On the Noncommutative Neutrix Product of Distributions
Let f and g be distributions and let gn=(g*δn)(x), where δn(x) is a certain sequence converging to the Dirac-delta function δ(x). The noncommutative neutrix product f∘g of f and g is defined to be the neutrix limit of the sequence {fgn}, provided the limit h exists in the sense that N‐limn→∞〈f...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2007-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2007/81907 |
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| Summary: | Let f
and g
be distributions and let gn=(g*δn)(x), where δn(x) is a certain sequence converging to the Dirac-delta function δ(x).
The noncommutative neutrix product f∘g
of f
and g
is defined to be the neutrix limit of the sequence {fgn}, provided the limit h
exists in the sense that N‐limn→∞〈f(x)gn(x),φ(x)〉=〈h(x),φ(x)〉, for all test functions in 𝒟. In this paper, using the concept of the neutrix limit due to van der Corput (1960), the noncommutative neutrix products x+rlnx+∘x−−r−1lnx−
and x−−r−1lnx−∘x+rlnx+ are proved to exist and are evaluated for r=1,2,…. It is consequently seen that these two products are in fact equal. |
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| ISSN: | 1085-3375 1687-0409 |