I + = I I = I I is deened. It contains the classical derivatives as I ?1 = d dr and is intimately connected to the Fourier transform.… (More)
In this paper, we use a kind of univariate multiquadric (MQ) quasi-interpolation to solve partial differential equation (PDE). We obtain the numerical scheme, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the temporal derivative of the… (More)
In this paper, we study several radial basis function approximation schemes in Sobolev spaces. We obtain an optional error estimate by using a class of smoothing operators. We also discussed sufficient conditions for the smoothing operators to attain the desired approximation order. We then construct the smoothing operators by some compactly supported… (More)
Based on the definition of MQ-B-Splines, this article constructs five types of univariate quasi-interpolants to non-uniformly distributed data. The error estimates and the shape-preserving properties are shown in details. And examples are shown to demonstrate the capacity of the quasi-interpolants for curve representation.
Viewing the classical Bernstein polynomials as sampling operators, we study a generalization by allowing the sampling operation to take place at scattered sites. We utilize both stochastic and deterministic approaches. On the stochastic side, we consider the sampling sites as random variables that obey some naturally derived probabilistic distributions, and… (More)