Representation of an infinitely differentiable function as a sum of two functions belonging to quasianalytic classes

1984 ◽  
Vol 35 (6) ◽  
pp. 701-704
Author(s):  
V. G. Khryptun ◽  
B. S. Sikora
Keyword(s):  
Differentiable Function ◽  
Infinitely Differentiable Function

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.

scholarly journals Note on Hypoellipticity of a First Order Linear Partial Differential Operator

1968 ◽  
Vol 32 ◽  
pp. 323-330
Author(s):  
Yoshio Kato
Keyword(s):  
Analytic Function ◽  
Differential Operator ◽  
Differentiable Function ◽  
Partial Differential Operator ◽  
Linear Partial Differential Operator ◽  
Partial Differential ◽  
First Order ◽  
Open Set ◽  
Euclidian Space ◽  
Infinitely Differentiable Function

Let Ω be a domain in the (n + 1)-dimensional euclidian space Rn+1. A linear partial differential operator P with coefficients in C∞(Ω) (resp. in Cω(Ω)) will be termed hypoelliptic (resp. analytic-hypoelliptic) in Ω if a distribution u on Ω (i.e. u ∈ D′(Ω)) is an infinitely differentiable function (resp. an analytic function) in every open set of Ω where Pu is an infinitely differentiable function (resp. an analytic function).

scholarly journals Limit Probabilities for Random Sparse Bit Strings

10.37236/1308 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Katherine St. John
Keyword(s):  
Positive Integer ◽  
Differentiable Function ◽  
Positive Constant ◽  
Linear Order ◽  
Unary Predicate ◽  
First Order ◽  
Infinitely Differentiable Function

Let $n$ be a positive integer, $c$ a real positive constant, and $p(n) = c/n$. Let $U_{n,p}$ be the random unary predicate under the linear order, and $S_c$ the almost sure theory of $U_{n,{c\over n}}$. We show that for every first-order sentence $\phi$: $$ f_{\phi}(c) = \lim_{n\rightarrow\infty}{\Pr}[U_{n,{c\over n}} { has\ property\ } \phi] $$ is an infinitely differentiable function. Further, let $S = \bigcap_c S_c$ be the set of all sentences that are true in every almost sure theory. Then, for every $c>0$, $S_c = S$.

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