Hardy Spaces For The Unit Disk
For spaces of holomorphic functions on the open unit disk, the Hardy space H2 consists of the functions ƒ whose mean square value on the circle of radius r remains bounded as r → 1 from below.
More generally, the Hardy space Hp for 0 < p < ∞ is the class of holomorphic functions f on the open unit disk satisfying
This class Hp is a vector space. The number on the left side of the above inequality is the Hardy space p-norm for f, denoted by It is a norm when p ≥ 1, but not when 0 < p < 1.
The space H∞ is defined as the vector space of bounded holomorphic functions on the disk, with the norm
For 0 < p ≤ q ≤ ∞, the class Hq is a subset of Hp, and the Hp-norm is increasing with p (it is a consequence of Hölder's inequality that the Lp-norm is increasing for probability measures, i.e. measures with total mass 1).
Read more about this topic: Hardy Space
Famous quotes containing the words hardy, spaces, unit and/or disk:
“If all hearts were open and all desires known—as they would be if people showed their souls—how many gapings, sighings, clenched fists, knotted brows, broad grins, and red eyes should we see in the market-place!”
—Thomas Hardy (1840–1928)
“through the spaces of the dark
Midnight shakes the memory
As a madman shakes a dead geranium.”
—T.S. (Thomas Stearns)
“During the Suffragette revolt of 1913 I ... [urged] that what was needed was not the vote, but a constitutional amendment enacting that all representative bodies shall consist of women and men in equal numbers, whether elected or nominated or coopted or registered or picked up in the street like a coroner’s jury. In the case of elected bodies the only way of effecting this is by the Coupled Vote. The representative unit must not be a man or a woman but a man and a woman.”
—George Bernard Shaw (1856–1950)
“Unloved, that beech will gather brown,
This maple burn itself away;
Unloved, the sun-flower, shining fair,
Ray round with flames her disk of seed,
And many a rose-carnation feed
With summer spice the humming air;”
—Alfred Tennyson (1809–1892)