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Extensions of sampling theory : sampling on spaces of homogeneous type and sampling along curves

Abstract : In this thesis we study different variations of sampling inequalities. First,mirroring a result in [56], we give the conditions for sampling-like inequalities for Besov functions on compact Riemannian manifolds and spaces of homogeneous type. The techniques used to prove these results are based on the decomposition of smooth functions into wavelets available in both of these settings. Further, as in the euclidean case, this characterization through a wavelet expansion allows us to deepen the study of Besov spaces, obtaining a trace theorem and results about their local regularity (inspired in the strategies developed in [21, 54]). Finally we shift to work within the classic setting of sampling theory but changing the way samples are taken: instead of taking a discrete set of points we consider certain type of curves. In particular we determine the Nyquist rate for spirals when sampling bandlimited functions. We then show that, below this rate, the amount of undersampling that compressible signals admit when sampled along spirals is limited.
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Submitted on : Monday, October 5, 2020 - 12:46:19 PM
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Felipe Negreira. Extensions of sampling theory : sampling on spaces of homogeneous type and sampling along curves. Numerical Analysis [cs.NA]. Université de Bordeaux, 2020. English. ⟨NNT : 2020BORD0051⟩. ⟨tel-02957686⟩

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