Improved orders of approximation derived from interpolatory cubic splines

Let s be a cubic spline, with equally spaced knots on [a,b], interpolating a given function y at the knots. The parameters which determine s are used to construct a piecewise defined polynomial P of degree four. It is shown that P can be used to give better orders of approximation to y and its derivatives than those obtained from s. It is also shown that the known superconvergence properties of the derivatives of s, at specific points [a,b], are all special cases of the main result contained in the present paper

End conditions for interpolatory quintic splines

Accurate end conditions are derived for quintic spline interpolation at equally spaced knots. These conditions are in terms of available function values at the knots and lead to 0(h6) covergence uniformly on the interval of interpolation

Superconvergence properties of quintic interpolatroy splines

Let Q be a quintic spline with equi-spaced knots on [a,b] interpolating a given function y at the knots. The parameters which determine Q are used to construct a piecewise defined polynomial P of degree six. It is shown that P can be used to give at any point of [a,b] better orders of approximation to y and its derivatives than those obtained from Q. It is also shown that the superconvergence properties of the derivatives of Q, at specific points of [a,b], are all simple consequences of the properties of P

A class of piecewise cubic interpolatory polynomials

A new class of C1 piecewise—cubic interpolatory polynomials is defined, by generalizing the definition of cubic X-splines given recently by Clenshaw and Negus (1978). It is shown that this new class contains a number of interpolatory functions which present practical advantages, when compared with the conventional cubic spline

Structural and Optical Properties of TiO2/Ag/TiO2 Multi Layers

ABSTRACT Titanium dioxide nano layers of same thicknesses, and near normal deposition angle, and same deposition rate were deposited on glass substrates, at room temperature, under UHV conditions. Silver nano particles of different thicknesses also near normal deposition angle and same deposition rates were post deposited on Titanium dioxide nano layers at room temperature under UHV conditions. At least again Titanium dioxide nano layers as third nano layer of same thicknesses with near normal deposition rate, same deposition angle at room temperature under UHV conditions, were deposited by thermal evaporation. nano structure of these multi layers were determined by AFM and XRD methods. Roughness of the films changed due to different thicknesses of silver nano particles. . Optical Reflectance were studied by spectrophotometer method. Correlation between nano-structures and optical reflectance studied. Thickness of titanium dioxide and silver nano particles can play an important role in the nano-structure of the films