Exact formulas for coefficients and residual of optimal approximate spline of simplest wave equation

We define the parameter family of finite-dimensional spaces of special quadratic splines of Lagrange's type. In each space, the optimal spline which gives the smallest residual being a square of the norm in the space $\mathrm L_2$, is proposed as a solution to the initial-boundary problem for the simplest wave equation. The exact formulas for the coefficients of the spline and its residual are obtained. The formula for the coefficients of this spline is a linear form of finite differences of the discretely given initial and boundary conditions of the original problem. The formula for the residual $J$ is a positive definite quadratic form of these quantities. The coefficients of both forms are computable via Chebyshev's polynomials of the second kind. The explicit form of the formula for the residual allows to solve the inequality $J<\varepsilon^2$ for a given computing accuracy $\varepsilon>0$ and to receive a priori sufficient number of nodes of a difference scheme.
The investigations were carried out for one time layer, which has two sublayers. We obtained difference formulas of the initial condition for the partial derivative with respect to time. They allow to create a difference scheme for the new layer, which in turn allows to continue the iterative computational process in time as far as desired.