progressive function

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In mathematics, a progressive function ƒ ∈ L²(R) is a function whose Fourier transform is supported by positive frequencies only:{{cite book |last1=Klees |first1=Roland |last2=Haagmans |first2=Roger |title=Wavelets in the Geosciences |date=6 March 2000 |publisher=Springer Science & Business Media |isbn=978-3-540-66951-7 |url=https://books.google.com/books?id=6-xW6Bo-UA8C&dq=%22Progressive+function%22+-wikipedia+Fourier+transform&pg=PA13 |language=en}}

:$\backslash mathop\{\backslash rm\; supp\}\backslash hat\{f\}\; \backslash subseteq\; \backslash mathbb\{R\}\_+.$

It is called super regressive if and only if the time reversed function f(−t) is progressive, or equivalently, if

:$\backslash mathop\{\backslash rm\; supp\}\backslash hat\{f\}\; \backslash subseteq\; \backslash mathbb\{R\}\_-.$

The complex conjugate of a progressive function is regressive, and vice versa.

The space of progressive functions is sometimes denoted $H^2\_+(R)$, which is known as the Hardy space of the upper half-plane. This is because a progressive function has the Fourier inversion formula

:$f(t)\; =\; \backslash int\_0^\backslash infty\; e^\{2\backslash pi\; i\; st\}\; \backslash hat\; f(s)\backslash ,\; ds$

and hence extends to a holomorphic function on the upper half-plane $\backslash \{\; t\; +\; iu:\; t,\; u\; \backslash in\; R,\; u\; \backslash geq\; 0\; \backslash \}$

by the formula

:$f(t+iu)\; =\; \backslash int\_0^\backslash infty\; e^\{2\backslash pi\; i\; s(t+iu)\}\; \backslash hat\; f(s)\backslash ,\; ds$

= \int_0^\infty e^{2\pi i st} e^{-2\pi su} \hat f(s)\, ds.

Conversely, every holomorphic function on the upper half-plane which is uniformly square-integrable on every horizontal line

will arise in this manner.

Regressive functions are similarly associated with the Hardy space on the lower half-plane $\backslash \{\; t\; +\; iu:\; t,\; u\; \backslash in\; R,\; u\; \backslash leq\; 0\; \backslash \}$.