= High-frequency obstruction for nonperiodic exponential data
For a continuously differentiable periodic kernel, integration by parts and the <Riemann-Lebesgue lemma> give $c_n=o(1/|n|)$. Nonperiodic data $e^{\alpha x}$ with $\alpha\notin i\mathbb Z$ have <Fourier coefficients> asymptotic to a nonzero multiple of $1/n$. Their inverse coefficients $y_n/c_n$ therefore fail square summability, and there is no Hilbert-space inverse or least-squares minimizer. Periodic exponential data $\alpha=im$ instead give the single-mode solution $e^{imx}/c_m$.
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