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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 (see commutativity ...
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain).
Above, denotes the support of a function f (i.e., the closure of the complement of f-1 (0)) and and denote the infimum and supremum. This theorem essentially states that the well-known inclusion supp φ ∗ ψ ⊂ supp φ + supp ψ {\displaystyle \operatorname {supp} \varphi \ast \psi \subset \operatorname {supp} \varphi ...
In probability theory, the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density ...
This follows from the fact that the convolution of two log-concave functions is log-concave. The product of two log-concave functions is log-concave. This means that joint densities formed by multiplying two probability densities (e.g. the normal-gamma distribution , which always has a shape parameter ≥ 1) will be log-concave.
Young's inequality has an elementary proof with the non-optimal constant 1. [4]We assume that the functions ,,: are nonnegative and integrable, where is a unimodular group endowed with a bi-invariant Haar measure .
Dirichlet convolution is a special case of the convolution multiplication for the incidence algebra of a poset, in this case the poset of positive integers ordered by divisibility. The Dirichlet hyperbola method computes the summation of a convolution in terms of its functions and their summation functions.