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# Hilbert transform

In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). [1]

83 relations: Alberto Calderón, Almost everywhere, Analytic function, Analytic signal, Andrey Kolmogorov, Angle modulation, Anticommutativity, Antoni Zygmund, Banach space, Bandlimiting, Bounded mean oscillation, Bounded operator, Cauchy principal value, Cauchy's integral formula, Cauchy–Riemann equations, Causal filter, Complex number, Convolution, David Hilbert, Dawson function, Dense set, Dirac delta function, Discrete Fourier transform, Discrete series representation, Discrete-time Fourier transform, Distribution (mathematics), Edward Charles Titchmarsh, Euler's formula, Finite impulse response, Fourier transform, Frequency, Frequency domain, Frequency modulation, Göttingen, Grunsky matrix, H square, Hardy space, Harmonic conjugate, Hölder's inequality, Heterodyne, Hilbert spectroscopy, Hilbert–Huang transform, Holomorphic function, Hyperfunction, Improper integral, Indicator function, Integrable system, Inverse limit, Kramers–Kronig relations, Linear complex structure, ... Expand index (33 more) »

## Alberto Calderón

Alberto Pedro Calderón (September 14, 1920 – April 16, 1998) was an Argentinian mathematician.

## Almost everywhere

In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities.

## Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

## Analytic signal

In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components.

## Andrey Kolmogorov

Andrey Nikolaevich Kolmogorov (a, 25 April 1903 – 20 October 1987) was a 20th-century Soviet mathematician who made significant contributions to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.

## Angle modulation

Angle modulation is a class of carrier modulation that is used in telecommunications transmission systems.

## Anticommutativity

In mathematics, anticommutativity is a specific property of some non-commutative operations.

## Antoni Zygmund

Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician.

## Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

## Bandlimiting

Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency.

## Bounded mean oscillation

In harmonic analysis in mathematics, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded (finite).

## Bounded operator

In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M\ge 0 such that for all v in X The smallest such M is called the operator norm \|L\|_ \, of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v).

## Cauchy principal value

In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.

## Cauchy's integral formula

In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.

## Cauchy–Riemann equations

In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic.

## Causal filter

In signal processing, a causal filter is a linear and time-invariant causal system.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Convolution

In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.

## David Hilbert

David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician.

## Dawson function

In mathematics, the Dawson function or Dawson integral (named after H. G. Dawson) is either also denoted as F(x) or D(x), or alternatively The Dawson function is the one-sided Fourier–Laplace sine transform of the Gaussian function, It is closely related to the error function erf, as where erfi is the imaginary error function, Similarly, in terms of the real error function, erf.

## Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A, that is the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A &mdash; for instance, every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).

## Dirac delta function

In mathematics, the Dirac delta function (function) is a generalized function or distribution introduced by the physicist Paul Dirac.

## Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency.

## Discrete series representation

In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²(G).

## Discrete-time Fourier transform

In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to the uniformly-spaced samples of a continuous function.

## Distribution (mathematics)

Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis.

## Edward Charles Titchmarsh

Edward Charles "Ted" Titchmarsh (June 1, 1899 – January 18, 1963) was a leading English mathematician.

## Euler's formula

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.

## Finite impulse response

In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time.

## Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.

## Frequency

Frequency is the number of occurrences of a repeating event per unit of time.

## Frequency domain

In electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time.

## Frequency modulation

In telecommunications and signal processing, frequency modulation (FM) is the encoding of information in a carrier wave by varying the instantaneous frequency of the wave.

## Göttingen

Göttingen (Low German: Chöttingen) is a university city in Lower Saxony, Germany.

## Grunsky matrix

In mathematics, the Grunsky matrices, or Grunsky operators, are matrices introduced by in complex analysis and geometric function theory.

## H square

In mathematics and control theory, H2, or H-square is a Hardy space with square norm.

## Hardy space

In complex analysis, the Hardy spaces (or Hardy classes) Hp are certain spaces of holomorphic functions on the unit disk or upper half plane.

## Harmonic conjugate

In mathematics, a function u(x,\,y) defined on some open domain \Omega\subset\R^2 is said to have as a conjugate a function v(x,\,y) if and only if they are respectively real and imaginary parts of a holomorphic function f(z) of the complex variable z.

