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We prove the existence of tight frames whose elements lie on an arbitrary ellipsoidal surface within a real or complex separable Hilbert space H , and we analyze the set of attainable frame bounds. In the case where H is real and has finite dimension, we give an algorithmic proof. Our main tool in the infinite-dimensional case is a result we have proven which concerns the decomposition of a positive invertible operator into a strongly converging sum of (not necessarily mutually orthogonal) self-adjoint projections. This decomposition result implies the existence of tight frames in the ellipsoidal surface determined by the positive operator. In the real or complex finite dimensional case, this provides an alternate (but not algorithmic) proof that every such surface contains tight frames with every prescribed length at least as large as dim H . A corollary in both finite and infinite dimensions is that every positive invertible operator is the frame operator for a spherical frame.

Abstract. The construction of frames for a Hilbert space H can be equated to the decomposition of the frame operator as a sum of positive operators having rank one. This realization provides a different approach to questions regarding frames with particular properties and motivates our results. We find a necessary and sufficient condition under which any positive finite-rank operator B can be expressed as a sum of rank-one operators with norms specified by a sequence of positive numbers {ci}. Equivalently, this result proves the existence of a frame with B as it's frame operator and with vector norms { √ ci}. We further prove that, given a non-compact positive operator B on an infinite dimensional separable real or complex Hilbert space, and given an infinite sequence {ci} of positive real numbers which has infinite sum and which has supremum strictly less than the essential norm of B, there is a sequence of rank-one positive operators, with norms given by {ci}, which sum to B in the strong operator topology. These results generalize results by Casazza, Kovačević, Leon, and Tremain, in which the operator is a scalar multiple of the identity operator (or equivalently the frame is a tight frame), and also results by Dykema, Freeman, Kornelson, Larson, Ordower, and Weber in which {ci} is a constant sequence.

A shift-invariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications. This paper characterizes those Communicated by Karlheinz Groechenig. J Fourier Anal Appl (2010) 16: 60-75 61shift-invariant subspaces S that are also invariant under additional (non-integer) translations. For the case of finitely generated spaces, these spaces are characterized in terms of the generators of the space. As a consequence, it is shown that principal shift-invariant spaces with a compactly supported generator cannot be invariant under any non-integer translations.

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