- Fejér's theorem
In mathematics,

**Fejér's theorem**, named for Hungarianmathematician Lipót Fejér , states that if "f":**R**→**C**is acontinuous function with period 2π, then thesequence (σ_{"n"}) ofCesàro mean s of the sequence ("s"_{"n"}) ofpartial sum s of theFourier series of "f" converges uniformly to "f" on [-π,π] .Explicitly,:$s\_n(x)=sum\_\{k=-n\}^nc\_ke^\{ikx\},$where:$c\_n=frac\{1\}\{2pi\}int\_\{-pi\}^pi\; f(t)e^\{-int\}dt,$and:$sigma\_n(x)=frac\{1\}\{n\}sum\_\{k=0\}^\{n-1\}s\_k(x)=frac\{1\}\{2pi\}int\_\{-pi\}^pi\; f(x-t)F\_n(t)dt,$with "F"

_{"n"}being the "n"th orderFejér kernel .A more general form of the theorem applies to functions which are not necessarily continuous harv|Zygmund|1968|loc=Theorem III.3.4. Suppose that "f" is in "L"

^{1}(-π,π). If the left and right limits "f"("x"_{0}±0) of "f"("x") exist at "x"_{0}, or if both limits are infinite of the same sign, then:$sigma\_n(x\_0)\; o\; frac\{1\}\{2\}left(f(x\_0+0)+f(x\_0-0)\; ight).$

Existence or divergence to infinity of the Cesàro mean is also implied. By a theorem of

Marcel Riesz , Fejér's theorem holds precisely as stated if the (C, 1) mean σ_{"n"}is replaced with (C, α) mean of the Fourier series harv|Zygmund|1968|loc=Theorem III.5.1.**References***.

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**FEJÉR, LEOPOLD**— (1880–1959), Hungarian mathematician. Fejér was educated in Budapest and Berlin. He spent a year in Berlin where he met H.A. Schwarz who had a decisive influence on his mathematical career. He was appointed professor at Budapest in 1911 and… … Encyclopedia of Judaism**Fejér kernel**— In mathematics, the Fejér kernel is used to express the effect of Cesàro summation on Fourier series. It is a non negative kernel, giving rise to an approximate identity.The Fejér kernel is defined as :F n(x) = frac{1}{n} sum {k=0}^{n 1}D… … Wikipedia**Lipót Fejér**— Infobox Scientist box width = 300px name = Lipót (Leopold) Fejér image width = 200px caption = Lipót Fejér birth date = birth date|1880|2|9|mf=y birth place = Pécs, Hungary death date = death date and age|1959|10|15|1880|2|9|mf=y death place =… … Wikipedia**Riemann mapping theorem**— In complex analysis, the Riemann mapping theorem states that if U is a simply connected open subset of the complex number plane Bbb C which is not all of Bbb C, then there exists a biholomorphic (bijective and holomorphic) mapping f, from U, onto … Wikipedia**Convergence of Fourier series**— In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily a given… … Wikipedia**List of mathematics articles (F)**— NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… … Wikipedia**Fourier series**— Fourier transforms Continuous Fourier transform Fourier series Discrete Fourier transform Discrete time Fourier transform Related transforms … Wikipedia**Least squares**— The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. Least squares means that the overall solution minimizes the sum of… … Wikipedia**Trigonometric polynomial**— In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n a natural number. The coefficients may be taken as real numbers, for… … Wikipedia**Cesàro summation**— For the song Cesaro Summability by the band Tool, see Ænima. In mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. If the series converges in the usual sense to a sum A, then the series is… … Wikipedia