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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.

We will show that the famous, intractible 1959 Kadison-Singer problem in C * -algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well as explaining why each of these areas has volumes of literature on their respective problems without a satisfactory resolution. In each of these areas we will reduce the problem to the minimum which needs to be proved to solve their version of Kadison-Singer. In some areas we will prove what we believe will be the strongest results ever available in the case that Kadison-Singer fails. Finally, we will give some directions for constructing a counter-example to Kadison-Singer.

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Abstract. We show that if the canonical dual of an affine frame has the affine structure, with the same number of generators, then the core subspace V 0 is shift invariant. We demonstrate, however, that the converse is not true. We apply our results in the setting of oversampling affine frames, as well as in computing the period of a Riesz wavelet, answering in the affirmative a conjecture of Daubechies and Han. Additionally, we completely characterize when the canonical dual of a quasi-affine frame has the quasi-affine structure.The study of affine systems in L 2 (R d ) which gives rise to frames is currently under vigorous research. Such systems can have better properties than systems which are bases [16], and can be utilized in denoising [2]. Since frames can have many duals which yield reconstruction, from a practical print of view it is important to know what structure the duals have. In particular, one wishes to know if a given affine frame has any duals with the affine structure.From a strictly theoretical viewpoint, one can ask if an affine frame generates a generalized multiresolution analysis (GMRA). A GMRA, as defined by Baggett, Medina, and Merrill [1], is a sequence {V j : j ∈ Z} of closed subspaces of L 2 (R d ) with the following five properties:Here, T z is a translation operator given by T z f (x) = f (x − z) and D is a dilation operator given by Df (x) = |det A| 1/2 f (Ax) for some d×d expansive integer-valued matrix A. In particular, for any l ∈ Z d we have D * T l D = T k for some k ∈ Z d .