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In a different thread, we stumbled upon the following question:

Given a continuous finite measure $w(x)dx$ on some interval $(a,b)$, $-\infty \leq a <b \leq \infty$, and the set of respective orthogonal polynomials $\{p_n \}_{n=0}^{\infty}$. We follow Szego and define the kernel as $$ K_N (t,x) = \sum\limits_{\nu =0}^N \bar{p_{\nu} }(t) p_{\nu}(x).$$

Question 1: Since for every polynomial $q$ of small enough a degree, we have $\int\limits_a^b K_N(t,x)q(t) \, dt = q(x)$, can say that $K_N(t,x)$ converges in $w^*$ topology? If so, that what is its limit?

Edit: In Aronszajn (1950), section 9, it seems clear that $K_n$ themselves converge in $L^2$ norm.

Question 2: If (1) is true, does $K_N$ converges in a stronger sense? pointwise, almost everywhere etc.

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It seems that both questions receive some answers in this paper.

In th $L^2$ norm, Section 9 seems to stipulate convergence and a limit $K$.

For the uniform and pointwise convergence, we have on p. 374, Theorem II, that since the orthogonal projections converge, we have pointwise convergence of $K_n(x,y)$ to $K$. Also, at least for some polynomial sets (Jacobi and Legendre specifically), we have uniform convergence of the projection, which implies uniform convergence of $K_n$.

Edit: After further searching, I've found out that this is the Christoffel-Darboux kernel. Here is a broad survey about this family of kernels.

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