As this question Fresnel Approximation of δ Function notes, Fresnel integrals can be used to make a Dirac sequence, but no proof is given. Specifically, I'm interested in knowing when the following is true $$\lim_{s\to0}\int_{-\infty}^\infty\frac{1}{\sqrt{\pi i}}e^{ix^2}f(sx)dx=f(0)$$ I know this is the case for a constant function, but I suppose it is the case for many more functions. For my own purpose, I would be happy if it is true for square-integrable complex functions $f(x)$ (i.e. $\int_{-\infty}^\infty|f(x)|^2dx<\infty$) that are holomorphic on an open disc $K(0,\delta)$. I have tried hard to prove it using complex analysis ideas but have failed.
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