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.

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.

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?

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

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.