Fejér polynomials

In mathematics , for a - periodic , continuous function , that is , the -th Fejér polynomial is defined by ${\displaystyle 2\pi }$ ${\displaystyle f}$ ${\displaystyle f\in C_{2\pi }}$ ${\displaystyle n}$ ${\displaystyle \sigma _{n}(f)}$

${\displaystyle \sigma _{n}(f)(x):=\sum _{k=-n}^{n}\left(1-{\frac {\left|k\right|}{n+1}}\right){\hat {f}}(k)e^{ikx},}$

in which

${\displaystyle {\hat {f}}(k):={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(t)e^{-ikt}\mathrm {d} t}$

is the -th Fourier coefficient . With the help of these trigonometric polynomials Fejér provided a constructive proof for the Weierstraß theorem , which states that every -periodic, continuous function can be approximated uniformly by trigonometric polynomials . This statement is also known as Fejér's theorem. ${\displaystyle k}$ ${\displaystyle 2\pi }$

Convergence Statements - Fejér's Theorem

→ Main article : Fejér's theorem

Fejér gave the proof of the (first) arithmetic mean of the partial sums of the Fourier series

${\displaystyle \sigma _{n}(f)(x)={\frac {1}{n+1}}\sum _{k=0}^{n}S_{k}(f)(x),}$

in which

${\displaystyle S_{k}(f)(x):=\sum _{j=-k}^{k}{\hat {f}}(j)e^{ijx}}$

is the -th partial sum by showing: ${\displaystyle k}$

For every - periodic , continuous function the sequence of Fejér polynomials converges uniformly to , i. H. ${\displaystyle 2\pi }$ ${\displaystyle f}$ ${\displaystyle \sigma _{n}(f)}$ ${\displaystyle f}$

${\displaystyle f\in C_{2\pi }\Rightarrow \lim \limits _{n\to \infty }\|\sigma _{n}(f)-f\|_{C_{2\pi }}=\lim \limits _{n\to \infty }\left(\max \limits _{x\in [-\pi ,\pi ]}|\sigma _{n}(f)(x)-f(x)|\right)=0.}$

Fejér core

The nth Fejér kernel is defined by ${\displaystyle \sigma _{n}(x)}$

${\displaystyle \sigma _{n}(x):=\sum _{k=-n}^{n}\left(1-{\frac {\left|k\right|}{n+1}}\right)e^{ikx}}$.

folding

The Fejér polynomials can be represented as a convolution with the Fejér kernel. It applies

${\displaystyle \sigma _{n}(f)(x)=(\sigma _{n}*f)(x):={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(t)\sigma _{n}(x-t)\mathrm {d} t}$

Arithmetic mean of the Dirichlet kernel

From the interpretation of the Fejér polynomials as the (first) arithmetic mean of the partial sums, the representation of the Fejér kernel as the arithmetic mean of the Dirichlet kernel follows

${\displaystyle \sigma _{n}(x)={\frac {1}{n+1}}\sum _{k=0}^{n}D_{k}(x)}$

where the Dirichlet core is defined via

${\displaystyle D_{n}(x):=\sum _{k=-n}^{n}e^{ikx}}$

Positive real core

In addition to the sums notation for complex functions, the Fejér kernel can also be represented in a closed form. For this purpose, the Dirichlet core is used to represent

${\displaystyle D_{k}(x)=1+2\sum _{j=1}^{k}\cos(jx)={\frac {\sin \left({\frac {2k+1}{2}}x\right)}{\sin(x/2)}}}$

owns. With the help of the above connection of the Fejér kernel with the Dirichlet kernels and the rule

${\displaystyle \sum _{k=0}^{n}\sin \left({\frac {2k+1}{2}}x\right)={\frac {\sin ^{2}\left({\frac {n+1}{2}}x\right)}{\sin \left(x/2\right)}}}$

the following closed representation of the Fejér core results.

${\displaystyle \sigma _{n}(x)={\begin{cases}{\frac {1}{n+1}}\left({\frac {\sin \left({\frac {n+1}{2}}x\right)}{\sin({\frac {x}{2}})}}\right)^{2}&,x\neq 2j\pi \\n+1&,x=2j\pi \end{cases}},j\in \mathbb {Z} }$

Due to the positivity of the Fejér kernel, which can be seen from this, the Bohman-Korowkin theorem can be used to demonstrate the uniform convergence of the Fejér polynomials , which says that the uniform convergence of the test functions and the uniform convergence for all functions result . ${\displaystyle \sin }$ ${\displaystyle \cos }$ ${\displaystyle f\in C_{2\pi }}$

Convergence in other function spaces

Also for non-continuous functions of other function spaces, e.g. B. of the Lebesgue integrable functions, statements about the approximability can be given.

Quantitative statements

Direct estimates of the convergence behavior of Fejér polynomials can be given for Hölder continuous functions . ${\displaystyle f}$

Belongs to a class of Hölder continuous functions , d. H. ${\displaystyle f}$ ${\displaystyle 0<\alpha \leq 1}$ ${\displaystyle C^{\alpha }}$

${\displaystyle \|f(\cdot +h)-f(\cdot )\|_{C_{2\pi }}={\mathcal {O}}(|h|^{\alpha }),h\to 0,}$

the following quantitative approximation statements apply:

${\displaystyle \|\sigma _{n}(f)-f\|_{C_{2\pi }}={\begin{cases}{\mathcal {O}}(|{\frac {1}{n}}|^{\alpha })&,0<\alpha <1\\{\mathcal {O}}\left({\frac {\log(n)}{n}}\right)&,\alpha =1\end{cases}},n\to \infty }$

literature

• NI Achieser : Lectures on approximation theory. Akademie-Verlag, Berlin 1953.
• PL Butzer , RJ Nessel: Fourier Analysis And Approximation, Vol. 1: One-Dimensional Theory. Birkhäuser, Basel 1971.
• Leopold Fejér : About trigonometric polynomials. In: J. Reine Angew. Math. Volume 146, 1916, pages 53-82.
• Leopold Fejér : Structural information about the partial sums and their mean values ​​in the Fourier series and the power series. In: Z. Angew. Math. Mech. Volume 13, 1933, pages 80-88.
• Antoni Zygmund : Trigonometric Series. Cambridge University Press, Cambridge 1968, 2nd Edition.