Error estimates for one local interpolation cubic spline with the maximum approximation order of O(h3) for W1[a, b] classes

It is known that local splines have been widely used for processing of various signals such as restoration of geophysical, biomedical signals. Moreover, it uses for the approximate calculation of regular and singular integrals, integrals of the Fourier types. Using such splines, we can build effective quadrature and cubature formulas. It uses for approximate solution of regular and singular integrals equations. In this note, error estimation of local spline functions is obtained in the class of function W1[a, b]. Moreover, comparisons with other splines are shown in graphical experiments. On the other hand, it is shown that the order of approximation of local spline function is higher than the Ryabenky and Grebennikov cubic splines. It is shown that local spline function in all cases achieved tangible results in comparison with the classical polynomial approximation. It is well known that the maximum order of approximation by the Ryabenky and Grebennikov cubic splines is O(h2) and maximum order of approximation of the local cubic spline function is O(h3), and the interpolation condition is satisfied. Numerical experiments are also in the line of theoretical findings.

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