Set of Uniqueness
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High Quality Content by WIKIPEDIA articles! In mathematics, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series. Their study is a relatively pure branch of harmonic analysis. The empty set is a set of uniqueness. This is just a fancy way to say that if a trigonometric series converges to zero everywhere then it is trivial. This was proved by Riemann, using a delicate technique of double formal integration; and showing that the resulting sum has some generalized kind of second derivative using Toeplitz operators. Later on, Cantor genera...
High Quality Content by WIKIPEDIA articles! In mathematics, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series. Their study is a relatively pure branch of harmonic analysis. The empty set is a set of uniqueness. This is just a fancy way to say that if a trigonometric series converges to zero everywhere then it is trivial. This was proved by Riemann, using a delicate technique of double formal integration; and showing that the resulting sum has some generalized kind of second derivative using Toeplitz operators. Later on, Cantor generalized Riemann's techniques to show that any countable, closed set is a set of uniqueness, a discovery which led him to the development of set theory. Interestingly, Paul Cohen, another great innovator in set theory, started his career with a thesis on sets of uniqueness.
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