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

Abstract

For an orthogonal polynomial system $$p = \left( {p_n } \right)_{n \in N_0 } $$ and a sequence $$d = \left( {d_n } \right)_{n \in N} 0$$ of nonzero numbers,let $$S_{p,d} $$ be the linear operator defined on the linear spaceof all polynomials via $$S_{p,d} p_n = d_n p_n $$ for all $$n \in N_0 $$ .We investigate conditions on $$p$$ and $$d$$ under which $$S_{p,d} $$ can simultaneously preserve the orthogonality ofdifferent polynomial systems. As an application, we get that for $$p = \left( {L_n^\alpha } \right)$$ , a generalized Laguerre polynomial system, no $$d$$ can simultaneously preserve the orthogonality of twoadditional Laguerre systems, $$\left( {L_n^{\alpha + t_1 } } \right)$$ and $$\left( {L_n^{\alpha + t_2 } } \right)$$ , where $$t_1 ,t_2 \ne 0$$ and $$t_1 \ne t_2 $$ . On the other hand, for $$p = \left( {T_n } \right)$$ ,the Chebyshev polynomial system and $$d = \left( {\left( { - 1} \right)^n } \right)$$ , $$S_{p,d} $$ simultaneously preserves the orthogonality of uncountablymany kernel polynomial systems associated with p. We study manyother examples of this type.