A Bernstein--Ganzburg limit theorem for best weighted approximation

We prove a Bernstein--Ganzburg type limit relation $$\lim_{n\to\infty} \Bigl(\frac{n}{\sigma}\Bigr)^{(2a+1)/p}E_{n,\sigma}(f)_{p,a,b} =A_{\sigma}(f)_{p,a},$$ where $E_{n,\sigma}(f)_{p,a,b}$ is the error of best approximation of $f(nt/\sigma)$ by trigonometric polynomials of degree at most $n$ in $L^{p}((-\pi,\pi],|2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}\,dt)$, and $A_{\sigma}(f)_{p,a}$ is the error of best approximation of $f$ by entire functions of exponential type at most $\sigma$ in $L^{p}(\mathbb{R},|x|^{2a}\,dx)$. For $a=b=0$, this result was obtained by M.~I.~Ganzburg. The proof uses ideas from the Bernstein--Ganzburg limit theorems and a localization method with the Fej\'er kernel from the proof of the limit relation for Nikol'skii constants. As an application, using known results for polynomial approximation, we compute the exact value of $A_{\pi}(\mathbf{1}_{(-1,1)})_{1,a}$ for $a=0$ and $a=1/2$.