Approximation by local parabolic splines with arbitrary knots

For the class of the functions $W_{\infty}^2$ with almost bounded second derivatives, a new linear local method of parabolic spline approximation on an arbitrary grid is constructed. This method has some smoothing properties and inherits the monotonicity and the convexity of the initial data (values of a function at the grid points). On this class the error of approximation by the splines constructed is exactly calculated.