Spectral and pseudospectral functions of various dimensions for symmetric systems

The main object of the paper is a symmetric system $J{y}^{\mathrm{\prime }}-B(t)y=\text{l}\text{D}(t)y$ defined on an interval $\text{cI}=[a,b)$ with the regular endpoint $a$. Let $\text{f}(\text{cd},\text{l})$ be a matrix solution of this system of an arbitrary dimension and let $(Vf)(s)=\underset{\text{cI}}{\int }{\text{f}}^{\ast }(t,s)\text{D}(t)f(t)dt$ be the Fourier transform of the function $f(\text{cd})\in L_{\text{D}}^2(\text{cI})$. We define a pseudospectral function of the system as a matrix-valued distribution function $\text{s}(\text{cd})$ of the dimension $n_{\text{s}}$ such that $V$ is a partial isometry from $L_{\text{D}}^2(\text{cI})$ to $L^2(\text{s};{\text{bC}}^{n_{\text{s}}})$ with the minimally possible kernel. Moreover, we find the minimally possible value of $n_{\text{s}}$ and parameterize all spectral and pseudospectral functions of every possible dimensions $n_{\text{s}}$ by means of a Nevanlinna boundary parameter. The obtained results develop the results by Arov and Dym; A.~Sakhnovich, L.~Sakhnovich and Roitberg; Langer and Textorius.