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We investigate the asymptotic behavior of a generalized sine kernel acting ona finite size interval [-q,q]. We determine its asymptotic resolvent as well asthe first terms in the asymptotic expansion of its Fredholm determinant.Further, we apply our results to build the resolvent of truncated Wiener--Hopfoperators generated by holomorphic symbols. Finally, the leading asymptotics ofthe Fredholm determinant allows us to establish the asymptotic estimates ofcertain oscillatory multidimensional coupled integrals that appear in the studyof correlation functions of quantum integrable models.

Riemann-Hilbert approach to a generalized sine kernel and applications