Quasi-Interpolation by Means of Filter-Banks

摘要

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 $varphi (t)$, specifically $f^{T}_{{rm approx}}(t)=sum a_{f}[n]varphi (t/T-n)$, 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 $O(T^{L})$ where $T$ is the sampling period. Recently, quasi-interpolation schemes using a discrete prefiltering of the samples $f(nT)$ to obtain the coefficients $a_{f}[n]$, have been proposed. They provide tight approximation with a low computational cost. In this work, we generalize considering rational filter bank-ns 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.