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For any $f\in L^2(\mathbb{R}^d)$ prove \begin{align}\left\lVert \int_{\mathbb{R}^d} e^{i |x-y|^2}f(y) dy-\int_{\mathbb{R}^d}e^{i |x-y|^2} e^{-|y|^2/a}f(y) dy \right\rVert_{L^2} \rightarrow 0\ \ \ \text{as } \ a\rightarrow \infty \end{align}

I can easily prove pointwise convergence using dominated convergence with $|f(y)|$ as upper bound. Is there some sort of extension to dominated convergence, that would prove the $L^2$ convergence above?

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The integral $$ \int_{\mathbb{R}^d} e^{i|x-y|^2}f(y)\,dy $$ is not well defined for a general $f \in L^2(\mathbb{R}^d)$. I think you are missing a hypothesis.

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