Christoffel–Darboux Formula

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

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

Proof

Specific cases

Hermite

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

The $H$ polynomials are orthogonal with respect to $\frac e^$, and with $k_n = 2^n$.$\sum_^n \frac = \frac\,\frac.$The $He$ polynomials are orthogonal with respect to $\frac e^$, and with $k_n = 1$.$\sum_^n \frac = \frac\,\frac.$

Laguerre

The Laguerre polynomials $L_n$ are orthonormal with respect to the exponential distribution $e^, \quad x \in (0, \infty)$, with $k_n = (-1)^n/n!$, so\sum_^n L_k(x) L_k(y)=\frac\left _n(x) L_(y)-L_n(y) L_(x)\right/math>

Legendre

Associated Legendre polynomials:

: \begin (\mu-\mu')\sum_^L\,(2l+1)\frac\,P_(\mu)P_(\mu')=\qquad\qquad\qquad\qquad\qquad\\\frac\big _(\mu)P_(\mu')-P_(\mu)P_(\mu')\big\end

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$K_n(x,y) := \sum_^n f_j(x) f_j(y)/h_j, \quad n = 0, 1, \dots$which are called the Christoffel–Darboux kernels. By the orthogonality, the kernel satisfies $\int f(y) K_n(x, y) \mathrm \mu(y)= \beginf(x), & f \in \operatorname\left(p_0, p_1, \ldots, p_n\right) \\ 0, & \int_a^b f(x) p_(x) \mathrm \mu(x)=0(\ell=0,1, \ldots, n)\end$In other words, the kernel is an integral operator that orthogonally projects each polynomial to the space of polynomials of degree up to $n$.

See also

* Turán's inequalities
* Sturm Chain

References

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Orthogonal polynomials
Functional analysis

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