I'm following Szego's book on orthogonal polynomials.

In chapter III he considers $f \in L^2\left((a,b),d\alpha \right)$, with $-\infty \leq a < b \leq \infty $. There exist a set of orthogonal polynomials $\{p_n \}_{n=0}^{\infty}$ w.r. to the measure $d\alpha$.

We consider the kernel $K_n(t,x) := \sum\limits_{\nu=0}^{n} \bar{p_{\nu} (t)} p_{\nu} (x) $. Then one can show that $\int\limits_{a}^{b} K_n (t,x) f(t) \, d\alpha (t) = \sum\limits_{\nu = 0}^{n} \hat{f}(n)p_n (x) $, the spectral expension of order $n$.

So, if we settled that the polynomial expansion of $f$ is an integral transform, there's a detailed discussion in Szego's book, Davis "approximation theory" and many other books about its $L^2$ error.

Edit: If we look in Aronszajn (1950), we can have by Section 9 that the kernels converge in L^2 norm.

I'm following Szego's book on orthogonal polynomials.

In chapter III he considers $f \in L^2\left((a,b),d\alpha \right)$, with $-\infty \leq a < b \leq \infty $. There exist a set of orthogonal polynomials $\{p_n \}_{n=0}^{\infty}$ w.r. to the measure $d\alpha$.

We consider the kernel $K_n(t,x) := \sum\limits_{\nu=0}^{n} \bar{p_{\nu} (t)} p_{\nu} (x) $. Then one can show that $\int\limits_{a}^{b} K_n (t,x) f(t) \, d\alpha (t) = \sum\limits_{\nu = 0}^{n} \hat{f}(n)p_n (x) $, the spectral expension of order $n$.

So, if we settled that the polynomial expansion of $f$ is an integral transform, there's a detailed discussion in Szego's book, Davis "approximation theory" and many other books about its $L^2$ error.

Edit: If we look in Aronszajn (1950), we can have by Section 9 that the kernels converge in L^2 norm.

Edit 2: I recommend you to read the aforementioned paper, as it deals with the much broader class of reproducing kernels, and give a lot of conditions and results that are relevant to your question.

I'm following Szego's book on orthogonal polynomials.

In chapter III he considers $f \in L^2\left((a,b),d\alpha \right)$, with $-\infty \leq a < b \leq \infty $. There exist a set of orthogonal polynomials $\{p_n \}_{n=0}^{\infty}$ w.r. to the measure $d\alpha$.

We consider the kernel $K_n(t,x) := \sum\limits_{\nu=0}^{n} \bar{p_{\nu} (t)} p_{\nu} (x) $. Then one can show that $\int\limits_{a}^{b} K_n (t,x) f(t) \, d\alpha (t) = \sum\limits_{\nu = 0}^{n} \hat{f}(n)p_n (x) $, the spectral expension of order $n$.

So, if we settled that the polynomial expansion of $f$ is an integral transform, there's a detailed discussion in Szego's book, Davis "approximation theory" and many other books about its $L^2$ error.

I'm following Szego's book on orthogonal polynomials.

In chapter III he considers $f \in L^2\left((a,b),d\alpha \right)$, with $-\infty \leq a < b \leq \infty $. There exist a set of orthogonal polynomials $\{p_n \}_{n=0}^{\infty}$ w.r. to the measure $d\alpha$.

We consider the kernel $K_n(t,x) := \sum\limits_{\nu=0}^{n} \bar{p_{\nu} (t)} p_{\nu} (x) $. Then one can show that $\int\limits_{a}^{b} K_n (t,x) f(t) \, d\alpha (t) = \sum\limits_{\nu = 0}^{n} \hat{f}(n)p_n (x) $, the spectral expension of order $n$.

So, if we settled that the polynomial expansion of $f$ is an integral transform, there's a detailed discussion in Szego's book, Davis "approximation theory" and many other books about its $L^2$ error.

Edit: If we look in Aronszajn (1950), we can have by Section 9 that the kernels converge in L^2 norm.