Riesz?Fischer theorem

In mathematics, the Riesz?Fischer theorem in real analysis states that a function is square integrable if and only if the corresponding Fourier series converges uniformly in the space L^2.

Related Topics:
Mathematics - Real analysis - Square integrable - If and only if - Fourier series - Converges uniformly - Space

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This means that if the Nth partial sum of the Fourier series corresponding to a function f is given by

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:S_N f(x) = sum_{n=-N}^{N} F_n ,e^{inx},

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where F_n, the nth Fourier coefficient, is given by

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:F_n =rac{1}{2pi}int_{-pi}^pi f(x),e^{-inx},dx,

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then

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:lim_{n o infty} left Vert S_n f - f ight | = 0,

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where left Vert g ight | is the L^2-norm, expressed as

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:left Vert g ight | = int_{-2 pi}^{2 pi} g^2, dx.

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Conversely, if left { a_n ight } quad is a two-sided sequence of complex numbers (that is, its indices range from negative infinity to positive infinity) such that

Related Topics:
Sequence - Complex number - Indices - Infinity

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:sum_{n=1}^infty left | a_n ight ert^2 < infty,

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then there exists a function f such that f is square-integrable and the values a_n are the Fourier coefficients of f.

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The Riesz?Fischer theorem is a stronger form of Bessel's inequality, and can be used to prove Parseval's identity.

Related Topics:
Bessel's inequality - Parseval's identity

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Hungarian mathematician Frigyes Riesz and Austrian mathematician Ernst Fischer, working independently, both discovered the theorem in 1907.

Related Topics:
Hungarian - Frigyes Riesz - Austria - Ernst Fischer

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