Narrow Results

It is well known that orthonormal bases for a separable Hilbert space $H$ are precisely collections of the form ${\{\mathrm{\Theta }{e}_k\}}_{k\in 𝕀}$, where $\mathrm{\Theta }$ is a linear unitary operator acting on $H$ and ${\{e_k\}}_{k\in 𝕀}$ is a given orthonormal basis for $H$. We show that this is not true for the matrix-valued signal space $L^2(G,{ℂ}^{s\times r})$, $G$ is a locally compact abelian group which is $\sigma$-compact and metrizable, and $s$ and $r$ are positive integers. This problem is related to the adjointability of bounded linear operators on $L^2(G,{ℂ}^{s\times r})$. We show that any orthonormal basis of the space $L^2(G,{ℂ}^{s\times r})$ is precisely of the form ${\{{𝒰}{E}_k\}}_{k\in 𝕀}$, where ${𝒰}$ is a linear unitary operator acting on $L^2(G,{ℂ}^{s\times r})$ which is adjointable with respect to the matrix-valued inner product and ${\{E_k\}}_{k\in 𝕀}$ is a matrix-valued orthonormal basis for $L^2(G,{ℂ}^{s\times r})$.

Motivated by the analytic signal approach and general construction methods by Qian et al. (accepted by Adv. Comput. Math.), we construct a class of orthonormal bases for the real signal space , which have nonconstant physically meaningful instantaneous frequencies. We develop a fast algorithm for computing the Hilbert–Fourier expansion of a given function in terms of the orthonormal bases. Moreover, we study the approximation properties of the Hilbert–Fourier expansion. A numerical example is presented to demonstrate an adaptive Fourier expansion based on an optimal selection of the parameter a in the orthonormal bases according to the approximation error.

Building the mathematical foundation for the empirical mode decomposition is an important issue in adaptive data analysis. The task of building such a foundation consists of two stages. The first is to construct a large bank of basis functions for the time–frequency analysis of nonlinear and nonstationary signals. The second is to establish a fast adaptive decomposition algorithm. We survey recent mathematical progress on these two stages. Related results on piecewise linear spectral sequences and the Bedrosian identity are also reviewed.