Christoffel–Darboux formula explained

In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by and . It states that

\sum ^{n} _{j =0} (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)

where f_j(x) is the jth term of a set of orthogonal polynomials of squared norm h_j and leading coefficient k_j.

There is also a "confluent form" of this identity by taking

y\to x limit:$$\backslash sum\_^n\; \backslash frac\; =\; \backslash frac\; \backslash left[f\_\{n\; +\; 1\}\text{'}(x)f\_\{n\}(x)\; -\; f\_\{n\}\text{'}(x)\; f\_\{n\; +\; 1\}(x)\backslash right].$$

Proof

Let

p _{n} be a sequence of polynomials orthonormal with respect to a probability measure \mu , and define$$a\_=\backslash langle\; x\; p\_,p\_\backslash rangle,\backslash qquad\; b\_=\backslash langle\; x\; p\_,p\_\backslash rangle,\backslash qquad\; n\backslash geq0$$(they are called the "Jacobi parameters"), then we have the three-term recurrence^[1] $$\backslash begin\backslash \backslash \; \backslash end$$

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,} 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} to p _{n} .

Since

(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 (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

$$\backslash sum\_^n\; \backslash frac\; =\; \backslash frac\backslash ,\backslash frac.$$$$\backslash sum\_^n\; \backslash frac\; =\; \backslash frac\backslash ,\backslash frac.$$

Associated Legendre polynomials

\begin{align} (\mu - \mu')\sum ^{L( 2 l +1 ) ((l - m)!)/((l + m)!)} _{l = m} P _{lm} (\mu)P _{lm} (\mu') = \\ ((L - m +1 )!)/((L + m)!) [ P _{L+1m} (\mu)P _{Lm} (\mu') - P _{Lm} (\mu)P _{L+1m} (\mu') ] .\end{align}

See also

• Turán's inequalities
• Sturm Chain

References

• • • • • (Hardback, Paperback)

Notes and References

1. Świderski . Grzegorz . Trojan . Bartosz . 2021-08-01 . Asymptotic Behaviour of Christoffel–Darboux Kernel Via Three-Term Recurrence Relation I . Constructive Approximation . en . 54 . 1 . 49–116 . 10.1007/s00365-020-09519-w . 202677666 . 1432-0940. free . 1909.09107 .