Construction principles of tight wavelet frames in connection with linear system theory
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Date
2025
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Abstract
This dissertation explores the connection between two mathematical fields: wavelet analysis, a subfield of time-frequency analysis, and linear system theory, which has its origins in the modeling of real-world systems using dynamical systems. In the former, we focus on the construction of multivariate tight wavelet frames via the Unitary and Oblique Extension Principles. In the latter, we are particularly interested in the state-space approach and its ties to the Nevanlinna-Pick interpolation and operator theory.
In 2015, Scheiderer, Charina, Putinar, and Stöckler linked wavelet frames constructed via the Unitary Extension Principle to so-called realizations from linear system theory. This dissertation aims to generalize this connection to frames based on the Oblique Extension Principle, potentially aiding the construction and study of such frames.
After brief introductions to the two fields mentioned above and to results on the factorization of trigonometric polynomials, we first consider the problem in the univariate case. Here, we can solve the problem both by bringing it back to the setting of the Unitary Extension Principle and by using a generalized version of the Nevanlinna-Pick interpolation.
These univariate results lead to a multivariate result: Using the Kronecker product, we construct so-called separable multivariate frames and connect them to linear system theory for any dimension.
Finally, in the bivariate case, we have to restrict ourselves to a specific subset of frames. For this subset, we use a parameterized version of the univariate results to solve the bivariate problem and again describe connections to the Nevanlinna-Pick interpolation. We illustrate all constructions with examples.
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Keywords
Tight wavelet frames, Linear system theory, Oblique extension principle, Nevanlinna-Pick
Subjects based on RSWK
Wavelet-Analyse, Lineare Kontrolltheorie