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.

On the unit circle[edit]

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

whereas elements of H2 are given by

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

On the half-plane[edit]

The Laplace transform given by

can be understood as a linear operator

where is the set of square-integrable functions on the positive real number line, and 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

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

one then obtains an orthogonal decomposition of into two Hardy spaces

This is essentially the Paley-Wiener theorem.

See also[edit]

References[edit]

  • 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.