Abstract
In this work, we establish various functional-analytical features of these decompositions and demonstrate their applicability to wavelet and gabor systems. First, we demonstrate the stability of atomic decompositions and frames under minor perturbations. This is motivated by analogous classical perturbation outcomes for bases, such as the Kato perturbation theorem and the Paley-Wiener basis stability requirements. The methodological contributions are concentrated on creating confidence bands, change-point tests, and two-sample tests because these procedures seem appropriate for the suggested situation. The reason for its selection and analysis in the thesis is its connection to operators. Frame sequences are created, and an investigation is conducted into a class of operators connected to a specific Bessel sequence, which turns it into a frame for every operator in the class.
Published Version
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