Trigonometric polynomial
Encyclopedia
In the mathematical subfields of numerical analysis
and mathematical analysis
, a trigonometric polynomial is a finite linear combination
of functions
sin(nx) and cos(nx) with n a natural number
. The coefficients may be taken as real numbers, for real-valued functions. For 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 used also in the discrete Fourier transform
.
The term trigonometric polynomial for the real-valued case can be seen as using the analogy
: the functions sin(nx) and cos(nx) are similar to the monomial basis
for polynomial
s. In the complex case the trigonometric polynomials are spanned by the positive and negative powers of eix.
with an, bn in C for 0 ≤ n ≤ N, is called a complex trigonometric polynomial of degree N . Using Euler's formula
the polynomial can be rewritten as
Analogously let an, bn be in R, 0 ≤ n ≤ N and aN ≠ 0 or bN ≠ 0 then
is called real trigonometric polynomial of degree N .
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
and mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, a trigonometric polynomial is a finite linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
sin(nx) and cos(nx) with n a natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
. The coefficients may be taken as real numbers, for real-valued functions. For complex
Complex
A complex is a whole that comprehends a number of intricate parts, especially one with interconnected or mutually related parts; for example, a complex of buildings.Complex may refer to:-Biology:...
coefficients, there is no difference between such a function and a finite Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
.
Trigonometric polynomials are widely used, for example in trigonometric interpolation
Trigonometric interpolation
In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes through some given data points. For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and...
applied to the interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
of periodic function
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
s. They are used also in the discrete Fourier transform
Discrete Fourier transform
In mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...
.
The term trigonometric polynomial for the real-valued case can be seen as using the analogy
Analogy
Analogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process...
: the functions sin(nx) and cos(nx) are similar to the monomial basis
Monomial basis
In mathematics a monomial basis is a way to describe uniquely a polynomial using a linear combination of monomials. This description, the monomial form of a polynomial, is often used because of the simple structure of the monomial basis....
for polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s. In the complex case the trigonometric polynomials are spanned by the positive and negative powers of eix.
Formal definition
Any function T of the formwith an, bn in C for 0 ≤ n ≤ N, is called a complex trigonometric polynomial of degree N . Using Euler's formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
the polynomial can be rewritten as
Analogously let an, bn be in R, 0 ≤ n ≤ N and aN ≠ 0 or bN ≠ 0 then
is called real trigonometric polynomial of degree N .