Trigonometric polynomial

In real analysis, a trigonometric polynomial , also called a trigonometric sum , is a finite, real linear combination of the trigonometric functions and , the linear combination being defined as a function for . These real-valued functions also allow an unambiguous (formally) complex representation in which certain complex linear combinations are formed from the exponential functions instead of the cosine and sine functions. This representation is often used to simplify invoices. The real trigonometric polynomials are partial sums of real Fourier series and, among other things, play an important role in the solution of ordinary , linear differential equations with constant coefficients and for the discrete Fourier transformation . ${\displaystyle x\rightarrow \cos(kx)\;(k\in \mathbb {N} _{0})}$ ${\displaystyle x\rightarrow \sin(kx)\;(k\in \mathbb {N} \setminus \{0\})}$ ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle e^{ikx}\;(k\in \mathbb {Z} )}$

In function theory , functional analysis and in many applications, such as analytical number theory (see Winogradow's circle method in this article), any complex linear combination of functions with a fixed real is referred to as a complex trigonometric polynomial or complex trigonometric sum. ${\displaystyle x\rightarrow e^{ik\omega x}\;(k\in \mathbb {Z} )}$ ${\displaystyle \omega >0}$

Both the real and the complex trigonometric polynomials provide unambiguous best approximations - for every given degree there is exactly one best approximation among the trigonometric polynomials that have at most this degree - in the root mean square for each function of the function space, which the generating trigonometric functions in each case as an orthonormal basis ( Orthogonal system ). ${\displaystyle n}$

If one allows an infinite number of non-vanishing “summands” in the linear combinations, then one arrives at the terms of a real or complex trigonometric series .

Definitions

Real trigonometric polynomial

The real- valued function is defined as the real trigonometric polynomial for ${\displaystyle x\in \mathbb {R} }$

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

denotes, where is. The natural number is called the degree of , if or not vanishing. The function has the period . ${\displaystyle a_{k},b_{k}\in \mathbb {R} }$ ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle a_{n}}$ ${\displaystyle b_{n}}$ ${\displaystyle f}$ ${\displaystyle 2\pi }$

Any period

A real trigonometric polynomial can also be defined more generally in such a way that the period of the polynomial is any positive, real number . If one sets , then the polynomials read: ${\displaystyle T}$ ${\displaystyle \omega ={\frac {2\pi }{T}}}$

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

the same requirements and designations apply to the other parameters as in the special case ${\displaystyle T=2\pi ,\omega =1.}$

Complex representation

The complex representation of the real trigonometric polynomial is:

${\displaystyle f(x)=\sum _{k=-n}^{k=n}c_{k}e^{ikx}}$in the case or in the case of any period. ${\displaystyle \omega =1}$ ${\displaystyle f(x)=\sum _{k=-n}^{k=n}c_{k}e^{ik\omega x}}$

The following applies and vice versa can be represented by the real part of the complex representation and by its imaginary part . The trigonometric polynomial is real if and only if holds. ${\displaystyle c_{0}={\frac {a_{0}}{2}},c_{k}={\frac {(a_{k}-ib_{k})}{2}},c_{-k}={\frac {(a_{k}+ib_{k})}{2}}}$ ${\displaystyle a_{k}=2\cdot \operatorname {Re} (c_{k})=2\cdot \operatorname {Re} (c_{-k})}$ ${\displaystyle b_{k}=-2\cdot \operatorname {Im} (c_{k})=2\cdot \operatorname {Im} (c_{-k})}$ ${\displaystyle c_{k}={\overline {c_{-k}}}}$

Complex trigonometric polynomial

If a family of complex coefficients, which vanish for all but a finite number of indices , and a positive, real number, then the sum becomes ${\displaystyle (c_{k})_{k\in \mathbb {Z} }}$ ${\displaystyle k\in \mathbb {Z} }$ ${\displaystyle \omega >0}$

${\displaystyle f(x)=\sum _{k\in \mathbb {Z} }c_{k}e^{ik\omega x}}$referred to as a complex trigonometric polynomial or complex trigonometric sum .

