Core of Fejér

In Mathematical, in analyzes functional and harmonic, the core of Fejér is a function making it possible to calculate the sums partial of Fejér.

Mathematical expression

One can establish the expression of the core of Fejér starting from the Noyau of Dirichlet D_n :

$D\_n\; \backslash \; left\; (U\; \backslash \; right)\; =\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; pi\}\; \backslash \; sum\_\; \{k=-n\}\; ^\; \{N\}\; e^\; \{-\; iku\}$

\ frac {1} {2 \ pi} \ frac {\ sin {\ left (\ left (N + \ frac {1} {2} \ right) U \ right)}}{\sin{u/2}}

Then the core of Fejér of order N is:

$F\_n\; \backslash \; left\; (U\; \backslash \; right)\; =\; \backslash \; frac\; \{1\}\; \{N\}\; \backslash \; sum\_\; \{k=0\}\; ^\; \{n-1\}\; D\_k\; \backslash \; left\; (U\; \backslash \; right)\; =\; \backslash \; frac\; \{1\}\; \{2\; N\; \backslash \; pi\}\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; sin\; \{N\; u/2\}\}\; \{\backslash \; sin\; \{u/2\}\}\; \backslash \; right)\; ^2$

It is shown whereas one obtains the sum partial of Fejér of order N of a function F (continuous by pieces and 2π-periodical) by carrying out a Produit convolution of F by the core of Fejér.

Properties

The function $F\_n$ is of constant Intégrale, equalizes to 1. For all ε>0, F_n (ε) is cancelled when $n\; \backslash \; to\; \backslash \; infty$. The continuation $\backslash \; left\; (F\_n\; \backslash \; right)\; \_\; \{N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\}$, restricted with $-\; \backslash \; pi;\; \backslash \; pi$ is thus a Suite of Dirac.

See too

• Somme of Fejér;
• Core of Dirichlet;
• Fourier series;
• Produces convolution;

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