Point evaluations, invariant subspaces and approximation in the mean by polynomials

Brennan, James E.

doi:10.1016/0022-1236(79)90084-3

Cited by 33 publications

(16 citation statements)

References 18 publications

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“…If α ∈ C and U is an annulus centered at α and such that | C ν| is bounded below on U , then (2.1) can easily be used to show that α is a bpe for P 1 (|ν|) and hence, by Hölder's inequality, for P t (µ). This observation was used by Brennan in [Bre79] to prove the existence of bpe's in some special cases, and it is the starting point for Thomson's proof. Thomson devised a coloring scheme on dyadic squares which, given a point a ∈ C and a positive integer m, starts with a dyadic square of side length 2 −m containing a and either terminates at some finite stage or produces an infinite sequence of annuli surrounding a.…”

Section: Bounded Point Evaluationsmentioning

confidence: 96%

Nontangential limits in 𝒫^t(μ)-spaces and the index of invariant subgroups

2009

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Let µ be a finite positive measure on the closed disk D in the complex plane, let 1 ≤ t < ∞, and let P t (µ) denote the closure of the analytic polynomials in L t (µ). We suppose that D is the set of analytic bounded point evaluations for P t (µ), and that P t (µ) contains no nontrivial characteristic functions. It is then known that the restriction of µ to ∂D must be of the form h|dz|. We prove that every function f ∈ P t (µ) has nontangential limits at h|dz|-almost every point of ∂D, and the resulting boundary function agrees with f as an element of L t (h|dz|).Our proof combines methods from James E. Thomson's proof of the existence of bounded point evaluations for P t (µ) whenever P t (µ) = L t (µ) with Xavier Tolsa's remarkable recent results on analytic capacity. These methods allow us to refine Thomson's results somewhat. In fact, for a general compactly supported measure ν in the complex plane we are able to describe locations of bounded point evaluations for P t (ν) in terms of the Cauchy transform of an annihilating measure.As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We show that for 1 < t < ∞ dim M/zM = 1 for every nonzero invariant subspace M of P t (µ) if and only if h = 0.We also investigate the boundary behaviour of the functions in P t (µ) near the points z ∈ ∂D where h(z) = 0. In particular, for 1 < t < ∞ we show that there are interpolating sequences for P t (µ) that accumulate nontangentially almost everywhere on {z : h(z) = 0}.

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Section: Bounded Point Evaluationsmentioning

confidence: 96%

Nontangential limits in 𝒫^t(μ)-spaces and the index of invariant subgroups

2009

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“…Since the L2(oS) and H2(E\Y) norms are equivalent for the polynomials, we can conclude that no point in d (E\Y) is an analytic bounded point evaluation for the polynomials with respect to the L2((l-\q>~x\2)dA) norm. Therefore, by [5,Theorem 4], the polynomials are dense in L2(E\Y, (l-\tp~x\2)dA), and thus are dense in H2(E\Y) by [4,Corollary 3.4]. D Let T be a Jordan arc that connects S to T. How pathological must Y be so that the polynomials have a chance of being dense in H2(E\Y)1 If (p is a conformai map from the unit disk D one-to-one and onto E\Y and the polynomials are dense in H2(E\Y), then by [4, Corollary 3.5] <p must be univalent almost everywhere on <9D.…”

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confidence: 95%

“…The numbering scheme is as follows: Ix(j) ç [I + i/4, 2 + i/4] for j = 1,5,9; I\U) Ç [1 -i/4, 2 -i/4] for ; = 3, 7; Re(z) < Re(C) < Re(?7) whenever z G /i(l), C G /i(5), and n e 7,(9), and Re(z) < Re(C) whenever z e /, (3) and C e 7[(7). Connect the right endpoint of /i(l) to the left endpoint of 7i(3), the right endpoint of 7j(3) to the left endpoint of 7i(5), the right endpoint of 7i (5) to the left endpoint of 7i (7), and the right endpoint of I\ (1) to the left endpoint or 7i(9), with segments 7^2), 7^4), 7i (6), and 7i(8), respectively. Let Yx = {fj=\ I\(j) (see Figure 1).…”