As a rule, is the independent variable in this sum remains a real number, and the sum then provides a -periodic function . Here, the amount is quantitatively the largest integer for which applies when the degree called the complex trigonometric polynomial. ${\displaystyle x}$ ${\displaystyle {\frac {2\pi }{\omega }}}$ ${\displaystyle f\colon \mathbb {R} \to \mathbb {C} }$ ${\displaystyle k\in \mathbb {Z} }$ ${\displaystyle c_{k}\neq 0}$ ${\displaystyle |k|}$

Trigonometric series

Analogous to the term trigonometric polynomial, the term (formal) trigonometric series can also be defined. These are used as Fourier series of periodic functions.

• Real trigonometric series can be represented as follows:

${\displaystyle {\frac {a_{0}}{2}}+\sum _{k=1}^{\infty }(a_{k}\cos k\omega x+b_{k}\sin k\omega x)}$ With ${\displaystyle a_{k},b_{k}\in \mathbb {R} }$

or in the complex representation

${\displaystyle \sum _{k=-\infty }^{k=\infty }c_{k}e^{ik\omega x}}$with . ${\displaystyle c_{k}={\overline {c_{-k}}}}$

• If you omit the condition for the coefficients , you get a complex trigonometric series : ${\displaystyle c_{k}}$

${\displaystyle \sum _{k=-\infty }^{k=\infty }c_{k}e^{ik\omega x}.}$

The definition range and the period are always the same as for the corresponding trigonometric polynomials ${\displaystyle \omega >0}$ ${\displaystyle D=\lbrace x\in \mathbb {R} \rbrace }$ ${\displaystyle T={\tfrac {2\pi }{\omega }}.}$

properties

Orthogonality

The trigonometric functions, from which the real trigonometric polynomials arise through linear combination, fulfill the following orthogonality relations : ${\displaystyle \left(k,l\in \mathbb {N} _{0}\right)}$

1. ${\displaystyle \int _{0}^{\frac {2\pi }{\omega }}\cos(k\omega x)\cdot \sin(l\omega x)\,dx=0}$,
2. ${\displaystyle \int _{0}^{\frac {2\pi }{\omega }}\cos(k\omega x)\cdot \cos(l\omega x)\,dx={\begin{cases}0\quad (k\neq l)\\{\frac {\pi }{\omega }}\quad (k=l\neq 0)\\{\frac {2\pi }{\omega }}\quad (k=l=0)\end{cases}}}$
3. ${\displaystyle \int _{0}^{\frac {2\pi }{\omega }}\sin(k\omega x)\cdot \sin(l\omega x)\,dx={\begin{cases}0\quad (k\neq l)\\{\frac {\pi }{\omega }}\quad (k=l\neq 0)\\0\quad (k=l=0).\end{cases}}}$

For the complex generators the orthogonality relation reads : ${\displaystyle \left(k,l\in \mathbb {Z} \right)}$

${\displaystyle \int _{0}^{\frac {2\pi }{\omega }}e^{ik\omega x}\cdot e^{-il\omega x}\,dx={\begin{cases}0\quad (k\neq l)\\{\frac {2\pi }{\omega }}\quad (k=l).\end{cases}}}$

Basic property

From the orthogonality relations it follows that the sequence of generating trigonometric polynomials is linearly independent . With suitable normalization, it forms an orthonormal basis of a real Hilbert space . This Hilbert space is the Lebesgue space . ${\displaystyle {\mathcal {B}}=\left((\cos(k\omega x))_{k\in \mathbb {N} _{0}},(\sin(l\omega x))_{l\in \mathbb {N} \setminus \lbrace 0\rbrace }\right)}$ ${\displaystyle L^{2}[0;T]}$

