Editing Polynomial (section)

=== Trigonometric polynomials ===
{{Main|Trigonometric polynomial}}
A '''trigonometric polynomial''' is a finite [[linear combination]] of [[function (mathematics)|functions]] sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more [[natural number]]s.<ref>{{cite book |last1=Powell |first1=Michael J. D. |author1-link=Michael J. D. Powell |title=Approximation Theory and Methods |publisher=[[Cambridge University Press]] |isbn=978-0-521-29514-7 |year=1981}}</ref> The coefficients may be taken as real numbers, for real-valued functions.

If sin(''nx'') and cos(''nx'') are expanded in terms of sin(''x'') and cos(''x''), a trigonometric polynomial becomes a polynomial in the two variables sin(''x'') and cos(''x'') (using the [[List of trigonometric identities#Multiple-angle formulae|multiple-angle formulae]]). Conversely, every polynomial in sin(''x'') and cos(''x'') may be converted, with [[List of trigonometric identities#Product-to-sum and sum-to-product identities|Product-to-sum identities]], into a linear combination of functions sin(''nx'') and cos(''nx''). This equivalence explains why linear combinations are called polynomials.

For [[complex number|complex coefficients]], there is no difference between such a function and a finite [[Fourier series]].

Trigonometric polynomials are widely used, for example in [[trigonometric interpolation]] applied to the [[interpolation]] of [[periodic function]]s. They are also used in the [[discrete Fourier transform]].