Generalized sinc-squared integrals: theoretical problems and examples

Résumé

\begin{abstract} We revisit the classical sinc-squared integral

\[
I(a) = \int_{0}^{\infty} \left(\frac{\sin(ax)}{x}\right)^{2} dx,
\]
whose well-known evaluation $I(a) = \tfrac{\pi a}{2}$ is a cornerstone of Fourier analysis.
Building on this foundation, we present a unified study of its generalizations and structural properties.
Our main contributions are: (i) exact formulas and asymptotic expansions for higher-power integrals
\(
I_{n}(a) = \int_{0}^{\infty} \big(\tfrac{\sin(ax)}{x}\big)^{n} dx
\),
including precise decay laws as $n \to \infty$;
(ii) weighted variants
\(
K_{\alpha}(a) = \int_{0}^{\infty} \big(\tfrac{\sin(ax)}{x}\big)^{2} x^{\alpha} dx
\)
with closed forms and asymptotics near critical indices; (iii) exponentially damped integrals $I_{n,\lambda}(a)$, providing a regularization framework with exact evaluations, limiting behavior, and convergence rates; and (iv) perspectives on multidimensional analogues, analytic continuation, and discrete series counterparts, which we formulate as open problems. Beyond the theoretical analysis, we demonstrate applications in signal processing, optics, quantum mechanics, and probability theory, showing how these integrals encode energy principles, diffraction patterns, wave-packet normalization, and uncertainty relations. The paper thus offers a unifying perspective on sinc-type integrals, combining rigorous analysis, explicit formulas, and practical interpretations, while identifying new directions for future research. \end{abstract}