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I want to study more formally the properties of the two sided Laplace transform $ \hat f(z)=\int_{-\infty}^{\infty} f(t)e^{zt}dt $ as a kind of generalization of the Fourier transform. I found some good references in the books by LePage and van der Pol, but these are operational calculus books, and I would like a more mathematical approach, well defined spaces, etc...

Where can I find something (books, articles) like that?

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    You made wrongly about the integral representation of Two-sided Laplace transform. It should be $\hat f(z)=\int_{-\infty}^\infty f(t)e^{-zt}~dt$ instead, especially the cases of $f(t)$ so that $\hat f(z)=\int_{-\infty}^\infty f(t)e^{zt}~dt$ is divergent but $\hat f(z)=\int_{-\infty}^\infty f(t)e^{-zt}~dt$ is convergent.2012-07-06

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I found it sometime ago. Advanced Mathematical Analysis by Richard Beals has a really good mathematical formulation of the two-sided Laplace transform, definition spaces and others.

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It seems that the books or articles that formally study the properties of the two sided Laplace transform is difficult to find, as the market about this is too low.

However, if you study clearly about the integral forms of two sided Laplace transform and Fourier transform, you will easily discover that if the output variable of the Fourier transform can hold in the complex world, in fact two sided Laplace transform is only making the output variable of the Fourier transform multiple with $i$ . So you can study the properties of the two sided Laplace transform by borrowing the properties of the Fourier transform. The books or articles that formally study the properties of the Fourier transform is relatively easy to find.