An application of the exponential spline for the approximation of a function and its derivatives in the presence of a boundary layer

The problem of exponential spline interpolation of functions having large gradients in the exponential boundary layer is considered. The spline of the class C²[0,1] is constructed as the sum of a polynomial of the second degree and a boundary-layer function on each grid interval. Estimates of the error in the approximation of a function and its derivatives are obtained. These estimates are uniform in small parameter. The limiting behavior of the exponential spline is investigated, when the perturbing parameter tends to infinity or to zero. In the first case, the spline becomes cubic, and in the second case it becomes parabolic. The results of numerical experiments are presented.