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confidence: 99%

Density of the Polynomials in the Hardy Space of Certain Slit Domains

Akeroyd¹

1992

Proceedings of the American Mathematical Society

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Abstract.In this article we construct a Jordan arc T in the complex plane, with endpoints 0 and 1, such that the polynomials are dense in the Hardy space //2(D\r); D:= {z e C : \z\ < 1} . In order to minimize technical details we do much of our work on a particular "annular" region which has rectilinear boundary. For the rest of the paper let E = {z = x+iy : 1 < max{|x|, \y\} < 2}, S = {z = x+iy : max{|jsc|, \y\} = 1}, and T -{z = x + iy : max{|x|, \y\} -2}. Let us say that a Jordan arc Y := It is well known that if

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“…analytic bounded] point evaluations for polynomials with respect to the Ls(oe) norm? Questions of this sort, phrased in terms of area measure rather than harmonic measure, have been answered by James Brennan [2]. While some of the techniques used in the area measure case have application to our problem, for the most part, new approaches are required in the context of harmonic measure.…”

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confidence: 99%

Point Evaluations and Polynomial Approximation in the Mean with Respect to Harmonic Measure

Akeroyd¹

1989

Proceedings of the American Mathematical Society

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Abstract.For 1 < s < oo and crescents ' G, with harmonic measure co, the author examines the collection of bounded point evaluations, bpe{Ps(a>)), (resp. analytic bounded point evaluations, abpe(/>i(w)) ) for polynomials with respect to the Ls(w) norm. If the polynomials are dense in the generalized Hardy space HS(G), then bpe(Ps(co)) = abpe(/>J(w)) = G (Theorem 4). If the polynomials are not dense in HS(G), then (with a mild restriction on dG) bpe(Ps(a>)) = abpe(/"(«)) = intiG*) (Theorem 7).If G is a crescent, oe = co(-,G,z0) denotes harmonic measure for G at some z0 in G, and 1 < s < oo, then what is the collection of bounded [resp. analytic bounded] point evaluations for polynomials with respect to the Ls(oe) norm? Questions of this sort, phrased in terms of area measure rather than harmonic measure, have been answered by James Brennan [2]. While some of the techniques used in the area measure case have application to our problem, for the most part, new approaches are required in the context of harmonic measure.1. Notation. If G is a bounded Dirichlet region (open) in the complex plane and z0 G G, then let p(-,G,z0) (or v(-,G,z0) etc.) denote harmonic measure for G at z0 .2. Definition. A crescent is a region (open) in the complex plane bounded by two Jordan curves which intersect in a single point (the multiple boundary point) such that one of the curves is internal to the other.For any crescent G, let QG denote the bounded component of the complement of G (the closure of G), ¿^G = 0(0*) and d0G = dGXd^G (where G* denotes the poly normally convex hull of the closure of G). Notice that d0G is the "inner boundary" of G with the multiple boundary point deleted.

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Point evaluations, invariant subspaces and approximation in the mean by polynomials

Cited by 33 publications

References 18 publications

Nontangential limits in 𝒫^t(μ)-spaces and the index of invariant subgroups

Nontangential limits in 𝒫^t(μ)-spaces and the index of invariant subgroups

Density of the Polynomials in the Hardy Space of Certain Slit Domains

Point Evaluations and Polynomial Approximation in the Mean with Respect to Harmonic Measure

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Point evaluations, invariant subspaces and approximation in the mean by polynomials

Cited by 33 publications

References 18 publications

Nontangential limits in 𝒫t(μ)-spaces and the index of invariant subgroups

Nontangential limits in 𝒫t(μ)-spaces and the index of invariant subgroups

Density of the Polynomials in the Hardy Space of Certain Slit Domains

Point Evaluations and Polynomial Approximation in the Mean with Respect to Harmonic Measure

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Nontangential limits in 𝒫^t(μ)-spaces and the index of invariant subgroups

Nontangential limits in 𝒫^t(μ)-spaces and the index of invariant subgroups