A trigonometric polynomial in the sum
of sines and cosines ,
(1)
where nonnegative integer and the coefficients
may be real or complex
(Constantin 2016, pp. 1-3). Equivalently,
(2)
so Laurent polynomial in
Every such differential operator identity
(3)
where sine
and cosine terms, and the remaining holonomic function .
In almost periodic function theory, the term also allows finite sums real
frequencies et al. 2006, p. 14). Such sums need
not be periodic .
See also Almost Periodic Function ,
Fourier Series ,
Holonomic
Function ,
Laurent Polynomial ,
Trigonometric
Functions
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References Bergelson, V.; Furstenberg, H.; and Weiss, B. "Piecewise-Bohr Sets of Integers and Combinatorial Number Theory." In Topics in Discrete
Mathematics (Ed. M. Klazar, J. Kratochvíl, M. Loebl, J. Matoušek,
P. Valtr, and R. Thomas). Berlin, Germany: Springer-Verlag, pp. 13-37,
2006. https://doi.org/10.1007/3-540-33700-8_2 . Constantin,
A. Fourier
Analysis: Part I--Theory. Trott, M. "Solution of Equations." §1.5
in The
Mathematica GuideBook for Symbolics. https://doi.org/10.1007/0-387-28815-5 .
https://www.mathematicaguidebooks.org/ .
Cite this as:
Weisstein, Eric W. "Trigonometric Polynomial."
From MathWorld https://mathworld.wolfram.com/TrigonometricPolynomial.html
Subject classifications
-periodic Fourier convention is a finite sum
of sines and cosines,
![T(x)=a_0+sum_(k=1)^n[a_kcos(kx)+b_ksin(kx)],](/images/equations/TrigonometricPolynomial/NumberedEquation1.svg)
is a nonnegative integer and the coefficients
may be real or complex
(Constantin 2016, pp. 1-3). Equivalently,
is a Laurent polynomial in 
.
satisfies the differential operator identity
.
Each quadratic factor annihilates the corresponding sine
and cosine terms, and the remaining 
annihilates the constant term. Thus every trigonometric polynomial
is a holonomic function.
with arbitrary real
frequencies 
(Bergelson et al. 2006, p. 14). Such sums need
not be periodic.
