Hilbert spaces of vector-valued functions generated by quadratic forms

We study Hilbert spaces $\mathfrak{L}^2(E, G)$, where $E\subset\mathbb{R}^d$ is a measurable set, $|E|>0$ and for almost every $t\in E$ the matrix $G(t)$ (see (3)) is a Hermitian positive-definite matrix. We find necessary and sufficient conditions for which the projection operators $T_k(f)(\cdot)=f_k(\cdot)\mathbf{e}_k$, $1\leqslant k\leqslant n$ are bounded. The obtained results allow us to translate various questions in the spaces $\mathfrak{L}^2(E, G)$ 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 $\{\varphi_m(t)\mathbf{e}_j, 1\leqslant j\leqslant n; m\in\mathbb{N}\}$ in the space $\mathfrak{L}^2(E, G)$, where $\Phi=\{\varphi_m\}_{m=1}^\infty$ is a complete orthonormal system defined on a measurable set $E\subset\mathbb{R}$. We concentrate our study on two classical systems: the Haar and the trigonometric systems. Simultaneous approximations of $n$ elements $F_1,\dots,F_n$ of some Banach spaces $X_1,\dots,X_n$ with respect to a system $\Psi$ which is a basis in any of those spaces are studied.