The map that associates an analytic function in the disc with its sequence of power series coefficients may be viewed as an extension of the Fourier transform on the circle. If the analytic function extends to be continuous on the circle then the two coincide.

The complex Laplace transform associates a (suitable) function defined on the half line with an analytic function in the right half plane. This may be viewed as an extension of

the Fourier transform on the line: If the analytic function extends to be continuous (and not too large) on the imaginary axis, then the Laplace transform is just the (Poisson extension of the) Fourier transform of the original function. It is therefore appropriate to view the power series representation of an analytic function in the disc and the Laplace representation of an analytic function in the half plane as analogous constructions. Moreover, both are intimately related to the Fourier transform.

In this talk I'll explore the relationship between the weighted Hardy-space norms of analytic functions in the disc and in the half plane and the weighted Lebegue-space norms of their power series and Laplace representations, respectively.