mathematics, the Weierstrass transform of a function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } , named after Karl Weierstrass, is a "smoothed"...

original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform. Marshall H. Stone considerably generalized the theorem...

transform Stieltjes transformation Sumudu transform Wavelet transform (integral) Weierstrass transform Binomial transform Discrete Fourier transform,...

solve heat equations and diffusion equations and to define the Weierstrass transform. They are also abundantly used in quantum chemistry to form basis...

In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration...

with a Gaussian function; this transformation is also known as the Weierstrass transform. The one-dimensional Gaussian filter has an impulse response given...

Hjalmar Mellin was among the first to study the Laplace transform, rigorously in the Karl Weierstrass school of analysis, and apply it to the study of differential...

mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class...

Weierstrass. Bolzano–Weierstrass theorem Casorati–Weierstrass theorem Weierstrass method Enneper–Weierstrass parameterization Lindemann–Weierstrass theorem...

substitution or half-angle substitution. It is sometimes misattributed as the Weierstrass substitution. Michael Spivak called it the "world's sneakiest substitution"...

with a Gaussian function. This is also known as a two-dimensional Weierstrass transform. By contrast, convolving by a circle (i.e., a circular box blur)...

for the Weierstrass transform W is eD2, we see that the Weierstrass transform of (√2)nHen(⁠x/√2⁠) is xn. Essentially the Weierstrass transform thus turns...

quasiprobability distributions. In fact, it can be understood as the Weierstrass transform of the Wigner quasiprobability distribution, i.e. a smoothing by...

Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: Lindemann–Weierstrass theorem—if...

operators. The map B may be computed explicitly as a modified double Weierstrass transform, ( B f ) ( z ) = ∫ R n exp ⁡ [ − 1 2 ( z ⋅ z − 2 2 z ⋅ x + x ⋅ x...

every complex number z. The definition for the gamma function due to Weierstrass is also valid for all complex numbers  z {\displaystyle z} except non-positive...

Ostrogradsky–Gauss theorem Gauss pseudospectral method Gauss transform, also known as Weierstrass transform. Gauss–Lucas theorem Gauss's continued fraction, an...

semigroup using the L 2 {\displaystyle L^{2}} norm (that is, the Weierstrass transform does not enlarge the L 2 {\displaystyle L^{2}} norm). Young's inequality...

Kurt Otto Friedrichs Non-analytic smooth function Sergei Sobolev Weierstrass transform That is, the mollified function is close to the original with respect...

The result is a Gaussian blur of the image, or equivalently the Weierstrass transform of the indicator function, with radius proportional to the square...

signature Minakshisundaram–Pleijel zeta function Mehler kernel Weierstrass transform § Generalizations Evans 1998, p. 48. Pinchover & Rubinstein 2005...

larger than ħ (e.g., convolving with a phase-space Gaussian, a Weierstrass transform, to yield the Husimi representation, below), results in a positive-semidefinite...

differential equation Relativistic heat conduction Schrödinger equation Weierstrass transform Evans 2010, p. 44. Stojanovic, Srdjan (2003), "3.3.1.3 Uniqueness...

unit circle and a closed Jordan curve) Kramers–Kronig relations Hilbert transform Breuer, Heinz-Peter; Petruccione, Francesco (2002). The Theory of Open...

inconvenient geometries. By choosing an appropriate mapping, the analyst can transform the inconvenient geometry into a much more convenient one. For example...

Removable singularity Essential singularity Branch point Principal branch Weierstrass–Casorati theorem Landau's constants Holomorphic functions are analytic...

are all interrelated through convolution by Gaussian functions, Weierstrass transforms, W ( α , α ∗ ) = 2 π ∫ P ( β , β ∗ ) e − 2 | α − β | 2 d 2 β {\displaystyle...

^{-1/4}\exp(-|y-x|^{2}/2)+i\,px).} (It can be understood as the Weierstrass transform of the Wigner quasi-probability distribution.) The Wehrl entropy...

generalizes the Fourier, fractional Fourier, Laplace, Gauss–Weierstrass, Bargmann and the Fresnel transforms as particular cases. The name "linear canonical transformation"...

introduced by Peter L. Montgomery in 1987, different from the usual Weierstrass form. It is used for certain computations, and in particular in different...