Duals of Frame Sequences

Frames provide unconditional basis-like, but generally nonunique, representations of vectors in a Hilbert space H. The redundancy of frame expansions allows the flexibility of choosing different dual sequences to employ in frame representations. In particular, oblique duals, Type I duals, and Type II duals have been introduced in the literature because of the special properties that they possess. A Type I dual is a dual such that the range of its synthesis operator is contained in the range of the synthesis operator of the original frame sequence, and a Type II dual is a dual such that the range of its analysis operator is contained in the range of the analysis operator of the original frame sequence. This paper proves that all Type I and Type II duals are oblique duals, but not conversely, and characterizes the existence of oblique and Type II duals in terms of direct sum decompositions of H, as well as characterizing when the Type I, Type II, and oblique duals will be unique. These results are also applied to the case of shift-generated sequences that are frames for shift-invariant subspaces of L2(ℝd).