ASYMPTOTIC EVALUATIONS FOR MULTIVARIATE MELLIN CONVOLUTION OPERATORS

Let n is an element of N and for every w > 0 let K-w : (0, infinity)(n)-> R be Borel measurable kernels. Under suitable assumptions, the multivariate Mellin convolution operator is defined by Z M-w (f) (s(1),..., s(n)) = (0,infinity)(n) K-w (t(1),..., t(n)) f (s(1)t(1), ..., sntn) dt /t(1) middot middot middot t(n.) In the paper we find the necessary and sufficient conditions for the convergence of the multivariate Mellin convolution operators. In the case of differentiable or twice differentiable functions we give the asymptotic evaluations for the multivariate Mellin convolution operators. We apply these general results for specific multivariate Mellin-Gauss-Weierstrass, Mellin-Poisson-Cauchy or Hadamard type convolution operators.