In this paper, we investigate the uniform convergence of the Fourier series expansions in terms of eigenfunctions for the spectral problemwhere is a spectral parameter, q.

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The paper is about investigating the uniform convergence conditions of spectral expansions of continuous functions in terms of root functions of a spectral problem with the same eigenparameter in the second-order differential equation and depending on quadratically in one of the boundary conditions on a closed interval.

The spectral problem −y ′′ + q(x)y = λy, 0 < x < 1, y(0) = 0, y ′ (0) = λ(ay(1) + by ′ (1)), is considered, where λ is a spectral parameter, q(x) ∈ L 1 (0, 1) is a complex-valued function, a and b are arbitrary complex numbers which satisfy the condition |a| + |b| ̸ = 0. We study the spectral properties (existence of eigenvalues, asymptotic formulae for eigenvalues and eigenfunctions, minimality and basicity of the system of eigenfunctions in L p (0, 1)) of the above-mentioned Sturm-Liouville problem.