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The book consists of six chapters based on the investigations of the theory of approximation and piecewise interpolatory functions. The first chapter consists of brief survey of introductory part of some basic results in connection with the matter considered in this book and also it includes some definitions and notations used in the relevant chapters for the sake of completeness. In Chapter II, we study existence, uniqueness and further error of approximation of Quartic convex type deficient splines by replacing continuity condition by some other kind of condition than the interpolation, through the selection of parameters. In Chapter III, we consider spline of general degree i.e., of degree m which has the same area as the function does in each partition of the interval. In the next Chapter we consider X-Spline by replacing the condition point interpolation by Area- Matching condition. In Chapter V, we investigated discrete cubic spline under the interpolatory condition under asort of area matching.The last Chapter deals with the existence, uniqueness and convergence of Complex cubic spline interpolation at general points.