The family of generators of the complex trigonometric polynomials is also linearly independent and, with suitable normalization, forms an orthonormal basis of the complex Hilbert space of the complex-valued functions defined on the unit circle , if they are viewed as parameterized Laurent series and otherwise a basis of the complex Hilbert space of complex-valued functions on . ${\displaystyle {\mathcal {C}}=\left(e^{ik\omega x}\right)_{k\in \mathbb {Z} }}$ ${\displaystyle L^{2}(\mathbb {S} ^{1})}$ ${\displaystyle L^{2}}$ ${\displaystyle L^{2}[0;T]}$ ${\displaystyle L^{2}}$ ${\displaystyle [0;T]}$

Convergence of the ranks

• A trigonometric series then converges almost everywhere and in the square mean if the series

${\displaystyle \sum _{k=1}^{\infty }\left(\left|c_{k}\right|^{2}+\left|c_{-k}\right|^{2}\right)}$ converges.

• For real trigonometric series this is equivalent to the series

${\displaystyle \sum _{k=1}^{\infty }\left(a_{k}^{2}+b_{k}^{2}\right)}$ converges.

Non-convergent series are also called formal trigonometric series .

Designation as a polynomial

The complex trigonometric polynomials make it clear why these functions are called polynomials : If you restrict the domain of any complex polynomial to the complex unit circle and parameterize it as a curve with a real parameter , then the ordinary polynomial becomes the trigonometric polynomial . In the case of complex trigonometric polynomials, terms with a negative "degree" that result from the parameterization generally also occur . Strictly speaking, trigonometric polynomials result from the aforementioned parameterization from Laurent series with the expansion point , which only have a finite number of non-vanishing coefficients. However, each trigonometric polynomial can also be understood as the sum of any two ordinary complex polynomials, whereby the unit circle is parameterized by for one polynomial and by for the other . ${\displaystyle c_{0}+c_{1}z+c_{2}z^{2}+\dotsb +c_{n}z^{n}}$ ${\displaystyle x}$ ${\displaystyle \left(z=e^{i\omega x},\;(\omega \in \mathbb {R} \setminus \lbrace 0\rbrace )\right)}$ ${\displaystyle c_{0}+c_{1}e^{i\omega x}+c_{2}e^{2i\omega x}+\dotsb +c_{n}e^{ni\omega x}}$ ${\displaystyle k}$ ${\displaystyle z^{k},k<0}$ ${\displaystyle z_{0}=0}$ ${\displaystyle x\mapsto e^{i\omega x}}$ ${\displaystyle x\mapsto e^{-i\omega x}}$

Application in number theory

In analytical number theory, certain trigonometric sums are used as solution-counting functions. This application is based on the orthogonality relation. For a clear presentation, abbreviations are used in number theory and the function is called the number-theoretical exponential function . The orthogonality relation, if formulated with the number-theoretic exponential function, reads: ${\displaystyle e(x)=e^{2\pi ix}}$ ${\displaystyle e}$

${\displaystyle \int _{0}^{1}e(R\alpha )\,d\alpha ={\begin{cases}0\quad (R\in \mathbb {Z} \setminus \lbrace 0\rbrace )\\1\quad (R=0).\end{cases}}}$

The function term of a Diophantine equation is now substituted for the function term . Then one can represent the number of solutions of the equation in a fixed finite set - e.g. the - tuples of natural numbers below a fixed bound - by an integral: ${\displaystyle R}$ ${\displaystyle g}$ ${\displaystyle g(x_{1},x_{2},\dotsc ,x_{s})=0}$ ${\displaystyle L=L(M)}$ ${\displaystyle M\subset \mathbb {Z} ^{s}}$ ${\displaystyle s}$

${\displaystyle L=\#\lbrace (x_{1},x_{2},\dotsc ,x_{s})\in M|g(x_{1},x_{2},\dotsc ,x_{s})=0\rbrace =\sum _{(x_{1},x_{2},\dotsc ,x_{s})\in M}\int _{0}^{1}e(g(x_{1},x_{2},\dotsc ,x_{s})\cdot \alpha )\,d\alpha .}$

Since the sum is finite, it can easily be exchanged for the integral and one obtains

