Generalized Interpolation

As an alternative algorithm, let us now consider the form f(x)=E <*?(* - k) VxeK*. (4)

The crucial difference between the classical formulation (1) and the generalized formulation (4) is the introduction of coefficients Ck in place of the sample values fk. This offers new possibilities, in the sense that interpolation can now be carried in two separate steps: firstly, the determination of coefficients Ck from the samples fk, and secondly, the determination of desired values f (x) from the coefficients Ck. The benefit of this separation is to allow for an extended choice of synthesis functions, some with better properties than those available in the restricted classical case where Ck = f^ The apparent drawback is the need for an additional step. We shall see later that this drawback is largely compensated by the gain in quality resulting from the larger selection of synthesis functions to choose from.

In the present approach, the synthesis function is not necessarily finite-support, nor is it required to satisfy the interpolation property; in return, it becomes essential to assign a specific value to those samples fk that are unknown because they are out of the range of our data. In practice, this assignment is implicit, and the unknown data are usually assumed to be mirrored from the known data.

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