Determination of the Coefficients

Suppose we want to enforce a condition akin to (2), in the context of generalized interpolation. Considering again only integer arguments x = k0, we write fk0 = E CkPK-k Vko eZ*, (5)

where pk = ^(k). Given some function ^ that is known a priori, this expression is nothing but a linear system of equations in terms of the unknown coefficients Ck. According to (5), the dimension of this system is infinite, both with respect to the number of equations (because all arguments k0 e Z* are considered) and with respect to the number of unknowns (because all indexes k e Z* are considered in the sum). One way to reduce this system to a manageable size is to remember that the number of known samples k0 is finite in practice (which limits the number of equations), and at the same time to constrain ^ to be finite-support (which limits the number of unknowns). We are now faced with a problem of the form c = P-1/, and a large part of the literature (e.g., [30]) is devoted to the development of efficient techniques for inverting the matrix P in the context of specific synthesis functions

Another strategy arises once it is recognized that (5) is again a discrete convolution equation that can be written as fko = (c *p)k0 Vko e Z*. (6)

It directly follows that the infinite sequence of coefficients {Ck} can be obtained by convolving the infinite sequence {/¿} by the convolution-inverse (p)-1. The latter is simply a sequence of numbers {(p)-1} such that (p*(p)-1)k = ^k. This sequence is uniquely defined and does generally exist in the cases of interest. Convolving both sides of (6) by (p)-1, we get that

Since discrete convolution is nothing but a digital filtering operation, this suggests that discrete filtering can be an alternative solution to matrix inversion for the determination of the sequence of coefficients {Ck} needed to enforce the desirable constraint (5). A very efficient algorithm for performing this computation for an important class of synthesis functions can be found in [19,20]; its computational cost for the popular cubic B-spline is two additions and three multiplications per produced coefficient.

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