This question arose during my exploration of convergence properties of Fourier series, which is a critical topic in my studies on harmonic analysis. The problem is interesting because it touches on how functions behave under scaling and how their Fourier coefficients influence convergence.
Show that for any $f \in L^1([-\pi, \pi])$ with a periodic extension outside the interval, and for any $g \in L^\infty([-\pi, \pi])$ with a periodic extension outside the interval:
$$ \lim_{N \to \infty} \int_{-\pi}^{\pi} f(Nx)g(x) \, dx = 2\pi \hat{f}(0)\hat{g}(0) $$
To tackle this, it might be useful to examine how the integral behaves as $N$ becomes large, possibly using properties of convolutions and the Fourier transform. Is there a specific approach or theorem that is particularly useful in this type of convergence?
