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Encyclopedia > Trigonometric polynomial

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. Hence the term trigonometric polynomial as the sin(nx)s and cos(nx)s are used similar to the monomial basis for a polynomial.

The trigonometric polynomials are used in trigonometric interpolation to interpolate periodic functions. They are used in the discrete Fourier transform which is a special kind of trigonometric interpolation.

## Definition

Let an be in C, 0 ≤ nN and aN ≠ 0 then

is called complex trigonometric polynomial of degree N. Using Euler's formula the polynomial can be rewritten as

Analogously let an, bn be in R, 0 ≤ nN and aN ≠ 0 or bN ≠ 0 then

is called real trigonometric polynomial of degree N.

## Notes

Using the relation

T2N(x) = eiNxtN(x)

we can construct a bijective mapping between the complex trigonometric polynomials and the real trigonometric polynomials. Thus a trigonometric polynomial can be considered a periodic function on the real line, with period some multiple of 2π, or as a function on the unit circle.

A trigonometric polynomial of degree N has a maximum of N roots in any open interval [a, a + 2π) with a in R.

A basic result is that the trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm. This is a special case, for example, of the Stone-Weierstrass theorem.

Results from FactBites:

 Reference.com/Encyclopedia/Trigonometric polynomial (317 words) Hence the term trigonometric polynomial as the sin(nx)s and cos(nx)s are used similar to the monomial basis for a polynomial. The trigonometric polynomials are used in trigonometric interpolation to interpolate periodic functions. A basic result is that the trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm.
 Trigonometric polynomial - Wikipedia, the free encyclopedia (252 words) Hence the term trigonometric polynomial as the sin( nx)s and cos( nx)s are used similar to the monomial basis for a polynomial. The trigonometric polynomials are used in trigonometric interpolation to interpolate periodic functions. A basic result is that the trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm.
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