This paper uses frame techniques to characterize the Schatten class properties of integral operators. The main result shows that if the coefficients {〈k,Φm,n〉} of certain frame expansions of the kernel k of an integral operator are in l, then the operator is Schatten p-class. As a corollary, we conclude that if the kernel or Kohn-Nirenberg symbol of a pseudodifferential operator lies in a parti...

Hilbert space frame theory has applications to various areas of pure mathematics, applied mathematics, and engineering. However, the question of how applying an invertible operator to a frame changes its properties has not yet been satisfactorily answered, and only partial results are known to date. In this paper, we will provide a comprehensive study of those questions, and, in particular, pro...

Let H be a Hilbert space. Given a bounded positive definite operator S on H, and a bounded sequence c = {ck}k∈N of non negative real numbers, the pair (S, c) is frame admissible, if there exists a frame {fk}k∈N on H with frame operator S, such that ‖fk‖ 2 = ck, k ∈ N. We relate the existence of such frames with the Schur-Horn theorem of majorization, and give a reformulation of the extended ver...

Spectral tetris is a flexible and elementary method to derive unit norm frames with a given frame operator having all of its eigenvalues ≥ 2. One important application of this method is to construct fusion frames. We will give necessary and sufficient conditions for a spectral tetris construction to give a fusion frame with prescribed eigenvalues for its fusion frame operator and with prescribe...

In this note we investigate the operators associated with frame sequences in a Hilbert space H, i.e., the synthesis operator T : l (N) → H , the analysis operator T ∗ : H → l (N) and the associated frame operator S = TT ∗ as operators defined on (or to) the whole space rather than on subspaces. Furthermore, the projection P onto the range of T , the projection Q onto the range of T ∗ and the Gr...

We present a comprehensive analysis of the convergence properties of the frame operators of Weyl-Heisenberg systems and shift-invariant systems, and relate these to the convergence of the Walnut representation. We give a deep analysis of necessary conditions and sufficient conditions for convergence of the frame operator. We show that symmetric, norm and unconditional convergence of the Walnut ...

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 whereH 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 posi...

If B < ∞ we call F = {fi}i∈I a Bessel sequence with Bessel bound B. If 0 < A ≤ B < ∞, then {fi}i∈I is a frame for K. If K 6= H we call {fi}i∈I a frame sequence in H. The largest A and the smallest B satisfying the above inequalities are called the optimal lower and upper frame bound and will be denoted A(F) and B(F) respectively. If A = B = λ we call this a λ-tight frame and if λ = 1 it is call...