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:

Some results on the product of distributions

1973 ◽  
Vol 73 (2) ◽  
pp. 317-325 ◽  
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(1) as the limit of the sequence {fngn}, provided this sequence is regular on (a, b), wherefor n = 1, 2, … and ρ is a fixed infinitely differentiable function having the following properties:(1) ρ(x) = 0 for |x| ≥ 1,(2) ρ(x) ≥ 0,(3) ρ(x) = ρ(−x),

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.

Quotient groups and realization of tight Riesz groups

1973 ◽  
Vol 16 (4) ◽  
pp. 416-430 ◽  
Author(s):  
John Boris Miller
Keyword(s):  
Normal Subgroup ◽  
Open Interval ◽  
Canonical Homomorphism ◽  
Interval Topology ◽  
Essential Step ◽  
Image Position ◽  
Quotient Groups

Let (G, ≼) be an l-group having a compatible tight Riesz order ≦ with open-interval topology U, and H a normal subgroup. The first part of the paper concerns the question: Under what conditions on H is the structure of (G, ≼, ∧, ∨, ≦, U) carried over satisfactorily to by the canonical homomorphism; and its answer (Theorem 8°): H should be an l-ideal of (G, ≼) closed and not open in (G, U). Such a normal subgroup is here called a tangent. An essential step is to show that ≼′ is the associated order of ≦′.

Linear measures of some sets of the Cantor type

1957 ◽  
Vol 53 (2) ◽  
pp. 312-317 ◽  
Author(s):  
Trevor J. Mcminn
Keyword(s):  
Lebesgue Measure ◽  
Measure Zero ◽  
Cartesian Product ◽  
Open Interval ◽  
Cantor Set ◽  
Unit Interval ◽  
Linear Measure ◽  
Closed Intervals ◽  
Image Position ◽  
Dense Set

1. Introduction. Let 0 < λ < 1 and remove from the closed unit interval the open interval of length λ concentric with the unit interval. From each of the two remaining closed intervals of length ½(1 − λ) remove the concentric open interval of length ½λ(1 − λ). From each of the four remaining closed intervals of length ¼λ(1 − λ)2 remove the concentric open interval of length ¼λ(l − λ)2, etc. The remaining set is a perfect non-dense set of Lebesgue measure zero and is the Cantor set for λ = ⅓. Let Tλr be the Cartesian product of this set with the set similar to it obtained by magnifying it by a factor r > 0. Letting L be Carathéodory linear measure (1) and letting G be Gillespie linear square(2), Randolph(3) has established the following relations:

Absolute summability of infinite series on a scale of Abel type summability methods

1968 ◽  
Vol 64 (2) ◽  
pp. 377-387 ◽  
Author(s):  
Babban Prasad Mishra
Keyword(s):  
Infinite Series ◽  
Open Interval ◽  
Absolute Summability ◽  
Finite Limit ◽  
Summability Methods ◽  
Image Position

Suppose that λ > − 1 and thatIt is easy to show thatWith Borwein(1), we say that the sequence {sn} is summable Aλ to s, and write sn → s(Aλ), if the seriesis convergent for all x in the open interval (0, 1)and tends to a finite limit s as x → 1 in (0, 1). The A0 method is the ordinary Abel method.

On the summability of Fourier integrals

1970 ◽  
Vol 67 (1) ◽  
pp. 23-28
Author(s):  
M. K. Nayak
Keyword(s):  
Open Interval ◽  
Finite Limit ◽  
Image Position ◽  
Fourier Integrals

We say a series is summable L iftends to a finite limit s as x → 1 in the open interval (0, 1) where

Some applications of renewal theory on the whole line

1970 ◽  
Vol 7 (03) ◽  
pp. 734-746
Author(s):  
Kenny S. Crump ◽  
David G. Hoel
Keyword(s):  
Distribution Function ◽  
Real Line ◽  
Renewal Theory ◽  
Open Interval ◽  
One Dimensional ◽  
The Real ◽  
Half Open Interval ◽  
Negative Function ◽  
Dimensional Distribution ◽  
Image Position

Suppose F is a one-dimensional distribution function, that is, a function from the real line to the real line that is right-continuous and non-decreasing. For any such function F we shall write F{I} = F(b)– F(a) where I is the half-open interval (a, b]. Denote the k-fold convolution of F with itself by Fk* and let Now if z is a non-negative function we may form the convolution although Z may be infinite for some (and possibly all) points.

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