We introduce a Stieltjes transform on the equivalence classes of
a new generalized function which has been successfully developed
by Colombeau. Subsets of rapid descent test functions, <mml:math alttext="$\mathcal{S}(\mathbb{R}^n)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#119982;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>&#8477;</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>, as well as tempered distributions, <mml:math alttext="$\mathcal{S}'(\mathbb{R}^n)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>&#119982;</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>&#8477;</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>, are used to preserve Fourier analysis techniques.

International Journal of Mathematics and Mathematical Sciences

Stieltjes transforms on new generalized functions

Abstract

Copyright