Laurent subspaces of the circle Sobolev spaces and holomorphic extension to the disc #
For a set S ⊆ ℤ of frequencies, the elements of H^s whose Fourier coefficients vanish outside
S form a closed subspace, the image of the frequency projection that discards the coefficients
outside S. Taking S to be the nonnegative or the negative frequencies gives the Laurent
splitting H^s = H^s_{≥0} ⊕ H^s_{<0} of a boundary circle, which is orthogonal in the Hilbert case.
These are the subspaces in which the Hardy decomposition of a cut Riemann surface is carried out:
the nonnegative part consists of boundary values of holomorphic functions on the disc bounded by
the circle (Robbin–Salamon, §§9–12).
The second half of the file makes this precise. The power series ∑_{n ≥ 0} a.coeff n w ^ n
converges on the open unit disc for every order s and defines a holomorphic function there,
the disc function of a. For s > 1/2 it converges on the closed disc and is continuous there,
and when the negative coefficients of a vanish its values on the unit circle are the continuous
function CircleSobolev.toContinuousMap attached to a. For every order s, an element of
H^s_{≥0} is determined by its holomorphic extension to the open disc, whose Taylor coefficients
at 0 are its nonnegative Fourier coefficients. Evaluation is jointly holomorphic in the
boundary data and the disc coordinate (CircleSobolev.analyticAt_diskFunction), for arbitrary
Sobolev order.
Main definitions #
TauCeti.CircleSobolev.supported s E S: the closed subspace ofH^sof elements with Fourier coefficients supported inS.TauCeti.CircleSobolev.indicator S: the frequency projection ontosupported s E S.TauCeti.CircleSobolev.indicatorProj S: the same projection, as a map tosupported s E S.TauCeti.CircleSobolev.diskFunction a: the functionw ↦ ∑_{n ≥ 0} a.coeff n w ^ n.
Main statements #
TauCeti.CircleSobolev.isCompl_supported:supported s E Sandsupported s E Sᶜare complementary;TauCeti.CircleSobolev.orthogonal_supportedidentifies the second as the orthogonal complement of the first in the Hilbert case.TauCeti.CircleSobolev.analyticOnNhd_diskFunction: the disc function is holomorphic on the open unit disc.TauCeti.CircleSobolev.continuousOn_diskFunction: fors > 1/2it is continuous on the closed unit disc.TauCeti.CircleSobolev.toContinuousMap_apply_eq_diskFunction: fors > 1/2andawith vanishing negative coefficients, the boundary values of the disc function are the continuous function attached toa.TauCeti.CircleSobolev.eq_of_eqOn_diskFunction: elements with vanishing negative coefficients are determined by their disc functions on the open disc.
References #
- J. W. Robbin and D. A. Salamon, A construction of the Deligne–Mumford orbifold, J. Eur. Math. Soc. 8 (2006), 611–699 (Hardy decompositions on boundary Sobolev spaces).
- P. L. Duren, Theory of Hᵖ Spaces, chapter 1.
Subspaces of prescribed frequency support #
The subspace of H^s of elements whose Fourier coefficients vanish outside S.
Instances For
Frequency projections #
The frequency projection onto S: it keeps the Fourier coefficients indexed by S and
replaces the others by zero. Its range is supported s E S and its kernel is
supported s E Sᶜ.
Instances For
The frequency projection indicator S, as a map onto its range supported s E S.
Instances For
The projection indicatorProj S restricts to the identity on supported s E S.
The subspaces of H^s supported on S and on its complement are complementary: this is the
Laurent splitting of H^s when S is the set of nonnegative frequencies.
In the Hilbert case, the Laurent splitting is orthogonal: the orthogonal complement of the
subspace supported on S is the subspace supported on Sᶜ.
The holomorphic disc function #
The function w ↦ ∑_{n ≥ 0} a.coeff n w ^ n determined by the nonnegative Fourier
coefficients of a. It is holomorphic on the open unit disc for every order s
(CircleSobolev.analyticOnNhd_diskFunction), and for s > 1/2 it is continuous on the closed disc
with boundary values CircleSobolev.toContinuousMap when the negative coefficients of a vanish
(CircleSobolev.toContinuousMap_apply_eq_diskFunction). Where the series does not converge its
value is a junk value.
Instances For
The disc function only depends on the nonnegative Fourier coefficients.
The disc function of the Fourier monomial x e^{inθ}, n ≥ 0, is the monomial w ^ n • x.
The disc function of a Fourier monomial of negative frequency vanishes.
The value of the disc function at the centre is the zeroth Fourier coefficient.
On the open unit disc the disc function is bounded by ∑ₙ ‖w‖ ^ n (1 + n) ^ (-s) times the
norm.
For s > 1/2, the disc function is bounded on the closed unit disc by the bound
√(∑ₙ (1 + |n|) ^ (-2 s)) ‖a‖ of CircleSobolev.norm_toContinuousMap_le.
On the open unit disc the disc function is the sum of its power series.
Disc evaluation is jointly holomorphic in the boundary data and the disc coordinate. The Sobolev order is arbitrary; only the coordinate must lie in the open unit disc.
The disc function is holomorphic on the open unit disc.
The disc function is analytic on the open unit disc.
An element of H^s with vanishing negative Fourier coefficients is determined by its disc
function on the open unit disc: its nonnegative coefficients are the Taylor coefficients of the
disc function at 0.
The closed disc and the boundary circle for s > 1/2 #
For s > 1/2, the power series of the disc function converges absolutely on the closed unit
disc.
For s > 1/2, the disc function is continuous on the closed unit disc.
Boundary values of the disc function. For s > 1/2 and a ∈ H^s with vanishing
negative Fourier coefficients, the continuous function on AddCircle T attached to a is the
restriction of the disc function to the unit circle.