In a [different thread](https://mathoverflow.net/a/236999/42864), 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)](http://www.jstor.org/stable/pdf/1990404.pdf?_=1461643206256), 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.



Various limits of the Christoffel Darboux Kernel
It seems that both questions receive some answers in [this paper](http://www.jstor.org/stable/pdf/1990404.pdf?_=1461643206256).


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](https://www.ma.utexas.edu/mp_arc/c/08/08-107.pdf) about this family of kernels.


