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H square

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

On the unit circle

In general, elements of L on the unit circle are given by

∑ n = − ∞ ∞ a n e i n φ {\displaystyle \sum _{n=-\infty }^{\infty }a_{n}e^{in\varphi }}

whereas elements of H are given by

∑ n = 0 ∞ a n e i n φ . {\displaystyle \sum _{n=0}^{\infty }a_{n}e^{in\varphi }.}

The projection from L to H (by setting an = 0 when n < 0) is orthogonal.

On the half-plane

The Laplace transform L {\displaystyle {\mathcal {L}}} given by

[ L f ] ( s ) = ∫ 0 ∞ e − s t f ( t ) d t {\displaystyle (s)=\int _{0}^{\infty }e^{-st}f(t)dt}

can be understood as a linear operator

L : L 2 ( 0 , ∞ ) → H 2 ( C + ) {\displaystyle {\mathcal {L}}:L^{2}(0,\infty )\to H^{2}\left(\mathbb {C} ^{+}\right)}

where L 2 ( 0 , ∞ ) {\displaystyle L^{2}(0,\infty )} is the set of square-integrable functions on the positive real number line, and C + {\displaystyle \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

‖ L f ‖ H 2 = 2 π ‖ f ‖ L 2 . {\displaystyle \|{\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 ( R ) = L 2 ( − ∞ , 0 ) ⊕ L 2 ( 0 , ∞ ) {\displaystyle L^{2}(\mathbb {R} )=L^{2}(-\infty ,0)\oplus L^{2}(0,\infty )}

one then obtains an orthogonal decomposition of L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} into two Hardy spaces

L 2 ( R ) = H 2 ( C − ) ⊕ H 2 ( C + ) . {\displaystyle 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

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, ISBN 0-521-54619-2.
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