Hilbert transform, the Glossary

Table of Contents

1. 80 relations: Alberto Calderón, Almost everywhere, Analytic function, Analytic signal, Andrey Kolmogorov, Angle modulation, Anticommutative property, Antoni Zygmund, Banach space, Bounded mean oscillation, Bounded operator, Cauchy principal value, Cauchy's integral formula, Causal filter, Conjugate index, 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, Fourier transform, Frequency, Frequency domain, Frequency modulation, Göttingen, Glossary of mathematical jargon, Grunsky matrix, H square, Hardy space, Harmonic conjugate, Heterodyne, Hilbert spectroscopy, Hilbert–Huang transform, Holomorphic function, Hyperfunction, Identity function, Indicator function, Integrable system, Inverse limit, Involution (mathematics), Kramers–Kronig relations, Linear complex structure, Lp space, Marcel Riesz, ... Expand index (30 more) »

2. Harmonic functions
3. Schwartz distributions
4. Singular integrals

Alberto Calderón

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

See Hilbert transform and Alberto Calderón

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.

See Hilbert transform and Almost everywhere

Analytic function

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

See Hilbert transform and Analytic function

Analytic signal

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

See Hilbert transform and Analytic signal

Andrey Kolmogorov

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

See Hilbert transform and Andrey Kolmogorov

Angle modulation

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

See Hilbert transform and Angle modulation

Anticommutative property

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

See Hilbert transform and Anticommutative property

Antoni Zygmund

Antoni Zygmund (December 26, 1900 – May 30, 1992) was a Polish mathematician.

See Hilbert transform and Antoni Zygmund

Banach space

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

See Hilbert transform and Banach space

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).

See Hilbert transform and Bounded mean oscillation

Bounded operator

In functional analysis and operator theory, a bounded linear operator is a linear transformation L: X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces (a special type of TVS), then L is bounded if and only if there exists some M > 0 such that for all x \in X, \|Lx\|_Y \leq M \|x\|_X.

See Hilbert transform and Bounded operator

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. Hilbert transform and Cauchy principal value are Schwartz distributions.

See Hilbert transform and Cauchy principal value

Cauchy's integral formula

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

See Hilbert transform and Cauchy's integral formula

Causal filter

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

See Hilbert transform and Causal filter

Conjugate index

In mathematics, two real numbers p, q>1 are called conjugate indices (or Hölder conjugates) if Formally, we also define q.

See Hilbert transform and Conjugate index

Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function (f*g).

See Hilbert transform and Convolution

David Hilbert

David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of his time.

See Hilbert transform and David Hilbert

Dawson function

In mathematics, the Dawson function or Dawson integral (named after H. G. Dawson) is the one-sided Fourier–Laplace sine transform of the Gaussian function.

See Hilbert transform and Dawson function

Dense set

In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).

See Hilbert transform and Dense set

Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Hilbert transform and Dirac delta function are Schwartz distributions.

See Hilbert transform and Dirac delta function

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.

See Hilbert transform and Discrete Fourier transform

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).

See Hilbert transform and Discrete series representation

Discrete-time Fourier transform

In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of discrete values.

See Hilbert transform and Discrete-time Fourier transform

Distribution (mathematics)

Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Hilbert transform and distribution (mathematics) are Schwartz distributions.

See Hilbert transform and Distribution (mathematics)

Edward Charles Titchmarsh

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

See Hilbert transform and Edward Charles Titchmarsh

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.

See Hilbert transform and Euler's formula

Fourier transform

In physics, engineering and mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. Hilbert transform and Fourier transform are integral transforms.

See Hilbert transform and Fourier transform

Frequency

Frequency (symbol f), most often measured in hertz (symbol: Hz), is the number of occurrences of a repeating event per unit of time.

See Hilbert transform and Frequency

Frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time series.

See Hilbert transform and Frequency domain

Frequency modulation

Frequency modulation (FM) is the encoding of information in a carrier wave by varying the instantaneous frequency of the wave.

See Hilbert transform and Frequency modulation

Göttingen

Göttingen (Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district.

