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, ..., Linear map, List of mathematical jargon, Lp space, Marcel Riesz, 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, Rectangular function, Regularization (physics), Riemann–Hilbert problem, 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. Expand index (33 more) » « Shrink index
Alberto Pedro Calderón (September 14, 1920 – April 16, 1998) was an Argentinian mathematician.
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.
In mathematics, an analytic function is a function that is locally given by a convergent power series.
In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components.
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 is a class of carrier modulation that is used in telecommunications transmission systems.
In mathematics, anticommutativity is a specific property of some non-commutative operations.
Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician.
In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.
Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency.
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).
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).
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.
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.
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.
In signal processing, a causal filter is a linear and time-invariant causal system.
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.
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 (23 January 1862 – 14 February 1943) was a German mathematician.
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.
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 — for instance, every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
In mathematics, the Dirac delta function (function) is a generalized function or distribution introduced by the physicist Paul Dirac.
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.
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).
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.
Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis.
Edward Charles "Ted" Titchmarsh (June 1, 1899 – January 18, 1963) was a leading English mathematician.
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.
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.
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 is the number of occurrences of a repeating event per unit of time.
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.
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 (Low German: Chöttingen) is a university city in Lower Saxony, Germany.
In mathematics, the Grunsky matrices, or Grunsky operators, are matrices introduced by in complex analysis and geometric function theory.
In mathematics and control theory, H2, or H-square is a Hardy space with square norm.
In complex analysis, the Hardy spaces (or Hardy classes) Hp are certain spaces of holomorphic functions on the unit disk or upper half plane.
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.
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.
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 uses Hilbert transforms to analyze broad spectrum signals from gigahertz to terahertz frequency radio.
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.
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.
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.
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.
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.
In the context of differential equations to integrate an equation means to solve it from initial conditions.
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.
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.
In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I.
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.
The language of mathematics has a vast vocabulary of specialist and technical terms.
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.
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.
In mathematics, the Marcinkiewicz interpolation theorem, discovered by, is a result bounding the norms of non-linear operators acting on ''L''p spaces.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment and proprietary programming language developed by MathWorks.
In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions.
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 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.
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.
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 is the position of a point in time (an instant) on a waveform cycle.
Phase modulation (PM) is a modulation pattern for conditioning communication signals for transmission.
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.
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).
In quantum physics, quantum state refers to the state of an isolated quantum system.
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|.
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.
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^*.
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 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.
In mathematics, physics and engineering, the cardinal sine function or sinc function, denoted by, has two slightly different definitions.
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.
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations.
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.
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.
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.
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.
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.
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.