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Results on Colombeau product of distributions. (English) Zbl 0937.46030
Let \(x_+ = \max (x,0)\), \(x_- = (-x)_+\). The Colombeau products of the distributions \(x^a_+,x^b_-\) and \(\delta ^{(p)}\) are determined in the paper. More exactly, the following formulas are proved: \(x^p_+ \cdot \delta ^{(p)}=(-1)^p p!\delta /2\), \(x^p_- \cdot \delta ^{(p)} = p!\delta /2\), \(x^p\delta ^{(p)} = (-1)^p p!\delta \) for any \(p\in \mathbb N_0\), \(x^a_+ \cdot x^b_- = \Gamma (a+1)\Gamma (b+1)\delta /2\), if \(-a, - b \notin \mathbb N\), \(a+b+1=0\). The same formulas hold if the regularized model product of distributions is considered. The quoted formulas are known with a more special product. The results are generalized for distributions in \(\mathbb R^m\).

MSC:
46F10 Operations with distributions and generalized functions
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