Browsing by Author "Fisher, B."

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On the composition of the distributions x(+)(-r) and x(+)(mu)
(2005) Taş, Kenan; Taş, Kenan; 4971
Let F be a distirbution and let f be a locally summable function. The distribution F (f) is defined as the neutrix limit of the sequence {F-n(f)}, where F-n(x) = F(x) (*) delta(n)(x) and {delta(n)(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta function delta(x). The distribution (x(+)(mu))(-r)(+) and (1 x 1(mu))(-r)(+) are evaluated for mu > 0, r = 1, 2,..., and k mu not equal 1, 2,....
On the composition of the distributions x(-1) ln vertical bar x vertical bar and x(+)(r)
(2005) Taş, Kenan; Taş, Kenan; 4971
Let F be a distribution and let f be a locally summable function. The distribution F(f) is defined as the neutrix limit of the sequence {F-n(f)}, where F-n(x) = F(x) * delta(n)(x) and {delta(n)(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function (x). The distribution (x(+)(r))(-1) ln \x(+)(r)\ is evaluated for r = 1, 2.....
On the non-commutative neutrix product of the distributions x(+)(lambda) and x(+)(mu)
(Springer Heidelberg, 2006) Fisher, B.; Taş, Kenan; Tas, K.; 4971
Let f and g be distributions and let g(n) = (g * delta(n))(x), where delta(n)(x) is a certain sequence converging to the Dirac delta function. The non-commutative neutrix product f circle g of f and g is defined to be the limit of the sequence {fg(n)}, provided its limit h exists in the sense that [GRAPHICS] for all functions p in D. It is proved that (x(+)(lambda)ln(p)x(+)) circle (x(+)(mu)ln(q)x(+)) = x(+)(lambda+mu)ln(p+q)x(+), (x(-)(lambda)ln(p)x(-)) circle (x(-)mu ln(q)x(-)) = x(-)(lambda+mu)ln(p+q)x(-), for lambda + mu < -1; lambda,mu,lambda+mu not equal -1,-2,... and p,q = 0,1,2.....