On the uniform convergence of spectral expansions for a spectral problem with boundary conditions of the third kind one of which contains the spectral parameter

On the uniform convergence of the Fourier series for one spectral problem with a spectral parameter in a boundary condition

“…The system v n .x/ .n D 0, 1, : : : ; n ¤ r/ is defined by (7). Let Then according to (7), we obtain…”

“…where the system v n .x/ .n D 0, 1, : : : ; n ¤ r/ is defined by (7). Note that the series (61) is uniformly convergent on OE0, 1 if and only if the series…”

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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.x/ is a real-valued continuous function on the interval OE0, 1, and a 1 , b 0 , b 1 , c 1 , d 0 , and d 1 are real constants that satisfy the conditionswhere is a spectral parameter, q.x/ is a real-valued continuous function on the interval OE0, 1, and a 1 , b 0 , b 1 , c 1 , d 0 , and d 1 are real constants that satisfy the following conditions:In this article, we study the uniform convergence of the expansions in terms of eigenfunctions of the boundary value problem (1)-(3) for the functions that belong to C OE0, 1. There are many articles that investigate the uniform convergence of the expansions for the functions in terms of root functions of some differential operators with a spectral parameter in the boundary conditions (for example, [1][2][3][4][5][6][7][8]). The condition D a 1 d 1 b 1 c 1 > 0 is essential.

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“…In this article, we study the uniform convergence of the expansions in terms of eigenfunctions of the boundary value problem (1)-(3) for the functions that belong to C OE0, 1. There are many articles that investigate the uniform convergence of the expansions for the functions in terms of root functions of some differential operators with a spectral parameter in the boundary conditions (for example, [1][2][3][4][5][6][7][8]). The condition D a 1 d 1 b 1 c 1 > 0 is essential.…”

On two spectral problems with the same characteristic equation

“…Let the function f (x) be zero at the point x = a and be smooth enough on [a, b] to ensure that its derivative provides the uniform convergence of the Fourier series on [a, b] in the orthonormal basis of problem (4), (5). Then the function f (x) can be expanded in the uniformly convergent Fourier series We transform the Fourier coefficient by using the relation between the eigenfunctions of the con-…”