Harmonic Analysis and Qualitative Uncertainty Principle

TL;DR

This paper explores the mathematical foundations of the qualitative uncertainty principle (QUP) in harmonic analysis, linking it to integral kernel properties and examining its validity across various transforms and potential bypass methods.

Contribution

It establishes a complete point theory for integral kernels related to QUP, analyzing conditions for its validity and limitations in harmonic analysis.

Findings

QUP depends on the existence of complete points in integral kernels.

QUP holds only for well-behaved integral operators.

Sparse representation and nonlinear methods may bypass QUP limitations.

Abstract

This paper investigates the mathematical nature of qualitative uncertainty principle (QUP), which plays an important role in mathematics, physics and engineering fields. Consider a 3-tuple (K, H1, H2) that K: H1 -> H2 is an integral operator. Suppose a signal f in H1, {\Omega}1 and {\Omega}2 are domains on which f, Kf define respectively. Does this signal f vanish if |{\Sigma}(f)|<|{\Omega}1|and|{\Sigma}(Kf)|<|{\Omega}2|? The excesses and deficiencies of integral kernel K({\omega}, t) are found to be greatly related to this general formulation of QUP. The complete point theory of integral kernel is so established to deal with the QUP. This theory addresses the density and linear independence of integral kernels. Some algebraic and geometric properties of complete points are presented. It is shown that the satisfaction of QUP depends on the existence of some complete points. By recognizing complete points of their corresponding integral kernels, the QUP with Fourier transform, Wigner-Ville distribution, Gabor transform and wavelet are studied. It is shown the QUP only holds for good behaved integral operators. An investigation of full violation of QUP shows that L2 space is large for high resolution harmonic analysis. And the invertible linear integral transforms whose kernels are complete in L2 probably lead to the satisfaction of QUP. It indicates the performance limitation of linear integral transforms in harmonic analysis. Two possible ways bypassing uncertainty principle, nonlinear method and sparse representation, are thus suggested. The notion of operator family is developed and is applied to understand remarkable performances of recent sparse representation.