TRIGONOMETRIC POLYNOMIAL

In the mathematical subfields of numerical analysis and mathematical analysis, a 'trigonometric polynomial' is a finite 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.

Contents

Definition

Notes

Definition

Let ''a''_''n'' be in 'C', 0 ≤ ''n'' ≤ ''N'' and ''a''_''N'' ≠ 0 then
:$T\_N(x)\; =\; sum\_\{n=0\}^N\; a\_n\; cos\; (nx)\; +\; mathrm\{i\}sum\_\{n=0\}^N\; a\_n\; sin(nx)\; qquad\; (x\; in\; mathbf\{R\})$
is called 'complex trigonometric polynomial' of degree ''N''. Using Euler's formula the polynomial can be rewritten as
:$T\_N(x)\; =\; sum\_\{n=0\}^N\; a\_n\; e^\{mathrm\{i\}nx\}\; qquad\; (x\; in\; mathbf\{R\})$
Analogously let ''a''_''n'', ''b''_''n'' be in 'R', 0 ≤ ''n'' ≤ ''N'' and ''a''_''N'' ≠ 0 or ''b''_''N'' ≠ 0 then
:
is called 'real trigonometric polynomial' of degree ''N''.

Notes

Using the relation
:$T\_\{2N\}(x)\; =\; e^\{mathrm\{i\}Nx\}\; t\_N(x)\; ,!$
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.