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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
Posted in Analysis
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Invariant objects on compact groups
A compact space may not be 2nd countable, e.g. an uncountable space with finitecomplement 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
Posted in Spaces
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FubiniTonelli principle
FubiniTonelli principle: Imagine that you have a sawshaped “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
Posted in Analysis
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Fourier analysis on LCA
Pontryagin duality compactnessdiscreteness duality timefrequency considerations Haar measure BochnerKhinchin theorem, etc. Fourier inversion theorem Plencherel theorem decaysmoothness duality PaleyWienerSchwartz theorem uncertainty principle maximal ideals of Banachalgebra considerations Reference: Walter Rudin, Fourier analysis on groups.
Posted in Fourier analysis
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