Simultaneous Preservation of Orthogonality of Polynomials by Linear Operators Arising from Dilation of Orthogonal Polynomial Systems

摘要

For an orthogonal polynomial system p = (p_n)_(n∈IN_0) and a sequence d = (d_n)_(n∈IN)0 of nonzero numbers, let S_(p,d) be the linear operator defined on the linear space of all polynomials via S_(p,d) p_n = d_np_n for all n ∈ IN_0. We investigate conditions on p and d under which S_(p,d) can simultaneously preserve the orthogonality of different polynomial systems. As an application, we get that for p = (L_n~α), a generalized Laguerre polynomial system, no d can simultaneously preserve the orthogonality of two additional Laguerre systems, (L_n~(α+t_1) and (L_n~(α+t_2)), where t_1, t_2 ≠ 0 and t_1 ≠ t_2. On the other hand, for p = (T_n), the Chebyshev polynomial system and d = ((-1)~n), S_(p,d) simultaneously preserves the orthogonality of uncountably many kernel polynomial systems associated with p. We study many other examples of this type.