Hilbert spaces of vector-valued functions generated by quadratic forms

We study Hilbert spaces (sic)(2)(E, G), where E subset of R-d is a measurable set, vertical bar E vertical bar > 0 and for almost every t is an element of E the matrix G(t) (see (3)) is a Hermitian positive-definite matrix. We find necessary and sufficient conditions for which the projection operators T-k(f)(.) = f(k)(.)e(k), 1 <= k <= n are bounded. The obtained results allow us to translate various questions in the spaces (sic)(2)(E, G) to weighted norm inequalities with weights which are the diagonal elements of the matrix G(t). In Section 3 we study the properties of the system (phi(m)(t)e(j), 1 <= j <= n; m is an element of N) in the space, (sic)(2)(E, G), where Phi = (phi(m))(m=1)(infinity), is a complete orthonormal system defined on a measurable set E subset of R. We concentrate our study on two classical systems: the Haar and the trigonometric systems. Simultaneous approximations of n elements F-1, ..., F-n of some Banach spaces X-1, ..., X-n with respect to a system psi which is a basis in any of those spaces are studied. (C) 2013 Elsevier Inc. All rights reserved.