On minimal scalings of scalable frames

Abstract

A tight frame in Rⁿ is a redundant system which has a reconstruction formula similar to that of an orthonormal basis. For a unit-norm frame F = {f_i}^k_i=1, a scaling is a vector c = (c(l),..., c(k)) ε R^k≥0 such that {c(i)f_i}^k_i=1 is a tight frame in Rⁿ. If a scaling c exists, we say that F is a scalable frame. A scaling c is a minimal scaling if {fi : c{i) > 0} has no proper scalable subframes. In this paper, we present the uniqueness of the orthogonal partitioning property of any set of minimal scalings and provide a construction of scalable frames by extending the standard orthonormal basis of Rⁿ.