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The Theory of Functional Connections (TFC) is a mathematical framework designed for functional interpolation.It introduces a method to derive a functional— a function that operates on another function—capable of transforming constrained optimization problems into equivalent unconstrained problems.
The Theory of Functional Connections (TFC) is a mathematical framework specifically developed for functional interpolation.Given any interpolant that satisfies a set of constraints, TFC derives a functional that represents the entire family of interpolants satisfying those constraints, including those that are discontinuous or partially defined.
Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing high-order accurate interpolants of unstructured data, possibly in high-dimensional spaces. The interpolant takes the form of a weighted sum of radial basis functions .
Nevanlinna–Pick interpolation — interpolation by analytic functions in the unit disc subject to a bound Pick matrix — the Nevanlinna–Pick interpolation has a solution if this matrix is positive semi-definite; Multivariate interpolation — the function being interpolated depends on more than one variable
In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points).
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The process of interpolation maps the function f to a polynomial p. This defines a mapping X from the space C ([ a , b ]) of all continuous functions on [ a , b ] to itself. The map X is linear and it is a projection on the subspace P ( n ) {\displaystyle P(n)} of polynomials of degree n or less.
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures.