ChristoffelDarbouxKernel - crowlogic/arb4j GitHub Wiki

The Christoffel-Darboux kernel is a mathematical concept primarily used in the context of orthogonal polynomials and their applications. It's particularly important in approximation theory and spectral analysis. Here's a detailed explanation:

  1. Orthogonal Polynomials: To define the Christoffel-Darboux kernel, we first need to understand orthogonal polynomials. Suppose ${p_n(x)}$ is a sequence of polynomials where each $p_n(x)$ is of degree $n$ and they are orthogonal with respect to some weight function $w(x)$ on an interval $I$. This means:

$$\int_I p_m(x)p_n(x)w(x)dx = 0\quad \text{for } m \neq n. $$

  1. Recurrence Relation: These polynomials satisfy a three-term recurrence relation of the form:

$$xp_n(x) = a_np_{n+1}(x) + b_np_n(x) + c_np_{n-1}(x) $$

where $a_n$, $b_n$, and $c_n$ are constants.

  1. Christoffel-Darboux Formula: The Christoffel-Darboux formula provides a compact way to express the sum of products of these orthogonal polynomials. It states that for $n \geq 0$

$$K_n(x,y) = \sum_{k=0}^n p_k(x)p_k(y) = \frac{a_n[p_{n+1}(x)p_n(y) - p_n(x)p_{n+1}(y)]}{x - y} $$

where $K_n(x, y)$ is the Christoffel-Darboux kernel, and $x \neq y$. When $x = y$, the formula has a limit form obtained by L'Hôpital's rule.

  1. Significance: This kernel simplifies many calculations in the theory of orthogonal polynomials. For example, in the analysis of orthogonal expansions, it allows for efficient computation of sums involving orthogonal polynomials.

  2. Applications: The Christoffel-Darboux kernel finds applications in various areas like numerical analysis, approximation theory, quantum mechanics, and signal processing. Its properties are utilized in solving differential equations and in the study of random matrices.

The beauty of this kernel lies in its ability to elegantly summarize complex relationships between orthogonal polynomials and provide a useful tool in mathematical analysis and applied mathematics.