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Journal of Contemporary Applied Mathematics, vol.16, no.1, pp.94-109, 2026 (Scopus)
In this paper it is considered the trigonometric system {1; cos nx; x sin nx}n∈N, which is a collection of eigenfunctions of one nonlocal spectral problem for an ordinary second order differential operator. Let X (-π, π) be a Banach Function Space (by Luxembourg classification) on (-π, π) with Lebesgue measure. A criterion is obtained for the trigonometric system{1/2; cosnt; sinnt}n∈N to have the Riesz Property in X (-π, π). It is proved that if the trigonometric system has the Riesz Property n∈N in X (-π, π), then the system (T) also forms a basis for X (-π, π). Some concrete spaces, such as the weighted Lebesgue space Lp;w (-π, π), the weighted grand Lebesgue space Lp);w (-π, π), Lebesgue space with variable exponent Lp(·) (-π, π), Morrey space Lp;λ (-π, π), symmetric space X (-π, π) with Boyd indices αX; βX ∈ (0, 1) are presented.