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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)] = ...

For a general second-order linear recurrence equation f_(n+1)=xf_n+yf_(n-1), (1) define a multiplication rule on ordered pairs by (A,B)(C,D)=(AD+BC+xAC,BD+yAC). (2) The ...

Can be used to invert a Laplace transform.

For a delta function at (x_0,y_0), R(p,tau) = int_(-infty)^inftyint_(-infty)^inftydelta(x-x_0)delta(y-y_0)delta[y-(tau+px)]dydx (1) = ...

Plancherel's theorem states that the integral of the squared modulus of a function is equal to the integral of the squared modulus of its spectrum. It corresponds to ...

A test for the convergence of Fourier series. Let phi_x(t)=f(x+t)+f(x-t)-2f(x), then if int_0^pi(|phi_x(t)|dt)/t is finite, the Fourier series converges to f(x) at x.

The Bump-Ng theorem (and also the title of the paper in which it was proved) states that the zeros of the Mellin transform of Hermite functions have real part equal to 1/2.

The engineering terminology for one use of Fourier transforms. By breaking up a wave pulse into its frequency spectrum f_nu=F(nu)e^(2piinut), (1) the entire signal can be ...

Leakage, more explicitly called spectral leakage, is a smearing of power across a frequency spectrum that occurs when the signal being measured is not periodic in the sample ...

The Gibbs phenomenon is an overshoot (or "ringing") of Fourier series and other eigenfunction series occurring at simple discontinuities. It can be reduced with the Lanczos ...