| Summary: | In carrying out continuous spline interpolation of a function,derivatives of the function at some points are always needed.However, in the real world situation, not only that it may be difficult to compute the derivatives of a function, the derivatives may not even exist at some points. In such a situation, the usual continuous spline interpolation will not be suitable. We therefore introduce a discrete interpolation scheme that involves only differences. Since no derivatives are involved, the discrete interpolant can be constructed for a more general class of functions and therefore has a wider range of applications. In this thesis, we shall develop two kinds of discrete spline via a constructive approach, the first kind of discrete spline involves forward differences, while the second kind of discrete spline involves central differences. We recall that a quintic polynomial is a polynomial of degree five.In the first case where $f(t)$ defined on a discrete interval, we shall develop a class of quintic discrete Hermite interpolant and derive explicit error bounds in $\ell_\infty$ norm. We also establish, for a two-variable function $f(t,u)$ defined on a discrete rectangle, the biquintic discrete Hermite interpolant and perform the related error analysis. Based on the results of discrete Hermite interpolation, we then define the quintic discrete spline interpolant of the function $f(t)$, formulate its construction, and establish explicit error estimates between $f(t)$ and its spline interpolant. We also tackle the two-variable discrete spline interpolation and the corresponding error analysis for $f(t,u)$. As an application, we solve Fredholm integral equations numerically by using biquintic discrete splines to degenerate the kernels, and furnish the related error analysis. In the second case where central differences are involved, for a periodic function $f(t)$ defined on a discrete interval, we construct the periodic quintic discrete spline interpolant and obtain the explicit error estimates between the function and its spline interpolant. The treatment is then extended to a periodic function $f(t,u)$ defined on a discrete rectangle, here we establish the two-variable periodic discrete spline interpolant and also provide the error analysis. As applications, we solve second order and fourth order boundary value problems by discrete splines involving central differences. Not only that we tackle the related convergence and error analysis, comparisons with other known methods in the literature are also illustrated by several examples.
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