Interpolation Constraint

Consider the evaluation of (1) in the specific case when all coordinates of x = k0 are integer:

This equation is known as the interpolation constraint. Perhaps the single most important point of this whole chapter about interpolation is to recognize that (2) is a discrete convolution. Then, we can rewrite Eq. (2) as

where we have introduced the notation pk = ^¡nt(k) to put a heavy emphasis on the fact that we only discuss convolution between sequences that have the crucial property of being discrete. By contrast, (1) is not a convolution in this sense, because there ^int is evaluated at possibly noninteger values. From now on, we shall use /k to describe the samples of /, and pk to describe the values taken by for integer argument. Clearly we have that (2) is equivalent to pk = ¿k, where Sk is the

Kronecker symbol that is characterized by a central value = 1, and by zero values everywhere else. This function of integer argument takes the role of the neutral element for a discrete convolution.

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