On linear summability methods of Fourier series in polynomials orthogonal in a discrete Sobolev space
The paper considers a discrete Sobolev space, i.e., the Sobolev space with the following nonstandard inner product \[ \int_{-1}^1 f(x)g(x)w(x)dx + A_1 f(1)g(1)+ B_1 f(-1)g(-1) + A_2 f'(1)g'(1) + B_2 f'( - 1)g'( - 1) \] where \(w(x)\) is an almost everywhere positive weight function on \([-1,1]\) and \(A_1\), \(B_1\), \(A_2\), \(B_2\) are nonnegative reals. The Gram-Schmidt orthogonalization generates a sequence of polynomials \(q_n(x)= q_n( x; A_1, B_1, A_2, B_2 )\) (\( n=0,1,2,\dots\); \(x \in [-1,1] \)) of power \(n\) which is orthonormal with respect to the inner product. These polynomials are in connection with various problems of mathematics and engineering. Some recent results on the convergence of Fourier series in this system of polynomials of this author are also known; this paper contains some other results concerning Toeplitz regular summation methods of Fourier series in \(q_n(x)\).
scientific article; zbMATH DE number 6526924
scientific article