In this thesis we extend the problem of interpolating one variable functions to interpolating two variable functions. For a simpler description of the Hermite interpolation problem, we introduce some notation so that the interpolation problem can be described in terms of a tree structure which can be further arranged in a blockwise structure. We derive the Vandermond matrix and look at some aspects of poisedness, using the notion of blockwise structure. We write down a number of conditions for never poisedness, poisedness and almost poisedness. We consider the computation of Hermite basis polynomials and derive a Newton basis which, together with finite differences, allows us to write the Hermite interpolation polynomials in a closed form. The whole thesis is supported by many practical examples to make it easier to understand.
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