Exact solution of optimization task generated by simplest heat conduction equation

In the previous paper of the authors the parameter family of finite-dimensional spaces of special quadratic splines of Lagrange's type has been defined. In each space, as a solution to the initial-boundary problem for the simplest heat conduction equation, we have proposed the optimal spline, which gives the smallest residual. We have obtained exact formulas for coefficients of this spline and its residual. The formula for coefficients of this spline is a linear form of initial finite differences. The formula for the residual is a positive definite quadratic form of these quantities, but because of its bulkiness it is ill-suited for analyzing of the approximation quality of the input problem at the variation with the parameters.
For the purposes of the present paper, we have obtained an alternative representation for the residual, which is the sum of two positive definite quadratic forms of the new finite differences defined on the boundary. The matrix of the first form has second order and the apparent spectrum. The elements of the second matrix of order $N+1$ are expressed in terms of Chebyshev's polynomials, the matrix is invertible and the inverse matrix has a tridiagonal form. This feature allows us to obtain, for the spectrum of the matrix, upper and lower bounds that are independent of the dimension $N$. Said fact allows us to make a study of the quality of approximation for different dimensions $N$ and weights $\omega\in[-1,1]$. It is shown that the parameter $\omega=0$ gives the best approximation and the residual tends to zero as $N$ increasing.