Dirichlet core

The first four Dirichlet kernels. (The functions are 2π-periodic.)

The Dirichlet core is a sequence of functions examined by Peter Gustav Lejeune Dirichlet . This is used in analysis in the sub-area of Fourier analysis . Dirichlet found the first strict proof of the convergence of the Fourier series from a periodic, piecewise continuous and piecewise monotonic function in 1829 . The convergence of Fourier series has been discussed since Leonhard Euler . This sequence of functions found by Dirichlet is an important part of this proof and is used there as an integral kernel. That is why they are called Dirichlet kernels.

definition

The sequence of functions is called the Dirichlet kernel

${\displaystyle D_{n}(x)=\sum _{k=-n}^{n}e^{ikx}=1+2\sum _{k=1}^{n}\cos(kx)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin(x/2)}}.}$

The meaning of the Dirichlet kernel is related to the relationship to the Fourier series . The convolution of with a function of the period is the Fourier -th degree approximation for . For example is ${\displaystyle D_{n}(x)}$ ${\displaystyle f}$ ${\displaystyle 2\pi }$ ${\displaystyle n}$ ${\displaystyle f}$

${\displaystyle (D_{n}*f)(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(y)D_{n}(x-y)\,dy=\sum _{k=-n}^{n}{\hat {f}}(k)e^{ikx},}$

in which

${\displaystyle {\hat {f}}(k)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)e^{-ikx}\,dx}$

is the -th Fourier coefficient of . From this it can be concluded that in order to study the convergence of Fourier series it is sufficient to study the properties of the Dirichlet kernel. From the fact that the L ¹ norm of for is logarithmic , it can be deduced that there are continuous functions that are not represented by their Fourier series. The following applies explicitly: ${\displaystyle k}$ ${\displaystyle f}$ ${\displaystyle D_{n}}$ ${\displaystyle n\to \infty }$ ${\displaystyle \infty }$

${\displaystyle \int |D_{n}(t)|\,dt={\frac {4}{\pi ^{2}}}\log n+{\mathcal {O}}(1)}$

For the -notation see Landau symbols . ${\displaystyle {\mathcal {O}}}$

Relationship to delta distribution

The periodic delta distribution is the neutral element for the convolution with - periodic functions: ${\displaystyle 2\pi }$

${\displaystyle f*(2\pi \delta )=f\,}$

for each function with a period . The Fourier series is represented by the following "function": ${\displaystyle f}$ ${\displaystyle 2\pi }$

${\displaystyle 2\pi \delta (x)\sim \sum _{k=-\infty }^{\infty }e^{ikx}=\left(1+2\sum _{k=1}^{\infty }\cos(kx)\right).}$

Proof of the trigonometric identity

The trigonometric identity

${\displaystyle \sum _{k=-n}^{n}e^{ikx}={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin(x/2)}}}$

can be proved as follows. To do this, consider the finite sum of the geometric series :

${\displaystyle \sum _{k=0}^{n}ar^{k}=a{\frac {1-r^{n+1}}{1-r}}.}$

In particular,

${\displaystyle \sum _{k=-n}^{n}r^{k}=r^{-n}\cdot {\frac {1-r^{2n+1}}{1-r}}.}$

If you multiply the numerator and denominator by , you get ${\displaystyle r^{-{1 \over 2}}}$

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

In the case of , one obtains ${\displaystyle r=e^{ix}}$

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

and finally abbreviates . ${\displaystyle -2i}$

literature

• Kurt Endl, Wolfgang Luh: Analysis II. An integrated representation . 7th edition, Aula-Verlag, Wiesbaden 1989, p. 117.
• Andrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson: Real Analysis . ClassicalRealAnalysis.com 1996, ISBN 013458886X , p. 620 ( full online version (Google Books) )

Web links

• Dirichlet kernel at PlanetMath (Engl.)

Individual evidence

1. ↑ W. Rudin, Real and Complex Analysis . McGraw-Hill, London 1970. Section 5.11, p. 101