In the mathematical subfield of numerical analysis, trigonometric interpolation is a special form of interpolation on the unit circle in the complex plane using trigonometric polynomials. Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ... Illustration of a unit circle. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In the mathematical subfield of numerical analysis, a trigonometric polynomial is a finite linear linear combination of sin(nx) and cos(nx) with n a natural number. ...

Complex trigonometric interpolation

Given N real numbers of the form In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

and N complex numbers y_n with n = 0,...,N-1 we are trying to find a function f with In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

f(x_n) = y_n.

Due to the Stone-Weierstrass theorem this function exists and is unique. It is called complex trigonometric polynomial of degree N-1 and has the form In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. ... In the mathematical subfield of numerical analysis, a trigonometric polynomial is a finite linear linear combination of sin(nx) and cos(nx) with n a natural number. ...

with

where

is the i-th N-root of unity. In mathematics, the nth roots of unity or de Moivre numbers are all the complex numbers which yield 1 when raised to a given power n. ...