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 Lp spaces L p {\displaystyle L^{p}} from Lebesgue integration theory are complete.
Modern forms of the theorem The most common form of the theorem states that a measurable function on [ − π , π ] {\displaystyle [-\pi ,\pi ]} is square integrable if and only if the corresponding Fourier series converges in the Lp space L 2 . {\displaystyle L^{2}.} This means that if the Nth partial sum of the Fourier series corresponding to a square-integrable function f is given by
S N f ( x ) = ∑ n = − N N F n e i n x , {\displaystyle S_{N}f(x)=\sum _{n=-N}^{N}F_{n}\,\mathrm {e} ^{inx},}
where F n , {\displaystyle F_{n},} the nth Fourier coefficient, is given by
F n = 1 2 π ∫ − π π f ( x ) e − i n x d x , {\displaystyle F_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)\,\mathrm {e} ^{-inx}\,\mathrm {d} x,}
then
lim N → ∞ ‖ S N f − f ‖ 2 = 0 , {\displaystyle \lim _{N\to \infty }\left\Vert S_{N}f-f\right\|_{2}=0,}
where ‖ ⋅ ‖ 2 {\displaystyle \|\,\cdot \,\|_{2}} is the L 2 {\displaystyle L^{2}} -norm. Conversely, if { a n } {\displaystyle \{a_{n}\}\,} is a two-sided sequence of complex numbers (that is, its indices range from negative infinity to positive infinity) such that
∑ n = − ∞ ∞ | a n | 2 < ∞ , {\displaystyle \sum _{n=-\infty }^{\infty }\left|a_{n}\right\vert ^{2}<\infty ,}
then there exists a function f such that f is square-integrable and the values a n {\displaystyle a_{n}} are the Fourier coefficients of f. This form of the Riesz–Fischer theorem is a stronger form of Bessel's inequality, and can be used to prove Parseval's identity for Fourier series. Other results are often called the Riesz–Fischer theorem (Dunford & Schwartz 1958, §IV.16). Among them is the theorem that, if A is an orthonormal set in a Hilbert space H, and x ∈ H , {\displaystyle x\in H,} then
⟨ x , y ⟩ = 0 {\displaystyle \langle x,y\rangle =0}
for all but countably many y ∈ A , {\displaystyle y\in A,} and
∑ y ∈ A | ⟨ x , y ⟩ | 2 ≤ ‖ x ‖ 2 . {\displaystyle \sum _{y\in A}|\langle x,y\rangle |^{2}\leq \|x\|^{2}.} Furthermore, if A is an orthonormal basis for H and x an arbitrary vector, the series
∑ y ∈ A ⟨ x , y ⟩ y {\displaystyle \sum _{y\in A}\langle x,y\rangle \,y}
converges commutatively (or unconditionally) to x. This is equivalent to saying that for every ε > 0 , {\displaystyle \varepsilon >0,} there exists a finite set B 0 {\displaystyle B_{0}} in A such that
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