Since is finite, , so the result holds for other spaces, as well.
If is continuous, then the convergence is uniform, yielding a proof of the Weierstrass theorem.
One consequence of the pointwise a.e. convergence is the uniqueness of Fourier coefficients: If with , then a.e. This follows from writing , which depends only on the Fourier coefficients.
A second consequence is that if exists a.e., then a.e., since Cesàro means converge to the original sequence limit if it exists.
^Hoffman, Kenneth (1988). Banach Spaces of Analytic Functions. Dover. p. 17. ISBN0-486-45874-1.
^Konigsberger, Konrad. Analysis 1 (in German) (6th ed.). Springer. p. 322.
April 06, 2024
fejér, kernel, mathematics, summability, kernel, used, express, effect, cesàro, summation, fourier, series, negative, kernel, giving, rise, approximate, identity, named, after, hungarian, mathematician, lipót, fejér, 1880, 1959, plot, several, contents, defini. In mathematics the Fejer kernel is a summability kernel used to express the effect of Cesaro summation on Fourier series It is a non negative kernel giving rise to an approximate identity It is named after the Hungarian mathematician Lipot Fejer 1880 1959 Plot of several Fejer kernels Contents 1 Definition 2 Properties 2 1 Convolution 3 See also 4 ReferencesDefinition editThe Fejer kernel has many equivalent definitions We outline three such definitions below 1 The traditional definition expresses the Fejer kernel Fn x displaystyle F n x nbsp in terms of the Dirichlet kernel Fn x 1n k 0n 1Dk x displaystyle F n x frac 1 n sum k 0 n 1 D k x nbsp where Dk x s kkeisx displaystyle D k x sum s k k rm e isx nbsp is the kth order Dirichlet kernel 2 The Fejer kernel Fn x displaystyle F n x nbsp may also be written in a closed form expression as follows 1 Fn x 1n sin nx2 sin x2 2 1n 1 cos nx 1 cos x displaystyle F n x frac 1 n left frac sin frac nx 2 sin frac x 2 right 2 frac 1 n left frac 1 cos nx 1 cos x right nbsp This closed form expression may be derived from the definitions used above The proof of this result goes as follows First we use the fact that the Dirichlet kernel may be written as 2 Dk x sin k 12 xsin x2 displaystyle D k x frac sin k frac 1 2 x sin frac x 2 nbsp Hence using the definition of the Fejer kernel above we get Fn x 1n k 0n 1Dk x 1n k 0n 1sin k 12 x sin x2 1n1sin x2 k 0n 1sin k 12 x 1n1sin2 x2 k 0n 1 sin k 12 x sin x2 displaystyle F n x frac 1 n sum k 0 n 1 D k x frac 1 n sum k 0 n 1 frac sin k frac 1 2 x sin frac x 2 frac 1 n frac 1 sin frac x 2 sum k 0 n 1 sin k frac 1 2 x frac 1 n frac 1 sin 2 frac x 2 sum k 0 n 1 sin k frac 1 2 x cdot sin frac x 2 nbsp Using the trigonometric identity sin a sin b 12 cos a b cos a b displaystyle sin alpha cdot sin beta frac 1 2 cos alpha beta cos alpha beta nbsp Fn x 1n1sin2 x2 k 0n 1 sin k 12 x sin x2 1n12sin2 x2 k 0n 1 cos kx cos k 1 x displaystyle F n x frac 1 n frac 1 sin 2 frac x 2 sum k 0 n 1 sin k frac 1 2 x cdot sin frac x 2 frac 1 n frac 1 2 sin 2 frac x 2 sum k 0 n 1 cos kx cos k 1 x nbsp Hence it follows that Fn x 1n1sin2 x2 1 cos nx 2 1n1sin2 x2 sin2 nx2 1n sin nx2 sin x2 2 displaystyle F n x frac 1 n frac 1 sin 2 frac x 2 frac 1 cos nx 2 frac 1 n frac 1 sin 2 frac x 2 sin 2 frac nx 2 frac 1 n frac sin frac nx 2 sin frac x 2 2 nbsp 3 The Fejer kernel can also be expressed as Fn x k n 1 1 k n eikx displaystyle F n x sum k leq n 1 left 1 frac k n right e ikx nbsp Properties editThe Fejer kernel is a positive summability kernel An important property of the Fejer kernel is Fn x 0 displaystyle F n x geq 0 nbsp with average value of 1 displaystyle 1 nbsp Convolution edit The convolution Fn is positive for f 0 displaystyle f geq 0 nbsp of period 2p displaystyle 2 pi nbsp it satisfies 0 f Fn x 12p ppf y Fn x y dy displaystyle 0 leq f F n x frac 1 2 pi int pi pi f y F n x y dy nbsp Since f Dn Sn f j nf jeijx displaystyle f D n S n f sum j leq n widehat f j e ijx nbsp we have f Fn 1n k 0n 1Sk f displaystyle f F n frac 1 n sum k 0 n 1 S k f nbsp which is Cesaro summation of Fourier series By Young s convolution inequality Fn f Lp p p f Lp p p for every 1 p for f Lp displaystyle F n f L p pi pi leq f L p pi pi text for every 1 leq p leq infty text for f in L p nbsp Additionally if f L1 p p displaystyle f in L 1 pi pi nbsp then f Fn f displaystyle f F n rightarrow f nbsp a e Since p p displaystyle pi pi nbsp is finite L1 p p L2 p p L p p displaystyle L 1 pi pi supset L 2 pi pi supset cdots supset L infty pi pi nbsp so the result holds for other Lp displaystyle L p nbsp spaces p 1 displaystyle p geq 1 nbsp as well If f displaystyle f nbsp is continuous then the convergence is uniform yielding a proof of the Weierstrass theorem One consequence of the pointwise a e convergence is the uniqueness of Fourier coefficients If f g L1 displaystyle f g in L 1 nbsp with f g displaystyle hat f hat g nbsp then f g displaystyle f g nbsp a e This follows from writing f Fn j n 1 j n f jeijt displaystyle f F n sum j leq n left 1 frac j n right hat f j e ijt nbsp which depends only on the Fourier coefficients A second consequence is that if limn Sn f displaystyle lim n to infty S n f nbsp exists a e then limn Fn f f displaystyle lim n to infty F n f f nbsp a e since Cesaro means Fn f displaystyle F n f nbsp converge to the original sequence limit if it exists See also editFejer s theorem Dirichlet kernel Gibbs phenomenon Charles Jean de la Vallee PoussinReferences edit Hoffman Kenneth 1988 Banach Spaces of Analytic Functions Dover p 17 ISBN 0 486 45874 1 Konigsberger Konrad Analysis 1 in German 6th ed Springer p 322 Retrieved from https en wikipedia org w index php title Fejer kernel amp oldid 1181680155, wikipedia, wiki, book, books, library,