Core of Dirichlet

See also: Core

In the theory of Fourier series, the core of Dirichlet is a trigonometrical Polynôme which in particular makes it possible to improve convergence of Fourier series. It has as expressions:

$D\_n\; (X)\; =\; \backslash \; sum\_\; \{k=-n\}\; ^n\; e^\; \{ikx\}\; =1+2\; \backslash \; sum\_\; \{k=1\}\; ^n\; \backslash \; cos\; (kx)\; =\; \backslash \; frac\; \{\backslash \; sin\; \backslash \; left\; (\backslash \; left\; (n+\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; right)\; X\; \backslash \; right)\}\{\backslash \; sin\; (x/2)\}\backslash ,$. For this last expression, the fraction is not defined in the whole multiple points of $2\; \backslash \; pi$, but is prolonged by continuity (and in manner $C^\; \{\backslash \; infty\}$) by the values: $D\_n\; (2k\; \backslash \; pi)\; =2n+1\; \backslash ,$.

The core of Dirichlet also intervenes in Optique, to give an account of the fringes and the compositions of coherent waves.

Elementary considerations

Equivalence of the two writings of the core of Dirichlet

The trigonometrical Identité which appears at the beginning of the article can be established by the calculation of a sum of a geometrical Série of reason $e^\; \{ix\}$.

Let us recall that the sum partial with the row N of a geometrical Série of reason r≠1 is worth

$\backslash \; sum\_\; \{k=0\}\; ^n\; has\; r^k=a\; \backslash \; frac\; \{1-r^\; \{n+1\}\}\; \{1-r\}.$

Here it is a symmetrical sum which interests us

$\backslash \; sum\_\; \{k=-n\}\; ^n\; r^k=\; \backslash \; sum\_\; \{k=0\}\; ^\; \{2n\}\; r^\; \{-\; N\}\; r^k=r^\; \{-\; N\}\; \backslash \; cdot\; \backslash \; frac\; \{1-r^\; \{2n+1\}\}\; \{1-r\}.$

The expression on the left of the equal symbol encourages us to think that the sum is a symmetrical function of R and 1 R . But in the expression on the right of the equal symbol, it is difficult to diagnose such a symmetry compared to these two quantities. The remedy is to multiply at the same time the numerator and the denominator by R ^-1/2, to obtain

$\backslash \; frac\; \{r^\; \{-\; n-1/2\}\}\; \{r^\; \{-\; 1/2\}\}\; \backslash \; cdot\; \backslash \; frac\; \{1-r^\; \{2n+1\}\}\; \{1-r\}\; =\; \backslash \; frac\; \{r^\; \{-\; n-1/2\}\; -\; r^\; \{n+1/2\}\}\; \{r^\; \{-\; 1/2\}\; -\; r^\; \{1/2\}\}.$

If R = E ^ix we have:

$\backslash \; sum\_\; \{k=-n\}\; ^n\; e^\; \{ikx\}\; =\; \backslash \; frac\; \{e^\; \{-\; (n+1/2)\; ix\}\; -\; e^\; \{(n+1/2)\; ix\}\}\; \{e^\; \{-\; ix/2\}\; -\; e^\; \{ix/2\}\}\; =\; \backslash \; frac\; \{-\; 2i\; \backslash \; sin\; ((n+1/2)\; X)\}\{-\; 2i\; \backslash \; sin\; (x/2)\}$

and then “-2 I ” disappears.

Alternative

Another guide for calculation can be the following idea: the calculation of the sum or the difference in two of the same complexes modulates is done by introducing the angle half

$e^\; \{I\; \backslash \; alpha\}\; -\; e^\; \{I\; \backslash \; beta\}\; =2ie^\; \{\backslash \; frac\; i2\; (\backslash \; alpha+\; \backslash \; beta)\}\backslash \; sin\; \backslash \; left\; (\backslash \; frac12\; (\backslash \; alpha\; \backslash \; beta)\; \backslash \; right)$

One proceeds thus to the numerator and the denominator.

The case $e^\; \{ix\}\; 1$

The equality of the two expressions was established for $e^\; \{ix\}\; \backslash \; neq\; 1$, i.e. for values of X not multiples of $2\; \backslash \; pi$.

The expression $\backslash \; frac\; \{\backslash \; sin\; ((n+1/2)\; X)\}\{\backslash \; sin\; (x/2)\}$ can thus be prolonged by continuity in any multiple point of $2\; \backslash \; pi$

$\backslash \; lim\; \backslash \; limits\_\; \{X\; \backslash \; to\; 2k\; \backslash \; pi\}\; \backslash \; frac\; \{\backslash \; sin\; ((n+1/2)\; X)\}\{\backslash \; sin\; (x/2)\}\; =\; 2n+1$

Properties of the core of Dirichlet

• It is a trigonometrical polynomial, therefore a function $\backslash \; mathcal\; C^\; \backslash \; infty$, 2π periodic.

