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Let f(x) be a positive definite, measurable function on the interval (-infty,infty). Then there exists a monotone increasing, real-valued bounded function alpha(t) such that ...

The Fourier transform of e^(-k_0|x|) is given by F_x[e^(-k_0|x|)](k)=int_(-infty)^inftye^(-k_0|x|)e^(-2piikx)dx = ...

A one-dimensional transform which makes use of the Haar functions.

A fast Fourier transform-like algorithm which produces a hologram of an image.

The forward and inverse Kontorovich-Lebedev transforms are defined by K_(ix)[f(t)] = int_0^inftyK_(ix)(t)f(t)dt (1) K_(ix)^(-1)[g(t)] = ...

Expresses a function in terms of its Radon transform, f(x,y) = R^(-1)(Rf)(x,y) (1) = ...

A one-sided (singly infinite) Laplace transform, L_t[f(t)](s)=int_0^inftyf(t)e^(-st)dt. This is the most common variety of Laplace transform and it what is usually meant by ...

A one-sided (singly infinite) Z-Transform, Z[{a_n}_(n=0)^infty](z)=sum_(n=0)^infty(a_n)/(z^n). This is the most common variety of Z-transform since it is essentially ...

A generalized hypergeometric function _pF_q(a_1,...,a_p;b_1,...,b_q;x) is a function which can be defined in the form of a hypergeometric series, i.e., a series for which the ...

The prime counting function is the function pi(x) giving the number of primes less than or equal to a given number x (Shanks 1993, p. 15). For example, there are no primes ...