Connection between p-frames and p-Riesz bases in locally finite SIS of LP(R)

Let 1 less than or equal to p less than or equal to infinity and Phi = (phi (1),...,phi (r))(T) be a vector-valued compactly supported L-P function on R-d. Define V-p(Phi) = {Sigma (r)(i=1) Sigma (j is an element of Zd) d(i)(j)phi (i)(. - j) : (d(i)(j))(j is an element of Zd) is an element ofl(p), 1 less than or equal to i less than or equal to r}. In this paper, we consider the p-frame property of the space V-p(Phi) with Phi being compactly supported function in L-p boolean AND Lp/(p-1). Moreover, for the one-dimensional case, we show that {phi (i)(. - j) : 1 less than or equal to i less than or equal to r,j is an element of Z) is a p-frame for V-p(Phi), then there exist a positive integer s less than or equal to r and compactly supported functions psi (1),..., psi (s) is an element of V-p(Phi) such that {psi (i)(. - j) : 1 less than or equal to i less than or equal to s,j is an element of Z} is a p-Riesz basis of V-p(Phi), where Psi = (psi (1),..., psi (s))(T).