In mathematics, the scale convolution of two functions and , also known as their logarithmic convolution or log-volution [1] is defined as the function [2]
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when this quantity exists.
In mathematics, the scale convolution of two functions and , also known as their logarithmic convolution or log-volution [1] is defined as the function [2]
when this quantity exists.
The logarithmic convolution can be related to the ordinary convolution by changing the variable from to : [2]
Define and and let , then
In mathematics, convolution is a mathematical operation on two functions that produces a third function. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result. Graphically, it expresses how the 'shape' of one function is modified by the other.
In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n,
In mathematics, the Laplace transform, named after Pierre-Simon Laplace, is an integral transform that converts a function of a real variable to a function of a complex variable .
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, logex, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
In mathematical analysis, the Dirac delta function, 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. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions.
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function f(w) = wew, where w is any complex number and ew is the exponential function. The function is named after Johann Lambert, who considered a related problem in 1758. Building on Lambert's work, Leonhard Euler described the W function per se in 1783.
Euler's constant is a mathematical constant, usually denoted by the lowercase Greek letter gamma, defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region.
In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let be a measure space, let and let and be elements of Then is in and we have the triangle inequality with equality for if and only if and are positively linearly dependent; that is, for some or Here, the norm is given by: if or in the case by the essential supremum
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form and with parametric extension for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak, and c controls the width of the "bell".
In mathematics, a Dirichlet series is any series of the form where s is complex, and is a complex sequence. It is a special case of general Dirichlet series.
In number theory, the Mertens function is defined for all positive integers n as
In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). The Hilbert transform is given by the Cauchy principal value of the convolution with the function (see § Definition). The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° (π/2 radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see § Relationship with the Fourier transform). The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal u(t). The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions.
In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line:
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form
In mathematics, the Hankel transform expresses any given function f(r) as the weighted sum of an infinite number of Bessel functions of the first kind Jν(kr). The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor k along the r axis. The necessary coefficient Fν of each Bessel function in the sum, as a function of the scaling factor k constitutes the transformed function. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval.
In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field.
In mathematical analysis, and especially in real, harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes the Lp spaces. Like the Lp spaces, they are Banach spaces. The spaces are named for Władysław Orlicz, who was the first to define them in 1932.
In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as
In analytic number theory, a Dirichlet series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic functions in a meaningful way. A little known, or at least often forgotten about, way of expressing formulas for arithmetic functions and their summatory functions is to perform an integral transform that inverts the operation of forming the DGF of a sequence. This inversion is analogous to performing an inverse Z-transform to the generating function of a sequence to express formulas for the series coefficients of a given ordinary generating function.
This article incorporates material from logarithmic convolution on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
