Fourier interpolation on the real line
The Schwartz class of real functions is defined as \(\mathcal{S}=\{f\in C^\infty(\mathbb{R}): \|f\|_{\alpha,\beta}<\infty, \alpha,\beta\ge0\}\) with seminorm \(\|f\|_{\alpha,\beta}=\sup_{x\in\mathbb{R}} |x^\alpha(\frac{d}{dx})^\beta f(x)|\). If we denote the Fourier transform of a function \(h\) by \(\widehat{h}\), then the main theorem of the paper states that there exists an interpolating basis \(\{a_n\}_{n=0}^\infty\subset\mathcal{S}_{\text{even}}\) of even Schwartz functions such that for every \(f\in\mathcal{S}_{\text{even}}\) it holds that \(f(x)=\sum_{n=0}^\infty a_n(x)f(\sqrt{n})+\sum_{n=0}^\infty \widehat{a}_n(x)\widehat{f}(\sqrt{n})\) for all \(x\in\mathbb{R}\) with absolutely converging right-hand side. A similar result is possible for odd functions in \(f\in\mathcal{S}\). The constructive proof shows that \(a_n\) is the average of two functions \(b_n^+(x)\) and \(b_n^-(x)\) that are integral transforms of \(g_n^\pm(z)\) which are holomorphic modular forms on the \(\theta\)-group \(\Gamma_\theta\) of weight \(3/2\). Although this links the interpolating problem to a rather general theory, the approach taken is largely self contained, introducing the subgroup \(\Gamma_\theta\) of \(\mathrm{SL}_2(\mathbb{Z})\), the modular forms for \(\Gamma_\theta\), leading to the introduction of the \(g_n^\pm\) functions. It is suggested that perhaps the result can be extended to a larger class than Schwartz functions. The bottleneck for that is to estimate bounds for the growth of the functions \(b_n^\pm\).
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