Optimal coefficients under signal expansion by $B$-splines

The paper is devoted to the problem of expanding a function given by a table as a linear combination of $B$-splines. An approach to this problem is presented. It is called the extended convolution approach. It prescribes to compute the coefficients of $B$-spline expansion of a function as linear combinations of function values at nodes near the nodes to whose values the corresponding $B$-spline introduces a nonzero contribution. In the paper, a criterion for the coefficients of such linear combinations to be optimal is formulated. The criterion resembles the minimum of the error in the uniform norm but is somewhat different. A conjecture on the technique for obtaining such coefficients is made and the corresponding system of linear equations on them is explicitly presented. The system is overdetermined but in the paper it is shown that the system has a unique solution. The proof of the conjecture is reduced to a question of roots of Eulerian polynomials and this assertion is verified for small values of parameters. The paper uses the results due to W. Trench on the determinants and inverse matrices for banded Toeplitz matrices and a minimax criterion for a system of absolute values of linear functions due to V. K. Ivanov. To verify the assertion of Eulerian polynomial roots, the SYM package for the computer algebra system MAXIMA for Linux was used.