×

On changing the variable in distributions. (English) Zbl 0727.46021

The authors define the distributions \(x_+^{-1}=(\ln x_+)'\) by induction: \(x_+^{-s}=-(s-1)^{-1}(x_+^{-s+1})'\) for \(s=2,3,...\), and \(x_ -^{-s}=(-x)^{-s}\), \(| x|^{-s}=x_+^{-s}+x_ - ^{-s}\), \(sgn| x|^{-s}=x_+^{-s}-x_+^{-s}\) for \(s=1,2,...\), where ln \(x_+=\ln x\) when \(x>0\) and ln \(x_+=0\) when \(x<0\). They prove the existence of the above-mentioned distributions and show their corresponding expressions by means of changes of the independent variable and by using neutrix limits in distributions.

MSC:

46F10 Operations with distributions and generalized functions
PDFBibTeX XMLCite