Lesson plan
Progressive function
In mathematics, a progressive function ƒ ∈ L 2 (R) is a function whose Fourier transform is supported by positive frequencies only: : \mathop{\rm supp}\hat{f} \subseteq \mathbb{R}_+. It is called super regressive if and only if the time reversed function f(−t) is progressive, or equivalently, if : \mathop{\rm supp}\hat{f} \subseteq \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) = \int_0^\infty e^{2\pi i st} \hat f(s)\, ds and hence extends to a holomorphic function on the upper half-plane \{ t + iu: t, u \in R, u \geq 0 \} by the formula : f(t+iu) = \int_0^\infty e^{2\pi i s(t+iu)} \hat f(s)\, 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 \{ t + iu: t, u \in R, u \leq 0 \} .
Article summary
In mathematics, a progressive function ƒ ∈ L 2 (R) is a function whose Fourier transform is supported by positive frequencies only: : \mathop{\rm supp}\hat{f} \subseteq \mathbb{R}_+. It is called super regressive if and only if the time reversed function f(−t) is progressive, or equivalently, if : \mathop{\rm supp}\hat{f} \subseteq \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) = \int_0^\infty e^{2\pi i st} \hat f(s)\, ds and hence extends to a holomorphic function on the upper half-plane \{ t + iu: t, u \in R, u \geq 0 \} by the formula : f(t+iu) = \int_0^\infty e^{2\pi i s(t+iu)} \hat f(s)\, 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 \{ t + iu: t, u \in R, u \leq 0 \} .
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
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- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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