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We study a spectral problem related to the finite-dimensional characters ofthe groups $Sp(2N)$, $SO(2N+1)$, and $SO(2N)$, which form the classical series$C$, $B$, and $D$, respectively. The irreducible characters of these threeseries are given by $N$-variate symmetric polynomials. The spectral problem inquestion consists in the decomposition of the characters after theirrestriction to the subgroups of the same type but smaller rank $K<N$. The mainresult of the paper is the derivation of explicit determinantal formulas forthe coefficients in this decomposition.  In fact, we first compute these coefficients in a greater generality -- forthe multivariate symmetric Jacobi polynomials depending on two continuousparameters. Next, we show that the formulas can be drastically simplified forthe three special cases of Jacobi polynomials corresponding to the $C$-$B$-$D$characters. In particular, we show that then the coefficients are given bypiecewise polynomial functions. This is where a link with discrete splinesarises.  In type $A$ (that is, for the characters of the unitary groups $U(N)$),similar results were earlier obtained by Alexei Borodin and the author [Adv.Math., 2012], and then reproved by another method by Leonid Petrov [MoscowMath. J., 2014]. The case of the symplectic and orthogonal characters is moreintricate.

Characters of classical groups, Schur-type functions, and discrete splines