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Theory of Splines: Course Outline (MAT-637)

In mathematics, a spline is a numeric function that is piecewise-defined by polynomial functions, and which possesses a high degree of smoothness at the places where the polynomial pieces connect (which are known as nodes).

Course Outline

The Cubic Spline: Existence, Uniqueness, and Best Approximation, Convergence, Equal Intervals, Approximate Differentiation and Integration, Curve Fitting, Approximate Solution of Differential Equations, Approximate Solution of Integral Equations, Additional Existence and Convergence Theorems. Intrinsic Properties of Cubic Splines: The minimum Norm Property, The Best Approximation Property, The Fundamental Identity, The First Integral Relation, Uniqueness, Existence, General Equations, Convergence of Lower-Order Derivatives, The Second Integral Relation, Raising the Order of Convergence, Convergence of Higher-Order Derivatives, Limits on the order of Convergence, Hilbert Space Interpretation, Convergence in Norm, Canonical Mesh Bases and Their Properties, Remainder Formulas, Transformations Defined by a Mesh, A Connection with Space Technology.

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