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Numerical analysis and weighted residuals
Published in Ken P. Chong, Arthur P. Boresi, Sunil Saigal, James D. Lee, Numerical Methods in Mechanics of Materials, 2017
Ken P. Chong, Arthur P. Boresi, Sunil Saigal, James D. Lee
There exists a broad area of mathematics known as approximation theory (Shisha, 1968). The term is usually reserved for that branch of mathematics devoted to the approximation of general functions by means of simple functions. For example, in practice, we may wish to approximate a real arbitrary function f(x) by means of a polynomial p(x) in some finite interval of space, say, a ≤ x ≤ b. The motivation for such an approximation is often one of simplification, particularly when the function f(x) is too complicated to manipulate. Because the approximation, say F(x), ordinarily differs from f(x), questions arise immediately as to the manner in which we should proceed. As noted by Shisha, approximation theory considers such problems as the following:
RBF Network Approximation and Persistence of Excitation
Published in Cong Wang, David J. Hill, Deterministic Learning Theory, 2018
Approximation theory has undergone major advances during the past two decades. Fundamental approximation theory includes interpolation, least squares, and Chebyshev approximation by polynomials, splines, and orthogonal-polynomials, which are still important and interesting topics. Nonetheless, some significant developments have emerged, which include new approximating tools, nonlinear approximation, and multivariate approximation [31]. RBF approximation is one of the most often applied approaches for multivariate approximation in modern approximation theory and has been considered in many applications [23].
Representation, Approximation, and Identification
Published in Wai-Kai Chen, Feedback, Nonlinear, and Distributed Circuits, 2018
The mathematical term “approximation” used here refers to the theory and methodology of function (functional or operator) approximation. Mathematical approximation theory and techniques are important in engineering when one seeks to represent a set of discrete data by a continuous function, to replace a complicated signal by a simpler one, or to approximate an infinite-dimensional system by a finitedimensional model, etc., under certain optimality criteria.
Approximation properties of bivariate α-fractal functions and dimension results
Published in Applicable Analysis, 2021
Sangita Jha, A. K. B. Chand, M. A. Navascués, Abhilash Sahu
The most essential question in classical approximation theory is how to represent an arbitrary function or a dataset concerning traditional (piecewise) smooth functions. In most physical situations, the original data-generating functions are non-smooth in nature, and we cannot use classical approximation techniques to approximate them. Barnsley [1,2] introduced the concept of fractal interpolation functions (FIFs) that are constructed as attractors of suitable iterated function systems (IFSs) on complete metric spaces. In [3], authors have studied the attractors of IFSs in uniform spaces. Fractal functions constitute advancement in the technique of approximation, since all the traditional functions of real-world data interpolation, can be generalized through fractal methods. For a given continuous function f, a family of smooth or non-smooth α-fractal functions can be obtained depending on the choice of scaling parameters. Wang and Yu [4] introduced FIF with variable scaling functions and Serpa and Buescu [5] gave an explicit representation of FIF with this setting. The notion of FIF provides a bounded linear operator, termed the α-fractal operator. Navascués studied these operators in details in earlier works [6,7]. The concept of α-fractal operator links the theory of fractal function to the area Functional analysis, Operator theory, Harmonic analysis, and Approximation theory.