In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform.
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.
In mathematics, the term linear function refers to two distinct but related notions:
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset.
In numerical analysis, the Newton–Cotes formulas, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulas for numerical integration based on evaluating the integrand at equally spaced points. They are named after Isaac Newton and Roger Cotes.
In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics. This contrasts with classical analysis, which simply means analysis done according to the principles of classical mathematics.
Errett Albert Bishop was an American mathematician known for his work on analysis. He expanded constructive analysis in his 1967 Foundations of Constructive Analysis, where he proved most of the important theorems in real analysis by constructive methods.
In mathematics, a time series is a series of data points indexed in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.
A radial basis function (RBF) is a real-valued function whose value depends only on the distance between the input and some fixed point, either the origin, so that , or some other fixed point , called a center, so that . Any function that satisfies the property is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection which forms a basis for some function space of interest, hence the name.
Mergelyan's theorem is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951.
In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable ; when the variates are spatial coordinates, it is also known as spatial interpolation.
In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation. It is closely related to approximation theory. The term was coined by Sergei Bernstein.
Isidor Pavlovich Natanson was a Swiss-born Soviet mathematician known for contributions to real analysis and constructive function theory, in particular, for his textbooks on these subjects. His son, Garal'd Natanson (1930–2003), was also a known mathematician.
Lee Albert Rubel was a mathematician known for his contributions to analog computing.
Edward Barry Saff is an American mathematician, specializing in complex analysis, approximation theory, numerical analysis, and potential theory.
George Gunter Lorentz was a Russian-American mathematician.
Charles Anthony Micchelli is an American mathematician, with an international reputation in numerical analysis, approximation theory, and machine learning.
