Orthogonal polynomials

In mathematics, orthogonal polynomials are an infinite sequence of polynomials

${\displaystyle P_{0}(x),P_{1}(x),P_{2}(x),\dotsc }$

in one unknown , so that the degree is, the orthogonal with respect to a - scalar product are. ${\displaystyle x}$ ${\displaystyle P_{n}(x)}$ ${\displaystyle n}$ ${\displaystyle L^{2}}$

definition

Be a Borel measure on and look to the Hilbert space of respect square integrable functions with the scalar product ${\displaystyle \mu }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle L^{2}(\mathbb {R} ,d\mu )}$ ${\displaystyle \mu }$

${\displaystyle \langle f,g\rangle =\int _{\mathbb {R} }{\overline {f(x)}}g(x)d\mu (x)}$.

Next be for everyone . This is the case, for example, if the measure has a compact support . In particular, the measure is finite and one can demand without restricting the generality . In the simplest case, the measure is by a non-negative weight function given: . ${\displaystyle \textstyle \int _{\mathbb {R} }|x|^{n}d\mu (x)<\infty }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \mu (\mathbb {R} )=1}$ ${\displaystyle w(x)}$ ${\displaystyle d\mu (x)=w(x)dx}$

A sequence of polynomials , called sequence of orthogonal polynomials, if degree has and various polynomials are orthogonal in pairs: ${\displaystyle P_{n}}$ ${\displaystyle n\in \mathbb {N} _{0}}$ ${\displaystyle P_{n}(x)}$ ${\displaystyle n}$

${\displaystyle \langle P_{m},P_{n}\rangle =0,\qquad m\neq n.}$

construction

Is the measure given, the corresponding polynomials can clearly using the Gram-Schmidt's orthogonalization from the Monomen , can be constructed. Obviously, the moments are enough for that ${\displaystyle x^{n}}$ ${\displaystyle n\in \mathbb {N} _{0}}$

${\displaystyle m_{n}=\int _{\mathbb {R} }x^{n}d\mu (x)}$

to know. The converse is known as Stieltjes ' moment problem.

Normalization

Various normalization options are in use. To describe this, we introduce the following constants:

${\displaystyle h_{n}=\langle P_{n},P_{n}\rangle =\int _{\mathbb {R} }P_{n}(x)^{2}d\mu (x),\qquad {\tilde {h}}_{n}=\langle P_{n}(x),x\,P_{n}(x)\rangle =\int _{\mathbb {R} }x\,P_{n}(x)^{2}d\mu (x)}$

and

${\displaystyle P_{n}(x)=k_{n}x^{n}+{\tilde {k}}_{n}x^{n-1}+{\tilde {\tilde {k}}}_{n}x^{n-2}+\dotsb }$.

Then the polynomials are called orthonormal , if , and monic , if . ${\displaystyle h_{n}=1}$ ${\displaystyle k_{n}=1}$

Recursion relation

Orthogonal polynomials satisfy a three-level recursion relation

${\displaystyle P_{n+1}(x)=(A_{n}x+B_{n})P_{n}(x)-C_{n}P_{n-1}(x)}$

( to be set in the case ) with ${\displaystyle P_{-1}(x)=0}$ ${\displaystyle n=0}$

${\displaystyle A_{n}={\frac {k_{n+1}}{k_{n}}},\quad B_{n}=\left({\frac {{\tilde {k}}_{n+1}}{k_{n+1}}}-{\frac {{\tilde {k}}_{n}}{k_{n}}}\right)A_{n}=-{\frac {{\tilde {h}}_{n}}{h_{n}}}A_{n},\quad C_{n}={\frac {A_{n}{\tilde {\tilde {k}}}_{n}+B_{n}{\tilde {k}}_{n}-{\tilde {\tilde {k}}}_{n+1}}{k_{n-1}}}={\frac {A_{n}}{A_{n-1}}}{\frac {h_{n}}{h_{n-1}}},}$

and the constants from the previous section. ${\displaystyle h_{n},k_{n},{\tilde {k}}_{n},{\tilde {\tilde {k}}}_{n}}$

The recursion relation can also be equivalent in the form

${\displaystyle a_{n}P_{n+1}(x)+b_{n}P_{n}(x)+c_{n}P_{n-1}(x)=x\,P_{n}(x)}$

With

${\displaystyle a_{n}={\frac {k_{n}}{k_{n+1}}},\quad b_{n}={\frac {{\tilde {k}}_{n}}{k_{n}}}-{\frac {{\tilde {k}}_{n+1}}{k_{n+1}}}={\frac {{\tilde {h}}_{n}}{h_{n}}},\quad c_{n}={\frac {{\tilde {\tilde {k}}}_{n}-a_{n}{\tilde {\tilde {k}}}_{n+1}-b_{n}{\tilde {k}}_{n}}{k_{n-1}}}=a_{n-1}{\frac {h_{n}}{h_{n-1}}},}$

to be written.

In the case of orthonormal polynomials,, in particular , a symmetrical recursion relation is obtained and the orthonormal polynomials exactly satisfy the generalized eigenvector equation of the associated Jacobi operator . The measure is the spectral measure of the Jacobi operator for the first basis vector . ${\displaystyle h_{n}=1}$ ${\displaystyle c_{n}=a_{n-1}}$ ${\displaystyle d\mu }$ ${\displaystyle \delta _{1,n}}$

Christoffel – Darboux formula

It applies

${\displaystyle \sum _{m=0}^{n}{\frac {P_{m}(x)P_{m}(y)}{h_{m}}}={\frac {k_{n}}{h_{n}k_{n+1}}}{\frac {P_{n+1}(x)P_{n}(y)-P_{n}(x)P_{n+1}(y)}{x-y}}}$

and in the case one obtains by setting the limit ${\displaystyle x=y}$

${\displaystyle \sum _{m=0}^{n}{\frac {P_{m}(x)^{2}}{h_{m}}}={\frac {k_{n}}{h_{n}k_{n+1}}}{\left(P_{n+1}'(x)P_{n}(x)-P_{n}'(x)P_{n+1}(x)\right)}.}$

zeropoint

The polynomial has exactly zeros , all of which are simple and lie in the carrier of the measure. The zeros of lie strictly between the zeros of . ${\displaystyle P_{n}}$ ${\displaystyle n}$ ${\displaystyle P_{n}}$ ${\displaystyle P_{n+1}}$

List of sequences of orthogonal polynomials

• Gegenbauer polynomial
• Hahn polynomial
• Hermitian polynomial
• Jacobi polynomial
• Legendre polynomial
• Laguerre polynomials
• Chebyshev polynomial
• Zernike polynomial

literature

• Milton Abramowitz and Irene A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, Dover (1965), ISBN 978-0486612720 (Chapter 22)
• Gábor Szegő , Orthogonal Polynomials, Colloquium Publications - American Mathematical Society, 1939. ISBN 0-8218-1023-5 .
• Theodore S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, 1978. ISBN 978-0677041506 .

Web links

• Orthogonal polynomials in the NIST Digital Library of Mathematical Functions