Conjugate Fourier series

Last updated April 29, 2022

In the mathematical field of Fourier analysis, the conjugate Fourier series arises by realizing the Fourier series formally as the boundary values of the real part of a holomorphic function on the unit disc. The imaginary part of that function then defines the conjugate series. Zygmund (1968) studied the delicate questions of convergence of this series, and its relationship with the Hilbert transform.

In detail, consider a trigonometric series of the form

f(\theta )={\tfrac {1}{2}}a_{0}+\sum _{n=1}^{\infty }\left(a_{n}\cos n\theta +b_{n}\sin n\theta \right)

in which the coefficients a_n and b_n are real numbers. This series is the real part of the power series

F(z)={\tfrac {1}{2}}a_{0}+\sum _{n=1}^{\infty }(a_{n}-ib_{n})z^{n}

along the unit circle with $z=e^{i\theta }$ . The imaginary part of F(z) is called the conjugate series of f, and is denoted

{\tilde {f}}(\theta )=\sum _{n=1}^{\infty }\left(a_{n}\sin n\theta -b_{n}\cos n\theta \right).

Related Research Articles

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number $x$ :

In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

Laplaces equation Second order partial differential equation

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as

Fourier series Decomposition of periodic functions into sums of simpler sinusoidal forms

A Fourier series is a sum that represents a periodic function as a sum of sine and cosine waves. The frequency of each wave in the sum, or harmonic, is an integer multiple of the periodic function's fundamental frequency. Each harmonic's phase and amplitude can be determined using harmonic analysis. A Fourier series may potentially contain an infinite number of harmonics. Summing part of but not all the harmonics in a function's Fourier series produces an approximation to that function. For example, using the first few harmonics of the Fourier series for a square wave yields an approximation of a square wave.

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as $and . They can be defined in several equivalent ways; in this article the polynomials are defined by starting with trigonometric functions:$

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

In complex analysis, the Hardy spacesH^p are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz, who named them after G. H. Hardy, because of the paper. In real analysis Hardy spaces are certain spaces of distributions on the real line, which are boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the L^p spaces of functional analysis. For 1 ≤ p ≤ ∞ these real Hardy spaces H^p are certain subsets of L^p, while for p < 1 the L^p spaces have some undesirable properties, and the Hardy spaces are much better behaved.

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions.

In mathematics, the Clausen function, introduced by Thomas Clausen (1832), is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other special functions. It is intimately connected with the polylogarithm, inverse tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function.

In mathematics, theta functions are special functions of several complex variables. They are important in many areas, including the theories of Abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory.

In mathematics, the circle group, denoted by $or 1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers$

In mathematics, physics and engineering, the sinc function, denoted by $sinc(x)$ , has two forms, normalized and unnormalized.

The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.

In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle $, the sine and cosine functions are denoted simply as and .$

In analysis, a lacunary function, also known as a lacunary series, is an analytic function that cannot be analytically continued anywhere outside the radius of convergence within which it is defined by a power series. The word lacunary is derived from lacuna, meaning gap, or vacancy.

In mathematics, the Jacobi–Anger expansion is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics, and in signal processing. This identity is named after the 19th-century mathematicians Carl Jacobi and Carl Theodor Anger.

In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rⁿ and Tⁿ through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on L² is evident because the Fourier transform converts them into multiplication operators. Continuity on L^p spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed by a number of authors to give general criteria for continuity on L^p spaces. This article explains the theory for the classical operators and sketches the subsequent general theory.

Wiener–Lévy theorem is a theorem in Fourier analysis, which states that a function of an absolutely convergent Fourier series has an absolutely convergent Fourier series under some conditions. The theorem was named after Norbert Wiener and Paul Lévy.

References

Grafakos, Loukas (2008), Classical Fourier analysis, Graduate Texts in Mathematics, vol. 249 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09432-8, ISBN 978-0-387-09431-1, MR 2445437
Zygmund, Antoni (1968), Trigonometric Series (2nd ed.), Cambridge University Press (published 1988), ISBN 978-0-521-35885-9

This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

Conjugate Fourier series

See also

Related Research Articles

References