VECTOR-VALUED LP-CONVERGENCE OF ORTHOGONAL SERIES AND LAGRANGE INTERPOLATION

We give necessary and sufficient conditions for interpolation inequali ties of the type considered by Marcinkiewicz and Zygmund to be true in the case of Banach space-valued polynomials and Jacobi weights and no des. We also study the vector-valued expansion problem of L(p)-functio ns in terms of Jacobi polynomials and consider the question of uncondi tional convergence. The notion of type p with respect to orthonormal s ystems leads to some characterizations of Hilbert spaces. It is also s hown that various vector-valued Jacobi means are equivalent.