Editing Polynomial (section)

== Polynomial expressions ==
{{anchor|Generalizations of polynomials}}<!-- [[Polynomial expression]] redirects to this section -->
Polynomials where indeterminates are substituted for some other mathematical objects are often considered, and sometimes have a special name.

=== 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]].

=== Matrix polynomials ===
{{Main|Matrix polynomial}}
A [[matrix polynomial]] is a polynomial with [[square matrix|square matrices]] as variables.<ref>{{cite book |title=Matrix Polynomials |volume=58 |series=Classics in Applied Mathematics |first1=Israel |last1=Gohberg |first2=Peter |last2=Lancaster |first3=Leiba |last3=Rodman |publisher=[[Society for Industrial and Applied Mathematics]] |location=Lancaster, PA |year=2009 |orig-year=1982 |isbn=978-0-89871-681-8 |zbl=1170.15300}}</ref> Given an ordinary, scalar-valued polynomial
<math display="block">P(x) = \sum_{i=0}^n{ a_i x^i} =a_0 + a_1 x+ a_2 x^2 + \cdots + a_n x^n, </math>
this polynomial evaluated at a matrix ''A'' is
<math display="block">P(A) = \sum_{i=0}^n{ a_i A^i} =a_0 I + a_1 A + a_2 A^2 + \cdots + a_n A^n,</math>
where ''I'' is the [[identity matrix]].{{sfn|Horn|Johnson|1990|p=36}}

A '''matrix polynomial equation''' is an equality between two matrix polynomials, which holds for the specific matrices in question. A '''matrix polynomial identity''' is a matrix polynomial equation which holds for all matrices ''A'' in a specified [[matrix ring]] ''M<sub>n</sub>''(''R'').

=== Exponential polynomials ===
A bivariate polynomial where the second variable is substituted for an exponential function applied to the first variable, for example {{math|''P''(''x'', ''e''<sup>''x''</sup>)}}, may be called an [[exponential polynomial]].