See Hilbert transform and Göttingen

Glossary of mathematical jargon

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

See Hilbert transform and Glossary of mathematical jargon

Grunsky matrix

In complex analysis and geometric function theory, the Grunsky matrices, or Grunsky operators, are infinite matrices introduced in 1939 by Helmut Grunsky.

See Hilbert transform and Grunsky matrix

H square

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

See Hilbert transform and H square

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. Hilbert transform and Hardy space are Schwartz distributions.

See Hilbert transform and Hardy space

Harmonic conjugate

In mathematics, a real-valued function u(x,y) defined on a connected open set \Omega \subset \R^2 is said to have a conjugate (function) v(x,y) if and only if they are respectively the real and imaginary parts of a holomorphic function f(z) of the complex variable z. Hilbert transform and Harmonic conjugate are harmonic functions.

See Hilbert transform and Harmonic conjugate

Heterodyne

A heterodyne is a signal frequency that is created by combining or mixing two other frequencies using a signal processing technique called heterodyning, which was invented by Canadian inventor-engineer Reginald Fessenden. Hilbert transform and heterodyne are signal processing.

See Hilbert transform and Heterodyne

Hilbert spectroscopy

Hilbert Spectroscopy uses Hilbert transforms to analyze broad spectrum signals from gigahertz to terahertz frequency radio. Hilbert transform and Hilbert spectroscopy are signal processing.

See Hilbert transform and Hilbert spectroscopy

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. Hilbert transform and Hilbert–Huang transform are signal processing.

See Hilbert transform and Hilbert–Huang transform

Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space.

See Hilbert transform and Holomorphic function

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.

See Hilbert transform and Hyperfunction

Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged.

See Hilbert transform and Identity function

Indicator function

In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero.

See Hilbert transform and Indicator function

Integrable system

In mathematics, integrability is a property of certain dynamical systems.

See Hilbert transform and Integrable system

Inverse limit

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

See Hilbert transform and Inverse limit

Involution (mathematics)

In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, for all in the domain of.

See Hilbert transform and Involution (mathematics)

Kramers–Kronig relations

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

See Hilbert transform and Kramers–Kronig relations

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, - I d_V.

See Hilbert transform and Linear complex structure

Lp space

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

See Hilbert transform and Lp space

Marcel Riesz

Marcel Riesz (Riesz Marcell; 16 November 1886 – 4 September 1969) was a Hungarian 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.

See Hilbert transform and Marcel Riesz

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.

See Hilbert transform and Marcinkiewicz interpolation theorem

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Hilbert transform and Mathematics

MATLAB

MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks.

See Hilbert transform and MATLAB

Multiplier (Fourier analysis)

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

See Hilbert transform and Multiplier (Fourier analysis)

Negative frequency

In mathematics, the concept of signed frequency (negative and positive frequency) can indicate both the rate and sense of rotation; it can be as simple as a wheel rotating clockwise or counterclockwise.

See Hilbert transform and Negative frequency

Overlap–save method

In signal processing, 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 for m outside the region. Hilbert transform and overlap–save method are signal processing.

See Hilbert transform and Overlap–save method

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.

See Hilbert transform and Paley–Wiener theorem

Periodic summation

In mathematics, any integrable function s(t) can be made into a periodic function s_P(t) with period P by summing the translations of the function s(t) by integer multiples of P. This is called periodic summation: When s_P(t) is alternatively represented as a Fourier series, the Fourier coefficients are equal to the values of the continuous Fourier transform, S(f) \triangleq \mathcal\, at intervals of \tfrac. Hilbert transform and periodic summation are signal processing.

See Hilbert transform and Periodic summation

Phase (waves)

In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is expressed in such a scale that it varies by one full turn as the variable t goes through each period (and F(t) goes through each complete cycle).

See Hilbert transform and Phase (waves)

Phase modulation

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

See Hilbert transform and Phase modulation

Poisson kernel

In mathematics, and specifically 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 disk. Hilbert transform and Poisson kernel are harmonic functions.

See Hilbert transform and Poisson kernel

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.

See Hilbert transform and Principal series representation

Quadrature filter

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). Hilbert transform and quadrature filter are signal processing.

