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). It states that

∑ j = 0 n f j ( x ) f j ( y ) h j = k n h n k n + 1 f n ( y ) f n + 1 ( x ) − f n + 1 ( y ) f n ( x ) x − y {\displaystyle \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 fj(x) is the jth term of a set of orthogonal polynomials of squared norm hj and leading coefficient kj.

There is also a "confluent form" of this identity by taking y → x {\displaystyle y\to x} limit:

∑ j = 0 n f j 2 ( x ) h j = k n h n k n + 1 [ f n + 1 ′ ( x ) f n ( x ) − f n ′ ( x ) f n + 1 ( x ) ] . {\displaystyle \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

Let p n {\displaystyle p_{n}} be a sequence of polynomials orthonormal with respect to a probability measure μ {\displaystyle \mu } , and define

a n = ⟨ x p n , p n + 1 ⟩ , b n = ⟨ x p n , p n ⟩ , n ≥ 0 {\displaystyle 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]
p 0 ( x ) = 1 , p 1 ( x ) = x − b 0 a 0 , x p n ( x ) = a n p n + 1 ( x ) + b n p n ( x ) + a n − 1 p n − 1 ( x ) , n ≥ 1 {\displaystyle {\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, ⟨ x p n , p k ⟩ = ⟨ p n , x p k ⟩ {\displaystyle \langle xp_{n},p_{k}\rangle =\langle p_{n},xp_{k}\rangle } , so if k ≤ n − 2 {\displaystyle k\leq n-2} , then x p k {\displaystyle xp_{k}} is a linear combination of p 0 , . . . , p n − 1 {\displaystyle p_{0},...,p_{n-1}} , and thus ⟨ x p n , p k ⟩ = 0 {\displaystyle \langle xp_{n},p_{k}\rangle =0} . So, to construct p n + 1 {\displaystyle p_{n+1}} , it suffices to perform Gram-Schmidt process on x p n {\displaystyle xp_{n}} using p n , p n − 1 {\displaystyle p_{n},p_{n-1}} , which yields the desired recurrence.


Proof of Christoffel–Darboux formula:

Since both sides are unchanged by multiplying with a constant, we can scale each f n {\displaystyle f_{n}} to p n {\displaystyle p_{n}} .

Since k n + 1 k n x p n − p n + 1 {\displaystyle {\frac {k_{n+1}}{k_{n}}}xp_{n}-p_{n+1}} is a degree n {\displaystyle n} polynomial, it is perpendicular to p n + 1 {\displaystyle p_{n+1}} , and so ⟨ k n + 1 k n x p n , p n + 1 ⟩ = ⟨ p n + 1 , p n + 1 ⟩ = 1 {\displaystyle \langle {\frac {k_{n+1}}{k_{n}}}xp_{n},p_{n+1}\rangle =\langle p_{n+1},p_{n+1}\rangle =1} . Now the Christoffel-Darboux formula is proved by induction, using the three-term recurrence.

Specific cases

Hermite polynomials:

∑ k = 0 n H k ( x ) H k ( y ) k ! 2 k = 1 n ! 2 n + 1 H n ( y ) H n + 1 ( x ) − H n ( x ) H n + 1 ( y ) x − y . {\displaystyle \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}}.}
∑ k = 0 n H e k ( x ) H e k ( y ) k ! = 1 n ! H e n ( y ) H e n + 1 ( x ) − H e n ( x ) H e n + 1 ( y ) x − y . {\displaystyle \sum _{k=0}^{n}{\frac {He_{k}(x)He_{k}(y)}{k!}}={\frac {1}{n!}}\,{\frac {He_{n}(y)He_{n+1}(x)-He_{n}(x)He_{n+1}(y)}{x-y}}.}

Associated Legendre polynomials:

( μ − μ ′ ) ∑ l = m L ( 2 l + 1 ) ( l − m ) ! ( l + m ) ! P l m ( μ ) P l m ( μ ′ ) = ( L − m + 1 ) ! ( L + m ) ! [ P L + 1 m ( μ ) P L m ( μ ′ ) − P L m ( μ ) P L + 1 m ( μ ′ ) ] . {\displaystyle {\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}}}

See also

References

  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.
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  • 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


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