Sharpened Uncertainty Principle
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For any finite group G, any finite G-set X and any field F, we consider the vector space F^X of all functions from X to F. When the group algebra FG is semisimple and splitting, we find a specific basis X of F^X, construct the Fourier transform: F^X→ F^X, f↦f, and define the rank support (f); we prove that (f)= FGf, where FGf is the submodule of the permutation module FX generated by the element f=∑_x∈ Xf(x)x. Next, we extend a sharpened uncertainty principle for abelian finite groups by Feng, Hollmann, and Xiang [9] to the following extensive framework: for any field F, any transitive G-set X and 0≠ f∈ F^X we prove that | supp(f)|· FGf ≥ |X|+| supp(f)|-|X_ supp(f)|, where supp(f) is the support of f, and X_ supp(f) is a block of X associated with the subset supp(f) such that supp(f) is a disjoint union of some translations of the block. Then many (sharpened or classical) versions of finite-dimensional uncertainty principle are derived as corollaries.
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