Trigonometric polynomial

A trigonometric polynomial, also called a trigonometric sum, is in real analysis is a finite real linear combination of trigonometric functions and the linear combination is defined as a function of. This real-valued functions are to a unique ( formal) complex representation, are formed in the certain complex linear combinations of the exponential functions instead of the cosine and sine functions. With this representation, calculations are often simplified. The real trigonometric polynomials are partial sums of real Fourier series and play an important role, among others, in the solution of ordinary linear differential equations with constant coefficients and for the discrete Fourier transform.

In the theory of functions, functional analysis, and in many applications, such as analytic number theory (see # circle method according to Vinogradov in this article) any complex linear combination of functions with fixed real is referred to as a complex trigonometric polynomial or complex trigonometric sum.

Both the real and the complex trigonometric polynomials provide unique best approximations - at any given level there is exactly one best approximation under the trigonometric polynomials have at most this degree - in the mean square for each function of the feature space, which the generating trigonometric functions in each case as orthonormal basis ( orthogonal ) determine.

If we let the linear combinations of infinitely many non-vanishing " summands", then you get to the terms of a real or complex trigonometric series.

• 2.1 orthogonality
• 2.2 Basic Properties
• 2.3 Convergence of the series

• 4.1 circle method according to Vinogradov
• 4.2 circle method according to Hardy and Littlewood

• 6.1 Number theoretic applications

Definitions

Real trigonometric polynomial

As a real trigonometric polynomial is defined for the real-valued function

Referred to, where is. The natural number is referred to as the degree of, or if not disappear. The function has the period.

Any period

A real trigonometric polynomial can be somewhat more generally defined as the period of the polynomial is any positive real number. Substituting, then read the polynomials:

The same conditions and designations applicable to other parameters as in the special case

Complex representation

The complex representation of the real trigonometric polynomial is:

Where and vice versa can be represented by the real part of the complex representation and its imaginary part. The trigonometric polynomial if and only real if applies.

Complex trigonometric polynomial

Is a family of complex coefficients, which many indices vanish for all but finitely, and a positive real number, then the sum

In general, the independent variable is in this sum is still a real number and the sum then represents a periodic function dar. Here is the magnitude of the absolute greatest integer for which holds is referred to as the degree of the complex trigonometric polynomial.

Trigonometric series

Analogous to the concept of trigonometric polynomial can the notion of (formal) are defined trigonometric series. These are used as the Fourier series of the periodic functions.

• Real trigonometric series thus can be represented as follows:

• If you leave the condition for the coefficients gone, then you will receive a complex trigonometric series:

It is always of the definition area, and the period as the corresponding trigonometric polynomials

Properties

Orthogonality

The trigonometric functions, from which the real trigonometric polynomials, linear combination, meet the following orthogonality relations:

For generating the complex is the orthogonality relation:

Base property

From the orthogonality relations follows that the sequence of generating trigonometric polynomials is linearly independent. It forms, with a suitable normalization is an orthonormal basis of a real Hilbert space. This Hilbert space is the Lebesgue space.

The generators of the complex trigonometric polynomials family is also linearly independent and forms with a suitable normalization is an orthonormal basis of the complex Hilbert space defined on the unit circle, complex-valued functions, when viewed as a parameterized Laurent series and otherwise a basis of the complex Hilbert space of complex-valued functions on.

Convergence of the series

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

• For real trigonometric series is the equivalent to the fact that the series

Also, do not convergent series are referred to as formal trigonometric series.

Designation as a polynomial

At the complex trigonometric polynomials is clear why these functions are called polynomials: Restricting the domain of an arbitrary complex polynomial on the complex unit circle and parameterized this as a curve with a real parameter, then out of the ordinary polynomial trigonometric polynomial. In complex trigonometric polynomials generally occur even with negative terms "degree", resulting from the parameterization on. Trigonometric polynomials arise Strictly speaking through said parameterization of Laurent series with the expansion point, the only finitely many non-zero coefficients. It can also be regarded as the sum of any two ordinary complex polynomials each trigonometric polynomial but, wherein, when a polynomial of the unit circle by, the other is parameterized.

Application in number theory

In analytic number theory certain trigonometric sums are used as lösungszählende functions. This application is based on the orthogonality. For a clear presentation is written abbreviation in number theory and the function is called number theoretical exponential function. The orthogonality relation reads when formulating them with the number theoretical exponential function:

Is now put in the place of the function term of a Diophantine equation. Then you can see the number of solutions of the equation in a fixed finite set - represented by an integral of - about the tuples of natural numbers below a specified barrier:

Since the sum is finite, it may be easily interchanged by the integral and we obtain

Thus, a representation of the solution number as an integral over a trigonometric polynomial. In this integral lösungszählende all methods of function theory and functional analysis can now be applied. This can be derived for the solution number, for example, an asymptotic formula that specifies how the solution behaves number if the bounds of infinity strive against.

Circle method according to Vinogradov

The idea of ​​the lösungszählende integral of a trigonometric polynomial in the form given here apply to a number theoretical problem was developed by Vinogradov and 1937 on the ternary Goldbach's conjecture:

Applied. Then this is an odd natural number, the set of triples of prime numbers that are less than and. So he managed to show that for sufficiently large, odd is the lösungszählende integral. This may be false presumption for only finitely many " small " odd numbers. ( → See also the set of Vinogradov )

Circle method of Hardy and Littlewood

Vinogradov form of the circle method is a variant of the circle method, which was developed by Hardy and Littlewood and has been used by them in 1917 with success on the Waring problem. In their formulation, the lösungszählende function is a power series. The numbers of solutions of Diophantine direct drying are coefficients of this series - at the Goldbach 's conjecture would be the number of representations of the odd number as a sum of three primes. Unlike Vinogradov here not an a priori restriction on the Diophantine equation to a finite domain is performed. Lösungszählende the integral, which is used in the Hardy - Littlewood method in a form that is similar to the given by Vinogradov to calculate residuals, may generally have singularities on the unit circle. It is often therefore first estimated on a circle around the origin with a smaller radius or the singularities be circulated.