In this paper we construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set \(\{0, \pm\sqrt{1}, \pm\sqrt{2}, \pm\sqrt{3},\dots\}\). The functions in the interpolating basis are constructed in a closed form as an integral transform of weakly holomorphic modular forms for the theta subgroup of the modular group.

中文翻译：

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实线上的傅立叶插值

在本文中，我们为实线上的Schwartz函数构造了一个显式插值公式。该公式根据函数的值及其在集合\（\ {0，\ pm \ sqrt {1}，\ pm \ sqrt {2}，\ pm \ sqrt {3}，\ dots \} \）。插值基础上的函数以封闭形式构造，作为对模群的theta子群的弱全纯模形式的积分变换。