ABSTRACT
The classical Shannon sampling theorem given by the formula [Sh]
»/v v-> »/ sin ait — nT) m „. / ( * ) = £ / ( n r ) a ( > _ n T ) \ T = 7r/(T, (9.1)
holds for cr-band limited signals, i.e., continuous functions in L 2 ( M ) whose Fourier transform has support in [— a,a]. I f a is taken to be a — 2 m 7r , and cp(t) — siwRt/irt is the scaling function of Example 5, then (9.1) is a statement about the elements of the subspaces Vm in the multiresolution analysis. Many generalizations of this theorem have been proposed. They usually involve transforms other than the Fourier transform [B-S-S], [H], and can be put into a reproducing kernel Hilbert space setting [N-W].
