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