Quasi-Interpolation by Means of Filter-Banks

Abstract

We consider the problem of approximating a regular function f(t) from its samples, f(nT), taken in a uniform grid. Quasi-interpolation schemes approximate f(t) with a dilated version of a linear combination of shifted versions of a kernel ¿(t), specifically f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">approx</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> (t) = ¿a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> [n]¿(t/T - n), in a way that the polynomials of degree at most L-1 are recovered exactly. These approximation schemes give order L, i.e., the error is O(T <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sup> ) where T is the sampling period. Recently, quasi-interpolation schemes using a discrete prefiltering of the samples f(nT) to obtain the coefficients a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> [n], have been proposed. They provide tight approximation with a low computational cost. In this work, we generalize considering rational filter banks to prefilter the samples, instead of a simple filter. This generalization provides a greater flexibility in the design of the approximation scheme. The upsampling and downsampling ratio r of the rational filter bank plays a significant role. When r = 1, the scheme has similar characteristics to those related to a simple filter. Approximation schemes corresponding to smaller ratios give less approximation quality, but, in return, they have less computational cost and involve less storage load in the system.