scholarly journals Some Formulas for Legendre Functions Induced by the Poisson Transform

10.14311/1368 ◽  
2011 ◽  
Vol 51 (2) ◽  
Author(s):  
I. A. Shilin ◽  
A. I. Nizhnikov
Keyword(s):  
Differentiable Function ◽  
Integral Representations ◽  
Legendre Functions ◽  
Poisson Transform ◽  
Infinitely Differentiable Function

Using the Poisson transform, which maps any homogeneous and infinitely differentiable function on a cone into a corresponding function on a hyperboloid, we derive some integral representations of the Legendre functions.

scholarly journals The convolution x-r*xs

1976 ◽  
Vol 17 (1) ◽  
pp. 53-56 ◽  
Author(s):  
B. Fisher
Keyword(s):  
Differentiable Function ◽  
Convolution Of Distributions ◽  
Definition Of ◽  
Image Position ◽  
Infinitely Differentiable Function

In a recent paper [1], Jones extended the definition of the convolution of distributions so that further convolutions could be defined. The convolution w1*w2 of two distributions w1 and w2 was defined as the limit ofithe sequence {wln*w2n}, provided the limit w exists in the sense thatfor all fine functions ф in the terminology of Jones [2], wherew1n(x) = wl(x)τ(x/n), W2n(x) = w2(x)τ(x/n)and τ is an infinitely differentiable function satisfying the following conditions:(i) τ(x) = τ(—x),(ii)0 ≤ τ (x) ≤ l,(iii)τ (x) = l for |x| ≤ ½,(iv) τ (x) = 0 for |x| ≥ 1.

The product of the distributions x−r and δ(r−1)(x)

1972 ◽  
Vol 72 (2) ◽  
pp. 201-204 ◽  
Author(s):  
B. Fisher
Keyword(s):  
Differentiable Function ◽  
Open Interval ◽  
Image Position ◽  
Infinitely Differentiable Function

The product of two distributions f and g on the open interval (a, b), where −∞ ≤ a < b ∞, was defined in (1) as the limit of the sequence {fn·gn} provided this sequence is regular in (a, b), wherefor n = 1, 2, … and ρ is a fixed infinitely differentiable function having the following properties:

scholarly journals On defining the productr−k⋅∇lδ

2004 ◽  
Vol 2004 (16) ◽  
pp. 833-845 ◽  
Author(s):  
C. K. Li ◽  
V. Zou
Keyword(s):  
Differentiable Function ◽  
Regular Sequence ◽  
Distributional Product ◽  
Java Swing ◽  
Infinitely Differentiable Function

Letρ(s)be a fixed infinitely differentiable function defined onR+=[0,∞)having the properties: (i)ρ(s)≥0, (ii)ρ(s)=0fors≥1, and (iii)∫Rmδn(x)dx=1whereδn(x)=cmnmρ(n2r2)andcmis the constant satisfying (iii). We overcome difficulties arising from computing∇lδnand express this regular sequence by two mutual recursions and use a Java swing program to evaluate corresponding coefficients. Hence, we are able to imply the distributional productr−k⋅∇lδfork=1,2,…andl=0,1,2,…with the help of Pizetti's formula and the normalization.

Sign in / Sign up

Export Citation Format

Share Document