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In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. What is meant by best and simpler will depend on the application.
First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomials or rational functions) that often have desirable properties (inexpensive computation, continuity ...
Approximation theory is a branch of mathematics, and a quantitative part of functional analysis. Diophantine approximation deals with approximations of real numbers by rational numbers . Approximation usually occurs when an exact form or an exact numerical number is unknown or difficult to obtain.
In the mathematical theory of artificial neural networks, universal approximation theorems are theorems [1] [2] of the following form: Given a family of neural networks, for each function from a certain function space, there exists a sequence of neural networks ,, … from the family, such that according to some criterion.
Since a Padé approximant is a rational function, an artificial singular point may occur as an approximation, but this can be avoided by Borel–Padé analysis. The reason the Padé approximant tends to be a better approximation than a truncating Taylor series is clear from the viewpoint of the multi-point summation method.
In mathematics, approximation theory is concerned with how functions can be approximated with other, simpler, functions, and with characterizing in a quantitative way the errors introduced thereby. Subcategories
A best approximation for the second definition is also a best approximation for the first one, but the converse is not true in general. [4] The theory of continued fractions allows us to compute the best approximations of a real number: for the second definition, they are the convergents of its expression as a regular continued fraction.
The Thue–Siegel–Roth theorem says that, for algebraic irrational numbers, the exponent of 2 in the corollary to Dirichlet’s approximation theorem is the best we can do: such numbers cannot be approximated by any exponent greater than 2.