Author Archives: xchenn

Implicit function theorem

Implicit funtion theorem states that if (or nonsingular), then near we can uniquely solve from the equation . Moreover smoothness of determines smoothness of . Usually at least is . A deep result is that in order to solve a continuous … Continue reading

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Invariant objects on compact groups

A compact space may not be 2nd countable, e.g. an uncountable space with finite-complement topology. If a compact hausdorff space is 2nd countable, then it is a Polish space, namely, a seperable complete metric space. All Polish spaces are Borel isomorphic. In … Continue reading

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Fubini-Tonelli principle

Fubini-Tonelli principle: Imagine that you have a saw-shaped “square” with very deep sawteeth and small area. If we integrate iteratedly, then in one order, in the inner integral we can only bound with and thus the estimate would be , … Continue reading

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Fourier analysis on LCA

Pontryagin duality compactness-discreteness duality time-frequency considerations Haar measure Bochner-Khinchin theorem, etc. Fourier inversion theorem Plencherel theorem decay-smoothness duality Paley-Wiener-Schwartz theorem uncertainty principle maximal ideals of Banach-algebra considerations Reference: Walter Rudin, Fourier analysis on groups.

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