## Hölder's inequality

In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of ''Lp'' spaces.

## Heterodyne

Heterodyning is a signal processing technique invented in 1901 by Canadian inventor-engineer Reginald Fessenden that creates new frequencies by combining or mixing two frequencies.

## Hilbert spectroscopy

Hilbert Spectroscopy uses Hilbert transforms to analyze broad spectrum signals from gigahertz to terahertz frequency radio.

## Hilbert–Huang transform

The Hilbert–Huang transform (HHT) is a way to decompose a signal into so-called intrinsic mode functions (IMF) along with a trend, and obtain instantaneous frequency data.

## Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

## Hyperfunction

In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order.

## Improper integral

In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number, \infty, -\infty, or in some instances as both endpoints approach limits.

## Indicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.

## Integrable system

In the context of differential equations to integrate an equation means to solve it from initial conditions.

## Inverse limit

In mathematics, the inverse limit (also called the projective limit or limit) is a construction that allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects.

## Kramers–Kronig relations

The Kramers–Kronig relations are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane.

## Linear complex structure

In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, &minus;I.

## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

## List of mathematical jargon

The language of mathematics has a vast vocabulary of specialist and technical terms.

## Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.

## Marcel Riesz

Marcel Riesz (Riesz Marcell; 16 November 1886 – 4 September 1969) was a Hungarian-born mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations, and Clifford algebras.

## Marcinkiewicz interpolation theorem

In mathematics, the Marcinkiewicz interpolation theorem, discovered by, is a result bounding the norms of non-linear operators acting on ''L''p spaces.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## MATLAB

MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment and proprietary programming language developed by MathWorks.

## Multiplier (Fourier analysis)

In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions.

## Negative frequency

The concept of negative and positive frequency can be as simple as a wheel rotating one way or the other way: a signed value of frequency can indicate both the rate and direction of rotation.

## Overlap–save method

Overlap–save is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal x and a finite impulse response (FIR) filter h: where h.

## Paley–Wiener theorem

In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform.

## Periodic summation

In signal processing, any periodic function, s_P(t) with period P, can be represented by a summation of an infinite number of instances of an aperiodic function, s(t), that are offset by integer multiples of P. This representation is called periodic summation: When s_P(t) is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values (or "samples") of the continuous Fourier transform, S(f) \ \stackrel \ \mathcal\, at intervals of 1/P.

## Phase (waves)

Phase is the position of a point in time (an instant) on a waveform cycle.

## Phase modulation

Phase modulation (PM) is a modulation pattern for conditioning communication signals for transmission.

## Poisson kernel

In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc.

## Principal series representation

In mathematics, the principal series representations of certain kinds of topological group G occur in the case where G is not a compact group.

In signal processing, a quadrature filter q(t) is the analytic representation of the impulse response f(t) of a real-valued filter: q(t).

## Quantum state

In quantum physics, quantum state refers to the state of an isolated quantum system.

## Rectangular function

The rectangular function (also known as the rectangle function, rect function, Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as: 0 & \mbox |t| > \frac \\ \frac & \mbox |t|.

## Regularization (physics)

In physics, especially quantum field theory, regularization is a method of modifying observables which have singularities in order to make them finite by the introduction of a suitable parameter called regulator.

## Riemann–Hilbert problem

In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane.

In mathematics, an element x of a *-algebra is self-adjoint if x^*.

## Sign function

In mathematics, the sign function or signum function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number.

## Signal processing

Signal processing concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements.

## Sinc function

In mathematics, physics and engineering, the cardinal sine function or sinc function, denoted by, has two slightly different definitions.

## Single-sideband modulation

In radio communications, single-sideband modulation (SSB) or single-sideband suppressed-carrier modulation (SSB-SC) is a type of modulation, used to transmit information, such as an audio signal, by radio waves.

## Singular integral

In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations.

## Singular integral operators of convolution type

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.

## Sobolev space

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function itself and its derivatives up to a given order.

## Square-integrable function

In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.

## Support (mathematics)

In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero.

## Unitary representation

In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.

## Upper half-plane

In mathematics, the upper half-plane H is the set of complex numbers with positive imaginary part: The term arises from a common visualization of the complex number x + iy as the point (x,y) in the plane endowed with Cartesian coordinates.

## References

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