In mathematics and control theory, H2, or H-square is a Hardy space with square norm. It is a subspace of L2 space, and is thus a Hilbert space. In particular, it is a reproducing kernel Hilbert space.
On the unit circle In general, elements of L2 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 H2 are given by
∑ n = 0 ∞ a n e i n φ . {\displaystyle \sum _{n=0}^{\infty }a_{n}e^{in\varphi }.}
The projection from L2 to H2 (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 [{\mathcal {L}}f](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 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, ISBN 0-521-54619-2.
