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On the convolution product of the distributional kernel \(K_{\alpha,\beta,\gamma,\nu}\). (English) Zbl 1106.46303
Summary: We introduce a distributional kernel \(K_{\alpha,\beta,\gamma,\nu}\) which is related to the operator \(\oplus^k\) iterated \(k\) times and defined by \(\oplus^k = [(\sum_{r=1}^p {\partial^2}/{\partial x_r^2})^4 - (\sum_{j=p+1}^{p+q} {\partial^2}/{\partial x_j^2})^4]^k\), where \(p + q = n\) is the dimension of the space \({\mathbb R}^n\) of the \(n\)-dimensional Euclidean space, \(x = (x_1, x_2,\dotsc, x_n)\in {\mathbb R}^n\), \(k\) is a nonnegative integer, and \(\alpha\), \(\beta\), \(\gamma\), and \(\nu\) are complex parameters. It is found that the existence of the convolution \(K_{\alpha,\beta,\gamma,\nu}* K_{\alpha',\beta',\gamma',\nu'}\) is depending on the conditions of \(p\) and \(q\).
MSC:
46F10 Operations with distributions and generalized functions
46F12 Integral transforms in distribution spaces
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