Girish Mahajan (Editor)

H square

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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.

Contents

On the unit circle

In general, elements of L2 on the unit circle are given by

∑ n = − ∞ ∞ a n e i n φ

whereas elements of H2 are given by

∑ n = 0 ∞ a n e i n φ .

The projection from L2 to H2 (by setting an = 0 when n < 0) is orthogonal.

On the half-plane

The Laplace transform L given by

[ L f ] ( s ) = ∫ 0 ∞ e − s t f ( t ) d t

can be understood as a linear operator

L : L 2 ( 0 , ∞ ) → H 2 ( C + )

where L 2 ( 0 , ∞ ) is the set of square-integrable functions on the positive real number line, and 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 .

The Laplace transform is "half" of a Fourier transform; from the decomposition

L 2 ( R ) = L 2 ( − ∞ , 0 ) ⊕ L 2 ( 0 , ∞ )

one then obtains an orthogonal decomposition of L 2 ( R ) into two Hardy spaces

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

This is essentially the Paley-Wiener theorem.

References

H square Wikipedia
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