Construction of Frames for Shift-Invariant Spaces

We construct a sequence {phi(i)(. - j) vertical bar j is an element of Z, i = 1,...,r} which constitutes a p-frame for the weighted shift-invariant space V-mu(p)(Phi) = {Sigma(r)(i=1) Sigma(j is an element of z) c(i)(j)phi(i)(. - j)vertical bar {c(i)(j)}(j is an element of z) is an element of l(mu)(p), i = 1,...,r}, p is an element of [1, infinity], and generates a closed shift-invariant subspace of L-mu(p)(R). The first construction is obtained by choosing functions. phi(i), i = 1,...,r with compactly supported Fourier transforms (phi) over cap (i), i = 1,...,r. The second construction, with compactly supported phi(i), i = 1,...,r, gives the Riesz basis.