Riesz–Fischer theorem

In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L² of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer.

For many authors, the Riesz–Fischer theorem refers to the fact that the L^p spaces from Lebesgue integration theory are complete.

The most common form of the theorem states that a measurable function on [–π, π] is square integrable if and only if the corresponding Fourier series converges in the space L². This means that if the Nth partial sum of the Fourier series corresponding to a square-integrable function f is given by

where F_n, the nth Fourier coefficient, is given by

then

where ${\displaystyle \left\Vert \cdot \right\|_{2}}$ is the L²-norm.

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