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On multiplication of some generalized functions. (English) Zbl 0961.46032

Summary: We show that if we extend the classical definition of a product of functions to a larger class of distributions, then for the distributions of the form \({1\over(\cdot-i0)^\alpha}\) and \({1\over (\cdot+i0)^\alpha}\), where \(\alpha\) is complex number, we get formulas: \[ {1\over (\cdot-i0)^\alpha}\cdot {1\over (\cdot- i0)^\beta}= {1\over (\cdot- i0)^{\alpha+\beta}} \] and \[ {1\over (\cdot+ i0)^\alpha}\cdot {1\over (\cdot+ i0)^\beta}= {1\over (\cdot+ i0)^{\alpha+ \beta}}, \] when \(\alpha\) and \(\beta\) are complex numbers, such that \(\text{Re }\alpha> {1\over 2}\) and \(\text{Re }\beta> {1\over 2}\).

MSC:

46F10 Operations with distributions and generalized functions
46F12 Integral transforms in distribution spaces
44A15 Special integral transforms (Legendre, Hilbert, etc.)
44A20 Integral transforms of special functions
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References:

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