Asymptotic behavior of sums of the type$\sum\limits_{\kappa = m}^{n - 1} {\exp } (i\omega \sqrt {n\kappa } )$ as n, m → ∞

Abstract

The asymptotic behavior asn, m → ∞ of the sum

$$\sum\limits_{\kappa ,\ell = m}^{n - 1} {\exp \left[ {i\omega \sqrt n \left( {\sqrt \kappa + \sqrt \ell } \right)} \right]} \Phi \left( {1 - \frac{{\left| {\sqrt \kappa - \sqrt \ell } \right|}}{\Delta }} \right)$$

is studied where π(t)=0 for t⩽0 and φ(t)=t for t > 0.