H square

In mathematics and control theory, H 2 , or H-square is a Hardy space with square norm. It is a subspace of L 2 space, and is thus a Hilbert space. In particular, it is a reproducing kernel Hilbert space.

In mathematics and control theory, H 2 , or H-square is a Hardy space with square norm. It is a subspace of L 2 space, and is thus a Hilbert space. In particular, it is a reproducing kernel Hilbert space. \sum_{n=-\infty}^\infty a_n e^{in\varphi} whereas elements of H 2 are given by \sum_{n=0}^\infty a_n e^{in\varphi}. The projection from L 2 to H 2 (by setting a n  = 0 when n  [\mathcal{L}f](s)=\int_0^\infty e^{-st}f(t)dt can be understood as a linear operator \mathcal{L}:L^2(0,\infty)\to H^2\left(\mathbb{C}^+\right) where L^2(0,\infty) is the set of square-integrable functions on the positive real number line, and \mathbb{C}^+ is the right half of the complex plane. It is more; it is an isomorphism, in that it is invertible, and it isometric, in that it satisfies \|\mathcal{L}f\|_{H^2} = \sqrt{2\pi} \|f\|_{L^2}. The Laplace transform is "half" of a Fourier transform; from the decomposition L^2(\mathbb{R})=L^2(-\infty,0) \oplus L^2(0,\infty) one then obtains an orthogonal decomposition of L^2(\mathbb{R}) into two Hardy spaces L^2(\mathbb{R})= H^2\left(\mathbb{C}^-\right) \oplus H^2\left(\mathbb{C}^+\right). This is essentially the Paley-Wiener theorem.

See also

Hardy space H ∞ Unilateral shift operator

References

Jonathan R. Partington, "Linear Operators and Linear Systems, An Analytical Approach to Control Theory", London Mathematical Society Student Texts 60, (2004) Cambridge University Press, .

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