In this paper an improved error bound is obtained for the complete quartic spline with deficiency 2, in the less smooth class of continuous functions. In the case of Lipschitzian functions, the obtained estimate improves the constant from Theorem 3, in J. Approx. Theory 58 (1989) 58-67. Some applications of the complete quartic spline in the numerical integration and in the construction of an iterative numerical method for fourth order two-point boundary value problems with pantograph type delay are presented.
For the Hermite type quartic spline interpolating on the partition knots and at the midpoint of each subinterval, we consider the estimation of the derivatives on the knots, and the values of these derivatives are obtained by constructing an algorithm of Akima's type. For computing the derivatives on endpoints are also considered alternatives that request optimal properties near the endpoints. The error estimate in the interpolation with this quartic spline is generally obtained in terms of the modulus of continuity. In the case of interpolating smooth functions, the corresponding error estimate reveal the maximal order of approximation O(h^3). A numerical experiment is presented for making the comparison between the Akima's cubic spline and the Akima's variant quartic spline havingdeficiency 2 and natural endpoint conditions.
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