A trigonometric polynomial is a function of a real argument, which is a finite trigonometric sum, that is, a function represented as:
- ,
where the argument and the coefficients , but .
In a complex form according to the Euler formula, such a polynomial is written as follows:
- ,
Where .
This function is infinitely differentiable and -periodic - continuous on the unit circle.
Trigonometric polynomials are the most important means of approximation of functions, used for interpolation and solving differential equations .
According to the Weierstrass theorem, for any function continuous on a circle, there exists a sequence of trigonometric polynomials that converges uniformly to it.
The trigonometric polynomial is a partial sum of the Fourier series . According to Fejér's theorem, the sequence of arithmetic mean partial sums of a Fourier series uniformly converges to a function continuous on a circle. This provides a simple constructive method for constructing a uniformly convergent sequence of trigonometric polynomials.
Literature
- Mathematical Encyclopedic Dictionary. - M .: “Owls. Encyclopedia " , 1988. - p. 847.
- Zhuk V.V., Natanson G.I. Fourier trigonometric series and elements of the theory of approximation. - L .: Publishing House Leningrad. University, 1983. - p. 188.