Trigonometric polynomial

A trigonometric polynomial is a function of a real argument, which is a finite trigonometric sum, that is, a function represented as:

f(x)={\frac {a_{0}}{2}}+\sum _{k=1}^{n}(a_{k}\cos(kx)+b_{k}\sin(kx))

{\ displaystyle f (x) = {\ frac {a_ {0}} {2}} + \ sum _ {k = 1} ^ {n} (a_ {k} \ cos (kx) + b_ {k} \ sin (kx))}

{\ displaystyle f (x) = {\ frac {a_ {0}} {2}} + \ sum _ {k = 1} ^ {n} (a_ {k} \ cos (kx) + b_ {k} \ sin (kx))}

,

where the argument and the coefficients $x,a_{k},b_{k}\in \mathbb {R}$ ${\ displaystyle x, a_ {k}, b_ {k} \ in \ mathbb {R}}$ ${\ displaystyle x, a_ {k}, b_ {k} \ in \ mathbb {R}}$ , but $k=1,2,...,n$ ${\ displaystyle k = 1,2, ..., n}$ ${\ displaystyle k = 1,2, ..., n}$ .

In a complex form according to the Euler formula, such a polynomial is written as follows:

f(x)=\sum _{k=-n}^{k=n}c_{k}e^{ikx}

{\ displaystyle f (x) = \ sum _ {k = -n} ^ {k = n} c_ {k} e ^ {ikx}}

{\ displaystyle f (x) = \ sum _ {k = -n} ^ {k = n} c_ {k} e ^ {ikx}}

,

Where $c_{0}={\frac {a_{0}}{2}},c_{k}={\frac {(a_{k}-ib_{k})}{2}},c_{-k}={\frac {(a_{k}+ib_{k})}{2}}$ ${\ displaystyle c_ {0} = {\ frac {a_ {0}} {2}}, c_ {k} = {\ frac {(a_ {k} -ib_ {k})} {2}}, c_ { -k} = {\ frac {(a_ {k} + ib_ {k})} {2}}}$ ${\ displaystyle c_ {0} = {\ frac {a_ {0}} {2}}, c_ {k} = {\ frac {(a_ {k} -ib_ {k})} {2}}, c_ { -k} = {\ frac {(a_ {k} + ib_ {k})} {2}}}$ .

This function is infinitely differentiable and $2\pi$ ${\ displaystyle 2 \ pi}$ $2 \ pi$ -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.

Trigonometric polynomial

Literature

More articles: