Christoffel–Darboux formula

In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Elwin Bruno Christoffel (1858) and Jean Gaston Darboux (1878).

Christoffel–Darboux formula—if a sequence of polynomials $f_{0},f_{1},\dots$ are of degrees $0,1,\dots$ , and orthogonal with respect to a probability measure $\mu$ , then

\sum _{j=0}^{n}{\frac {f_{j}(x)f_{j}(y)}{h_{j}}}={\frac {k_{n}}{h_{n}k_{n+1}}}{\frac {f_{n}(y)f_{n+1}(x)-f_{n+1}(y)f_{n}(x)}{x-y}}

where $h_{j}=\|f_{j}\|_{\mu }^{2}$ are the squared norms, and $k_{j}$ are the leading coefficients.

There is also a "confluent form" of this identity by taking $y\to x$ limit:

Christoffel–Darboux formula, confluent form— $\sum _{j=0}^{n}{\frac {f_{j}^{2}(x)}{h_{j}}}={\frac {k_{n}}{h_{n}k_{n+1}}}\left[f_{n+1}'(x)f_{n}(x)-f_{n}'(x)f_{n+1}(x)\right].$

Proof

Lemma—Let $p_{n}$ be a sequence of polynomials orthonormal with respect to a probability measure $\mu$ , such that $p_{n}$ has degree $n$ , and define $a_{n}=\langle xp_{n},p_{n+1}\rangle ,\qquad b_{n}=\langle xp_{n},p_{n}\rangle ,\qquad n\geq 0$ (they are called the "Jacobi parameters"), then we have the three-term recurrence^[1] ${\begin{array}{l l}{p_{0}(x)=1,\qquad p_{1}(x)={\frac {x-b_{0}}{a_{0}}},}\\{xp_{n}(x)=a_{n}p_{n+1}(x)+b_{n}p_{n}(x)+a_{n-1}p_{n-1}(x),\qquad n\geq 1}\end{array}}$

Proof

By definition, $\langle xp_{n},p_{k}\rangle =\langle p_{n},xp_{k}\rangle$ , so if $k\leq n-2$ , then $xp_{k}$ is a linear combination of $p_{0},...,p_{n-1}$ , and thus $\langle xp_{n},p_{k}\rangle =0$ . So, to construct $p_{n+1}$ , it suffices to perform Gram-Schmidt process on $xp_{n}$ using $p_{n+1},p_{n},p_{n-1}$ , which yields the desired recurrence.

Proof of Christoffel–Darboux formula

For any sequence of nonzero constants $c_{0},c_{1},\dots$ , we can change each $f_{k}$ to $c_{k}f_{k}$ , and both sides of the equation would remain unchanged. Thus WLOG, scale each $f_{n}$ to $p_{n}$ .

Since ${\frac {k_{n+1}}{k_{n}}}xp_{n}-p_{n+1}$ is a degree $n$ polynomial, it is perpendicular to $p_{n+1}$ , and so $\langle {\frac {k_{n+1}}{k_{n}}}xp_{n},p_{n+1}\rangle =\langle p_{n+1},p_{n+1}\rangle =1$ . Thus, ${\textstyle a_{n}={\frac {k_{n}}{k_{n+1}}}}$ .

Base case:

${\frac {k_{0}}{k_{1}}}{\frac {p_{0}(y)p_{1}(x)-p_{1}(y)p_{0}(x)}{x-y}}={\frac {k_{0}}{k_{1}}}{\frac {p_{0}(y){\frac {x-b_{0}}{a_{0}}}-p_{0}(x){\frac {y-b_{0}}{a_{0}}}}{x-y}}=1$

Induction:

${\begin{aligned}&(x-y)\sum _{j=0}^{n}p_{j}(x)p_{j}(y)=(x-y)\sum _{j=0}^{n-1}p_{j}(x)p_{j}(y)+p_{n}(x)p_{n}(y)(x-y)\\&{\stackrel {\text{ (IH) }}{=}}{\frac {k_{n-1}}{k_{n}}}\left[p_{n}(x)p_{n-1}(y)-p_{n}(y)p_{n-1}(x)\right]+(x-y)p_{n}(x)p_{n}(y)\end{aligned}}$

By the three-term recurrence,

${\begin{aligned}&xp_{n}(x)=a_{n}p_{n+1}(x)+b_{n}p_{n}(x)+a_{n-1}p_{n-1}(x)\\&yp_{n}(y)=a_{n}p_{n+1}(y)+b_{n}p_{n}(y)+a_{n-1}p_{n-1}(y)\end{aligned}}$

Multiply the first by ${\textstyle p_{n}(y)}$ and the second by ${\textstyle p_{n}(x)}$ and subtract:

$(x-y)p_{n}(x)p_{n}(y)=a_{n}\left[p_{n}(y)p_{n+1}(x)-p_{n}(x)p_{n+1}(y)\right]+a_{n-1}\left[p_{n}(y)p_{n-1}(x)-p_{n}(x)p_{n-1}(y)\right]$

Now substitute in ${\textstyle a_{n}={\frac {k_{n}}{k_{n+1}}},\quad a_{n-1}={\frac {k_{n-1}}{k_{n}}}}$ and simplify.

Specific cases

Hermite

The Hermite polynomials are orthogonal with respect to the gaussian distribution.