• It is even
• Its value Moyenne is 1;
• asymptotic behavior of the standard of convergence on average

$\backslash |D\_n\; \backslash |\_1=\; \backslash \; frac1\; \{2\; \backslash \; pi\}\; \backslash \; int\_\; \{-\; \backslash \; pi\}\; ^\; \{\backslash \; pi\}\; |D\_n\; (T)|\; D\; T\; =\; \backslash \; frac4\; \{\backslash \; pi^2\}\; \backslash \; ln\; n+O\; (1)$

Associated operator

N - ième term of the Fourier series of a function F 2π periodic integrable is written:

$S\_n\; (F)\; (X)\; =\; \backslash \; frac1\; \{2\; \backslash \; pi\}\; \backslash \; int\_\; \{-\; \backslash \; pi\}\; ^\; \{\backslash \; pi\}\; F\; (T)\; D\_n\; (x-t)\; dt\; =\; \backslash \; frac1\; \{2\; \backslash \; pi\}\; \backslash \; int\_\; \{-\; \backslash \; pi\}\; ^\; \{\backslash \; pi\}\; D\_n\; (T)\; F\; (x-t)\; dt=\; (D\_n*f)\; (X)\; \backslash ,$; The preceding identity is a Produit convolution, or the application of a Opérateur to core.

It is starting from this expression and of the properties of the core of Dirichlet that one shows the Théorème of Dirichlet on the convergence of Fourier series.

This operator is a Opérateur limited on the space of the continuous functions, whose Norme of operator is raised by $\backslash |D\_n\; \backslash |\_1$.

By specializing the study in a point X particular, the application $x\; \backslash \; mapsto\; S\_n\; (F)\; (X)$ has as a standard of operator $\backslash |D\_n\; \backslash |\_1$ him even, which tends towards the infinite one with N . Using the Theorem of Banach-Steinhaus, one can deduce from it that there exist continuous functions from which the Fourier series diverge at the point X .

Introduction to the formalism of the distributions

The core of Dirichlet is 2π time the sum of order N of the development in Fourier series of a “function”, in distribution, of period 2π given by

$\backslash \; delta\_p\; (X)\; =\; \backslash \; sum\_\; \{k=-\; \backslash \; infty\}\; ^\; \backslash \; infty\; \backslash \; delta\; (x-2\; \backslash \; pi\; K)$

where δ is the function delta of Dirac, which is not really a function, in the direction of application of a unit towards another, but is rather “generalized function”, such a called a distribution . In other words, the development in Fourier series of this “function” is written

$\backslash \; delta\_p\; (X)\; =\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; pi\}\; \backslash \; sum\_\; \{k=-\; \backslash \; infty\}\; ^\; \backslash \; infty\; e^\; \{ikx\}\; =\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; pi\}\; \backslash \; left\; (1+2\; \backslash \; sum\_\; \{k=1\}\; ^\; \backslash \; infty\; \backslash \; cos\; (kx)\; \backslash \; right).$

This “periodic function delta” is the neutral element for the Produit convolution definite on the whole of the functions of period 2π by

$(f*g)\; (X)\; =\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; pi\}\; \backslash \; int\_\; \{-\; \backslash \; pi\}\; ^\; \backslash \; pi\; F\; (there)\; G\; (X\; there)\; \backslash ,\; dy.$

In other words,

for any function F of period 2π, $f*\; \backslash \; delta\_p=\; \backslash \; delta\_p*f=f$

The product of convolution of D_n with any function F of period 2π is equal to the sum of order N of the development in Fourier series of F , i.e., we have

$(D\_n*f)\; (X)\; =\; (f*D\_n)\; (X)\; =\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; pi\}\; \backslash \; int\_\; \{-\; \backslash \; pi\}\; ^\; \backslash \; pi\; F\; (there)\; D\_n\; (X\; there)\; \backslash ,\; dy=\; \backslash \; sum\_\; \{k=-n\}\; ^n\; \backslash \; hat\; \{F\}\; (K)\; e^\; \{ikx\},$

where

$\backslash \; hat\; \{F\}\; (K)\; =\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; pi\}\; \backslash \; int\_\; \{-\; \backslash \; pi\}\; ^\; \backslash \; pi\; F\; (X)\; e^\; \{-\; ikx\}\; \backslash ,\; dx$

is the K ème coefficient of Fourier of F .

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