Functions of one or more variables are usually approximated with a basis; a complete, linearly independent set of functions that spans an appropriate function space. The topic of this paper is the numerical approximation of functions using the more general notion of frames; that is, complete systems that are generally redundant but provide stable infinite representations. While frames are well-...

Several variations are given on the construction of orthonormal bases of wavelets with compact support. They have, respectively, more symmetry, more regularity, or more vanishing moments for the scaling function than the examples constructed in Daubechies [Comm.

This paper considers the design of wavelet tight frames based on iterated oversampled filter banks. The greater design freedom available makes possible the construction of wavelets with a high degree of smoothness, in comparison with orthonormal wavelet bases. In particular, this paper takes up the design of systems that are analogous to Daubechies orthonormal wavelets—that is, the design of mi...

We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75,1998, pp 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condit...

Two orthonormal bases B andB′ of a d-dimensional complex inner-product space are called mutually unbiased if and only if |〈b|b〉| = 1/d holds for all b ∈ B and b′ ∈ B′. The size of any set containing pairwise mutually unbiased bases of C cannot exceed d + 1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of thi...

We give characterizations of orthogonal families, tight frames and orthonormal bases of Gabor systems. The conditions we propose are stated in terms of equations for the Fourier transforms of the Gabor system’s generating functions.

We consider a sequence HN of finite-dimensional Hilbert spaces of dimensions dN → ∞. Motivating examples are eigenspaces, or spaces of quasi-modes, for a Laplace or Schrödinger operator on a compact Riemannian manifold. The set of Hermitian orthonormal bases of HN may be identified with U(dN), and a random orthonormal basis of is a choice of a random sequence UN∈U(dN) from the product of normal...