On $B_\varphi$-spline approximation

Evaluations of approach for function $u\in C^2(\alpha,\beta)$ with biorthogonal non-polynomial $B_\varphi$-spline approximation $\widetilde u$ of the first order are discussed. Spline grid $\{x_j\}_{j\in\mathbb Z}$ is defined on an interval $(\alpha, \beta)$ such that $\lim_{j\to -\infty}x_j=\alpha$, $\lim_{j\to +\infty}x_j=\beta$. Coordinate $B_\varphi$-splines are obtained by approximation relations with generating vector-function $\varphi=(\varphi_0,\varphi_1)^T$ under condition that absolute value of Wronskian for the functions $\varphi_0,\varphi_1$ isn't less than $c>0$. The method of integral representation of residual is applied; the last one differs from method of similarity, which is implicated in the case of polynomial splines. As a result the evaluations of norms $\|u^{(i)}-\widetilde u^{(i)}\|_{C[x_k,x_{k+1}]}$ are obtained by product of $2c^{-1}(x_{k+1}-x_{k})^{2-i}$ and
$$\sup_{\xi,\eta\in [x_k,x_{k+1}]} |\det(\Phi(x_k),\Phi\,'(\xi),\Phi\,''(\eta))|;$$
here $\Phi(t)= (\varphi_0(t),\varphi_1(t),u(t))^T$, $i=0,1,2$. The evaluations are exact for components of generating vector-functions $\varphi$. If $x_{k+1}-x_k\to 0$ then the determinant tends to the linear differential operator of the second order over function $u$, where fundamental solutions of the differential equation with mentioned operator and zero right part are functions $\varphi_0(t),\varphi_1(t)$. Bibliogr. 3.