See Hilbert transform and Quadrature filter

Quantum state

In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system.

See Hilbert transform and Quantum state

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 the regulator.

See Hilbert transform and Regularization (physics)

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.

See Hilbert transform and Riemann–Hilbert problem

Riesz transform

In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension d > 1. Hilbert transform and Riesz transform are integral transforms and singular integrals.

See Hilbert transform and Riesz transform

Self-adjoint

In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a.

See Hilbert transform and Self-adjoint

Sign function

In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that has the value, or according to whether the sign of a given real number is positive or negative, or the given number is itself zero.

See Hilbert transform and Sign function

Signal processing

Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing signals, such as sound, images, potential fields, seismic signals, altimetry processing, and scientific measurements.

See Hilbert transform and Signal processing

Sinc function

In mathematics, physics and engineering, the sinc function, denoted by, has two forms, normalized and unnormalized. Hilbert transform and sinc function are signal processing.

See Hilbert transform and Sinc function

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.

See Hilbert transform and Single-sideband modulation

Singular integral

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

See Hilbert transform and Singular integral

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. Hilbert transform and singular integral operators of convolution type are singular integrals.

See Hilbert transform and Singular integral operators of convolution type

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 together with its derivatives up to a given order.

See Hilbert transform and Sobolev space

Square-integrable function

In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.

See Hilbert transform and Square-integrable function

Support (mathematics)

In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. Hilbert transform and support (mathematics) are Schwartz distributions.

See Hilbert transform and Support (mathematics)

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 the case that G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.

See Hilbert transform and Unitary representation

Upper half-plane

In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with.

See Hilbert transform and Upper half-plane

Window function

In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval.

See Hilbert transform and Window function

See also

Harmonic functions

• Bôcher's theorem
• Cauchy–Riemann equations
• Differential forms on a Riemann surface
• Dirichlet's principle
• Edmund Schuster
• Harmonic conjugate
• Harmonic coordinates
• Harmonic function
• Harmonic map
• Harmonic morphism
• Harnack's inequality
• Harnack's principle
• Hilbert transform
• Kellogg's theorem
• Kelvin transform
• Laplace operator
• Laplace's equation
• Maximum principle
• Newtonian potential
• Pluriharmonic function
• Poisson kernel
• Positive harmonic function
• Schwarz alternating method
• Schwarz reflection principle
• Weakly harmonic function
• Weyl's lemma (Laplace equation)

Schwartz distributions

• Bump function
• Cauchy principal value
• Colombeau algebra
• Current (mathematics)
• Dirac delta function
• Distribution (mathematics)
• Fourier inversion theorem
• Fundamental solution
• Gelfand–Shilov space
• Green's function
• Hardy space
• Heaviside step function
• Hilbert transform
• Homogeneous distribution
• Laplacian of the indicator
• Limit of distributions
• Lobachevsky integral formula
• Malgrange–Ehrenpreis theorem
• Mollifier
• Oscillatory integral
• Parametrix
• Rigged Hilbert space
• Schwartz kernel theorem
• Schwartz space
• Spaces of test functions and distributions
• Support (mathematics)
• Weak solution

Singular integrals

• Bessel potential
• Hilbert transform
• Newtonian potential
• Oscillatory integral operator
• Riesz potential
• Riesz transform
• Singular integral
• Singular integral operators of convolution type

References

[1] https://en.wikipedia.org/wiki/Hilbert_transform

Also known as Discrete Hilbert transform, Hilbert kernel, Hilbert transforms, Iterated Hilbert Transform.

, Marcinkiewicz interpolation theorem, Mathematics, MATLAB, Multiplier (Fourier analysis), Negative frequency, Overlap–save method, Paley–Wiener theorem, Periodic summation, Phase (waves), Phase modulation, Poisson kernel, Principal series representation, Quadrature filter, Quantum state, Regularization (physics), Riemann–Hilbert problem, Riesz transform, Self-adjoint, Sign function, Signal processing, Sinc function, Single-sideband modulation, Singular integral, Singular integral operators of convolution type, Sobolev space, Square-integrable function, Support (mathematics), Unitary representation, Upper half-plane, Window function.