Synonyms for weierstrass_transform or Related words with weierstrass_transform

laplace_transform              laplacian              dirac_delta_function              hilbert_transform              poisson_kernel              frobenius_norm              mellin_transform              logarithmic_derivative              laplace_operator              heaviside_step_function              legendre_symbol              hankel_transform              moore_penrose_pseudoinverse              cauchy_schwarz_inequality              integrand              laplace_beltrami_operator              spectral_decomposition              digamma_function              symmetrization              wronskian              plancherel_theorem              fubini_theorem              lebesgue_integral              dirac_operator              radon_nikodym_derivative              adjoint_operator              discrete_fourier_transform              hypergeometric_function              skew_symmetric_matrix              legendre_transform              pseudoinverse              laplacian_operator              jacobian_matrix              cholesky_decomposition              euclidean_norm              riemann_tensor              zeta_function              jacobian              stochastic_differential_equation              poisson_equation              hessian_matrix              riemann_integral              geodesic_equation              quantile_function              riemann_curvature_tensor              ricci_scalar              liouville_equation              poisson_bracket              parametrization              euler_lagrange_equation             

Examples of "weierstrass_transform"
Since the formal expression for the Weierstrass transform is , we see that the Weierstrass transform of . Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding Maclaurin series.
The Weierstrass transform can also be defined for certain classes of distributions or "generalized functions". For example, the Weierstrass transform of the Dirac delta is the Gaussian formula_17.
As mentioned above, every constant function is its own Weierstrass transform. The Weierstrass transform of any polynomial is a polynomial of the same degree. Indeed, if denotes the (physicist's) Hermite polynomial of degree "n", then the Weierstrass transform of (/2) is simply . This can be shown by exploiting the fact that the generating function for the Hermite polynomials is closely related to the Gaussian kernel used in the definition of the Weierstrass transform.
formula_14, thus defining an operator , the generalized Weierstrass transform.
The Weierstrass transform of the function "e" (where "a" is an arbitrary constant) is "e" "e". The function "e" is thus an eigenfunction of the Weierstrass transform. (This is, in fact, more generally true for "all" convolution transforms.)
If "f" is sufficiently smooth, then the Weierstrass transform of the "k"-th derivative of "f" is equal to the "k"-th derivative of the Weierstrass transform of "f".
The map may be computed explicitly as a modified double Weierstrass transform,
to thus obtain the following formal expression for the Weierstrass transform "W",
The formal inverse of the Weierstrass transform is thus given by
The generalized Weierstrass transform provides a means to approximate a given integrable function arbitrarily well with analytic functions.
There is a formula relating the Weierstrass transform "W" and the two-sided Laplace transform "L". If we define
One may, alternatively, attempt to invert the Weierstrass transform in a slightly different way: given the analytic function
Setting "a"="bi" where "i" is the imaginary unit, and applying Euler's identity, one sees that the Weierstrass transform of the function cos("bx") is "e" cos("bx") and the Weierstrass transform of the function sin("bx") is "e" sin("bx").
More generally, the Weierstrass transform can be defined on any Riemannian manifold: the heat equation can be formulated there (using the manifold's Laplace–Beltrami operator), and the Weierstrass transform is then given by following the solution of the heat equation for one time unit, starting with the initial "temperature distribution" "f".
In particular, by choosing "a" negative, it is evident that the Weierstrass transform of a Gaussian function is again a Gaussian function, but a "wider" one.
The kernel formula_13 used for the generalized Weierstrass transform is sometimes called the Gauss–Weierstrass kernel, and is Green's function for the diffusion equation on R.
In his papers and , Ennio de Giorgi introduces the following smoothing operator, analogous to the Weierstrass transform in the one-dimensional case
Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function; this transformation is also known as the Weierstrass transform.
The Weierstrass transform is intimately related to the heat equation (or, equivalently, the diffusion equation with constant diffusion coefficient). If the function describes the initial temperature at each point of an infinitely long rod that has constant thermal conductivity equal to 1, then the temperature distribution of the rod "t" = 1 time units later will be given by the function "F". By using values of "t" different from 1, we can define the generalized Weierstrass transform of .
Using the above, one can show that for 0 < "p" ≤ ∞ and "f" ∈ L(R), we have "F" ∈ L(R) and ||"F"|| ≤ ||"f"||. The Weierstrass transform consequently yields a bounded operator W : L(R) → L(R).