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Solutions of Fredholm integral equations of the second kind with oscillatorykernels likely exhibit oscillation. Standard numerical methods applied tosolving equations of this type have poor numerical performance due to theinfluence of the highly rapid oscillation in the solutions. Understanding ofthe oscillation of the solutions is still inadequate in the literature and thusit requires further investigation. For this purpose, we introduce a notion todescribe the degree of oscillation of an oscillatory function based on thedependence of its norm in a certain function space on the wavenumber. Based onthis new notion, we construct structured oscillatory spaces with oscillatorystructures. The structured spaces with a specific oscillatory structure cancapture the oscillatory components of the solutions of Fredholm integralequations with oscillatory kernels. We then further propose oscillationpreserving Galerkin methods for solving the equations by incorporating thestandard approximation subspace of spline functions with a finite number ofoscillatory functions which capture the oscillation of the exact solutions ofthe integral equations. We prove that the proposed methods have the optimalconvergence order uniformly with respect to the wavenumber and they arenumerically stable. A numerical example is presented to confirm the theoreticalestimates.

Oscillation Preserving Galerkin Methods for Fredholm Integral Equations of the Second Kind with Oscillatory Kernels