Determine the Fuorier transform of a product between a periodic and a non-periodic function.

I have a function $\varphi \in L^2(\mathbb{R})$ such that $\text{supp}(\widehat{\varphi})\subset [0,1]$ and a $2\pi$-periodic Fourier series $R(x):=\sum_{k\in \mathbb{Z}} a_k e^{ikx}$ (of some continuous function) which convergences unconditionally in the $L^2([-\pi,\pi))$-norm. I need a proof that the Fourier coefficients of the product $f:=R\cdot \varphi$ are given by  $\hat{f}(\xi)=\sum_{k\in\mathbb{Z}} a_k \widehat{\varphi}(\xi-k)$ almost everywhere.

I cannot use any convolution theorem that I know of since $R$ is periodic (and not in $L^2(\mathbb{R})$) while $\varphi$ is not.

Fourier Analysis Fourier Series Fourier Transform
Ichbinanonym Ichbinanonym
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  • Ichbinanonym Ichbinanonym
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    f is not an L1 function, just L2. So I don‘t think that the formula for the fourier transform of f is valid.

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Kav10 Kav10
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  • Kav10 Kav10
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    If f(x) is an L2 but not necessarily in L1 , then the L2 Fourier transform can be used rather than the standard integral formula. It will be the same approach but using the Parseval’s theorem. Hope that helps.

  • Ichbinanonym Ichbinanonym
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    I still don't see how it is the same approach since you also used the integral formula for phi as well, which is not in L1 either. Thus by Parseval we can only approximate f and phi by Schwartz functions whose fourier transform has integral formula. But I don't think that we can exchange the integral and the limit that way since we still need pointwise convergence in the end. Sorry if I am missing something.

    • Kav10 Kav10
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      Not using the integrals, we can use approximations of those functions. Assume Phi_n and f_n are the Schwartz approximations, their Fourier Transforms are well-defined and follows from the standard convolution-type relationship for Schwartz functions. The L2 limit of f_n to f and the convergence phi^_n to phi^ in L2 allows us to pass the result to f. The final series for f^(ξ) converges pointwise almost everywhere due to the compact support of phi^(ξ).

    • Kav10 Kav10
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      This way we avoid directly using the integral formula for phi or f and rely on the L2 Fourier Transform and the approximations.

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