The $H$ polynomials are orthogonal with respect to ${\frac {1}{\sqrt {\pi }}}e^{-x^{2}}$ , and with $k_{n}=2^{n}$ . $\sum _{k=0}^{n}{\frac {H_{k}(x)H_{k}(y)}{k!2^{k}}}={\frac {1}{n!2^{n+1}}}\,{\frac {H_{n}(y)H_{n+1}(x)-H_{n}(x)H_{n+1}(y)}{x-y}}.$ The $\operatorname {He}$ polynomials are orthogonal with respect to ${\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}$ , and with $k_{n}=1$ . $\sum _{k=0}^{n}{\frac {\operatorname {He} _{k}(x)\operatorname {He} _{k}(y)}{k!}}={\frac {1}{n!}}\,{\frac {\operatorname {He} _{n}(y)\operatorname {He} _{n+1}(x)-\operatorname {He} _{n}(x)\operatorname {He} _{n+1}(y)}{x-y}}.$ The confluent form and the three-term recurrence gives $\sum _{k=0}^{n}{\frac {H_{k}(x)^{2}}{k!2^{k}}}={\frac {H_{n+1}(x)^{2}-H_{n+2}(x)H_{n}(x)}{n!2^{n+1}}}$ $\sum _{k=0}^{n}{\frac {\operatorname {He} _{k}(x)^{2}}{k!}}={\frac {\operatorname {He} _{n+1}(x)^{2}-\operatorname {He} _{n+2}(x)\operatorname {He} _{n}(x)}{n!}}$

Laguerre

The Laguerre polynomials $L_{n}$ are orthonormal with respect to the exponential distribution $e^{-x},\quad x\in (0,\infty )$ , with $k_{n}=(-1)^{n}/n!$ , so $\sum _{k=0}^{n}L_{k}(x)L_{k}(y)={\frac {n+1}{x-y}}\left[L_{n}(x)L_{n+1}(y)-L_{n}(y)L_{n+1}(x)\right]$

Legendre

Associated Legendre polynomials:

{\begin{aligned}(\mu -\mu ')\sum _{l=m}^{L}\,(2l+1){\frac {(l-m)!}{(l+m)!}}\,P_{lm}(\mu )P_{lm}(\mu ')=\qquad \qquad \qquad \qquad \qquad \\{\frac {(L-m+1)!}{(L+m)!}}{\big [}P_{L+1\,m}(\mu )P_{Lm}(\mu ')-P_{Lm}(\mu )P_{L+1\,m}(\mu '){\big ]}.\end{aligned}}

Christoffel–Darboux kernel

The summation involved in the Christoffel–Darboux formula is invariant by scaling the polynomials with nonzero constants. Thus, each probability distribution $\mu$ defines a series of functions $K_{n}(x,y):=\sum _{j=0}^{n}f_{j}(x)f_{j}(y)/h_{j},\quad n=0,1,\dots$ which are called the Christoffel–Darboux kernels. By the orthogonality, the kernel satisfies $\int f(y)K_{n}(x,y)\mathrm {d} \mu (y)={\begin{cases}f(x),&f\in \operatorname {Span} \left(p_{0},p_{1},\ldots ,p_{n}\right)\\0,&\int _{a}^{b}f(x)p_{\ell }(x)\mathrm {d} \mu (x)=0(\ell =0,1,\ldots ,n)\end{cases}}$ In other words, the kernel is an integral operator that orthogonally projects each polynomial to the space of polynomials of degree up to $n$ .

References

^ Świderski, Grzegorz; Trojan, Bartosz (2021-08-01). "Asymptotic Behaviour of Christoffel–Darboux Kernel Via Three-Term Recurrence Relation I". Constructive Approximation. 54 (1): 49–116. arXiv:1909.09107. doi:10.1007/s00365-020-09519-w. ISSN 1432-0940. S2CID 202677666.

Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, ISBN 978-0-521-62321-6, MR 1688958
Christoffel, E. B. (1858), "Über die Gaußische Quadratur und eine Verallgemeinerung derselben.", Journal für die Reine und Angewandte Mathematik (in German), 55: 61–82, doi:10.1515/crll.1858.55.61, ISSN 0075-4102, S2CID 123118038
Darboux, Gaston (1878), "Mémoire sur l'approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série", Journal de Mathématiques Pures et Appliquées (in French), 4: 5–56, 377–416, JFM 10.0279.01
Abramowitz, Milton; Stegun, Irene A. (1972), Handbook of Mathematical Functions, Dover Publications, Inc., New York, p. 785, Eq. 22.12.1
Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W. (2010), "NIST Handbook of Mathematical Functions", NIST Digital Library of Mathematical Functions, Cambridge University Press, p. 438, Eqs. 18.2.12 and 18.2.13, ISBN 978-0-521-19225-5 (Hardback, ISBN 978-0-521-14063-8 Paperback)
Simons, Frederik J.; Dahlen, F. A.; Wieczorek, Mark A. (2006), "Spatiospectral concentration on a sphere", SIAM Review, 48 (1): 504–536, arXiv:math/0408424, Bibcode:2006SIAMR..48..504S, doi:10.1137/S0036144504445765, S2CID 27519592

[1] Świderski, Grzegorz; Trojan, Bartosz (2021-08-01). "Asymptotic Behaviour of Christoffel–Darboux Kernel Via Three-Term Recurrence Relation I". Constructive Approximation. 54 (1): 49–116. arXiv:1909.09107. doi:10.1007/s00365-020-09519-w. ISSN 1432-0940. S2CID 202677666.

[1]