${\displaystyle L=\int _{0}^{1}\sum _{(x_{1},x_{2},\ldots x_{s})\in M}e(g(x_{1},x_{2},\dotsc ,x_{s})\cdot \alpha )\,d\alpha ,}$

thus a representation of the number of solutions as an integral over a trigonometric polynomial. All methods of function theory and functional analysis can now be applied to this solution-counting integral . With this, an asymptotic formula can be derived for the number of solutions, for example, which indicates how the number of solutions behaves when the bounds of strive towards infinity. ${\displaystyle L=L(M)}$ ${\displaystyle M}$

Vinogradov's circle method

The idea of ​​applying the solution-counting integral over a trigonometric polynomial in the form given here to a number-theoretical problem was developed by Winogradow and based on Goldbach's ternary conjecture in 1937 :

Every odd number greater than 5 can be represented as the sum of three prime numbers.

applied. Here is an odd natural number, the set of all triples of prime numbers that are smaller than and . In this way he succeeded in showing that for sufficiently large, odd is the solution-counting integral . This means that the conjecture can only be wrong for a finite number of “small”, odd numbers . (→ See also Vinogradov's theorem ) ${\displaystyle N}$ ${\displaystyle M_{N}}$ ${\displaystyle N}$ ${\displaystyle g(p_{1},p_{2},p_{3})=p_{1}+p_{2}+p_{3}-N}$ ${\displaystyle N}$ ${\displaystyle L(M_{N})>0{,}6}$ ${\displaystyle N}$

Hardy and Littlewood circle method

Winogradov's form of the circle method is a variant of the circle method developed by Hardy and Littlewood , which they successfully applied to Waring's problem in 1917 . In their formulation, the solution-counting function is a power series . The numbers of the solutions to a Diophantine equation are coefficients of this series - in Goldbach's conjecture, the number of representations of the odd number would be the sum of 3 prime numbers. In contrast to Vinogradow, the Diophantine equation is not restricted from the outset to a finite domain of definition. The solution-counting integral , which is used in the Hardy-Littlewood method in a form that is similar to that given by Winogradow, to compute residuals , can in general also have singularities on the unit circle. It is therefore often first estimated on a circle around the origin with a smaller radius or the singularities are revolved. ${\displaystyle a_{N}}$ ${\displaystyle N}$

See also

• Fourier series
• Trigonometric interpolation

literature

• Christian Blatter: Analysis 3 . In: Heidelberger Taschenbücher; Vol. 153 . 2., verb. u. exp. Edition. tape 3 . Springer, Berlin / Heidelberg / New York 1981, ISBN 3-540-10892-0 .
• Harro Heuser : Textbook of Analysis - Part 2 . 14th edition. Vieweg and Teubner, Stuttgart 2008, ISBN 978-3-8351-0208-8 ( table of contents [accessed on August 31, 2012]).
• Harro Heuser: Functional Analysis . 3rd, revised edition. Teubner, Stuttgart 1992, ISBN 3-519-22206-X ( table of contents [accessed on August 31, 2012]).

Number theoretic applications

• Jörg Brüdern : Introduction to analytical number theory . Springer, Berlin, Heidelberg, New York 1995, ISBN 3-540-58821-3 .
• Robert Charles Vaughan : The Hardy-Littlewood Method . 2nd Edition. Cambridge University Press, Cambridge 1997, ISBN 0-521-57347-5 .
• Ivan Matveevitch Vinogradov: The Method of Trigonometrical Sums in the Theory of Numbers . Translated from the Russian and annotated by Klaus Friedrich Roth and Anne Ashley Davenport. New York, Dover 2004.
• Ivan Matveevitch Vinogradov: Representation of an Odd Number as a Sum of Three Primes . In: Comptes rendus (Doklady) de l'Académie des Sciences de l'URSS . No. 15 , 1937, pp. 169-172 .

Individual evidence

1. ↑ Brüdern (1995) p. 20
2. ↑ All variable names in this section are based on informal conventions common in number theory.
3. ↑ Winogradov (1937) and Weisstein, Eric W. "Vinogradov's Theorem." From MathWorld - A Wolfram Web Resource.