[Paper Review] Adaptive orthonormal systems for matrix-valued functions

Daniel Alpay, Fabrizio Colombo+2 more|arXiv (Cornell University)|Nov 24, 2015

Mathematical Analysis and Transform Methods24 references3 citations

TL;DR

This paper extends the adaptive Fourier decomposition (AFD) to matrix-valued functions in the Hardy space $\mathbf{H}_2^{p\times q}$ using a generalized maximum selection principle that adaptively selects points in the unit disc and orthogonal projections to construct orthonormal systems of matrix-valued Blaschke products. The method ensures fast convergence and preserves the property of nonnegative analytic instantaneous frequency, enabling efficient rational approximation and interpolation of matrix signals.

ABSTRACT

In this paper we consider functions in the Hardy space $\mathbf{H}_2^{p imes q}$ defined in the unit disc of matrix-valued. We show that it is possible, as in the scalar case, to decompose those functions as linear combinations of suitably modified matrix-valued Blaschke product, in an adaptive way. The procedure is based on a generalization to the matrix-valued case of the maximum selection principle which involves not only selections of suitable points in the unit disc but also suitable orthogonal projections. We show that the maximum selection principle gives rise to a convergent algorithm. Finally, we discuss the case of real-valued signals.

Motivation & Objective

• To generalize the adaptive Fourier decomposition (AFD) from scalar to matrix-valued functions in the Hardy space $\mathbf{H}_2^{p\times q}$.
• To develop an adaptive algorithm based on a matrix-valued extension of the maximum selection principle, incorporating both point selection and orthogonal projections.
• To ensure convergence of the decomposition to the target matrix-valued function while preserving the property of nonnegative analytic instantaneous frequency.
• To extend the theory to real-valued matrix signals by leveraging symmetry and conjugate symmetry in Fourier coefficients.
• To provide a framework for rational approximation and interpolation of matrix-valued functions using orthonormal systems derived from Blaschke products.

Proposed method

• Adaptively select matrix-valued Blaschke factors $B_k(z)$ using a generalized maximum selection principle that maximizes $|\langle f_k, e_{a_k} \rangle|^2 = (1 - |a_k|^2) \|f_k(a_k)\|^2$ over $a_k \in \mathbb{D}$.
• Define the $k$-th reduced remainder $f_k(z)$ via a generalized backward-shift operation: $f_k(z) = \frac{f_{k-1}(z) - \langle f_{k-1}, e_{a_{k-1}} \rangle e_{a_{k-1}}(z)}{\frac{z - a_{k-1}}{1 - z\overline{a_{k-1}}}}$.
• Construct orthonormal systems of matrix-valued Blaschke products $B_k(z)$ that are unimodular on the unit circle and orthogonal in $\mathbf{H}_2^{p\times q}$.
• Use the Beurling-Lax theorem to characterize backward-shift invariant subspaces as $\mathcal{B} \mathbf{H}_2^{p\times q}$, where $\mathcal{B}$ is a matrix-valued Blaschke product.
• Apply Leech’s factorization theorem to show that if a function $F$ is orthogonal to the range of a Blaschke product $\mathcal{B}$, then $\mathcal{B} = \Theta \Theta_1$ with $\Theta_1$ contractive and $\ell = p$.
• For real matrix signals, exploit the conjugate symmetry $F_{-n} = \overline{F_n}$ to express $F(e^{it}) = F_+(e^{it}) + \overline{F_+(e^{it})} - F_0$, enabling use of the AFD algorithm on the analytic part $F_+$.

Experimental results

Research questions

• RQ1 Can the adaptive Fourier decomposition (AFD) be generalized to matrix-valued functions in $\mathbf{H}_2^{p\times q}$ using a maximum selection principle that includes orthogonal projections?
• RQ2 Does the resulting orthonormal system of matrix-valued Blaschke products converge to the target function, and under what conditions is the convergence fast?
• RQ3 How can the property of nonnegative analytic instantaneous frequency be preserved in the matrix-valued case, and what does it imply for signal decomposition?
• RQ4 What is the structure of backward-shift invariant subspaces in the matrix-valued Hardy space, and how does it relate to the range of a matrix-valued Blaschke product?
• RQ5 How can the AFD algorithm be adapted to real-valued matrix signals by exploiting conjugate symmetry in the Fourier coefficients?

Key findings

• The generalized maximum selection principle ensures that the algorithm converges to the target matrix-valued function $f$ in $\mathbf{H}_2^{p\times q}$, with $\|f_{n+1}\| \to 0$ as $n \to \infty$.
• The decomposition $f(z) = \sum_{k=1}^\infty \langle f_k, e_{a_k} \rangle B_k(z)$ holds, where each $B_k$ is a matrix-valued Blaschke product and the system is orthonormal in $\mathbf{H}_2^{p\times q}$.
• The method preserves the property of nonnegative analytic instantaneous frequency for each component $B_k$, ensuring each term is mono-component in the matrix-valued sense.
• The backward-shift invariant subspace generated by the algorithm is of the form $\mathcal{B} \mathbf{H}_2^{p\times q}$, where $\mathcal{B}$ is a matrix-valued Blaschke product constructed from the selected points $a_k$ and projections.
• For real matrix signals, the analytic part $F_+$ lies in $\mathbf{H}_2^{p\times q}$, and the full signal can be reconstructed via $F(e^{it}) = F_+(e^{it}) + \overline{F_+(e^{it})} - F_0$, enabling AFD application to the analytic part.
• The use of Leech’s factorization theorem implies that if $F$ is orthogonal to $\mathcal{B} \mathbf{H}_2^{p\times q}$, then $\mathcal{B} = \Theta \Theta_1$ with $\Theta_1$ analytic and contractive, and $\ell = p$.