Luzin-type harmonic approximation on subsets of non-compact Riemannian manifolds

Gauthier, Paul M.; Sharifi, Fatemeh

doi:10.1016/j.jmaa.2019.02.008

Cited by 7 publications

(3 citation statements)

References 8 publications

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“…As far as we know, their result is the first general result for nested Riemann surfaces, for L 2 approximability, as opposed to uniform approximation, e.g. P. Gauthier and F. Sharifi [6] (see also F. Sharifi [17] for a literature review).…”

Section: Letmentioning

confidence: 98%

Schiffer comparison operators and approximations on Riemann surfaces bordered by quasicircles

Schippers¹,

Shirazi²,

Staubach³

2019

Preprint

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We consider a compact Riemann surface R of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate R into two subsets: a connected Riemann surface Σ, and the union O of a finite collection of simply-connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on O to the Bergman space of holomorphic forms on Σ is an isomorphism. We then apply this to prove versions of the Plemelj-Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on Σ by elements of Bergman space and Dirichlet space on fixed regions in R containing Σ.

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Section: Letmentioning

confidence: 98%

Schiffer comparison operators and approximations on Riemann surfaces bordered by quasicircles

Schippers¹,

Shirazi²,

Staubach³

2019

Preprint

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“…Density results for L 2 spaces of differentials of general orders (automorphic forms) were obtained by N. Askaripour and T. Barron [2,3], using Poincaré series; see also references therein for density results for L 1 holomorphic quadratic differentials. For other theorems involving approximation on a larger region in other senses such as uniform approximation, see e.g., Gauthier and Sharifi [9]. For literature reviews see F. Sharifi [18] and Fornaess et al [8].…”

Section: Introductionmentioning

confidence: 99%

Faber series for $L^2$ holomorphic one-forms on Riemann surfaces with boundary

Schippers¹,

Shirazi²

2023

Preprint

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Consider a compact surface R with distinguished points z 1 , . . . , z n and conformal maps f k from the unit disk into non-overlapping quasidisks on R taking 0 to z k . Let Σ be the Riemann surface obtained by removing the closures of the images of f k from R. We define forms which are meromorphic on R with poles only at z 1 , . . . , z n , which we call Faber-Tietz forms. These are analogous to Faber polynomials in the sphere. We show that any L 2 holomorphic one-form on Σ is uniquely expressible as a series of Faber-Tietz forms. This series converges both in L 2 (Σ) and uniformly on compact subsets of Σ.

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“…It should be observed that there are very few results involving approximation in the L 2 norm by functions/forms on a larger region. The literature has so far been largely concerned with uniform approximation, e.g., Gauthier and Sharifi [7]; for literature reviews see F. Sharifi [21] and Fornaess et al [5]. To our knowledge the only result of this form is that of Askaripour and Barron [2].…”

Section: Introductionmentioning

confidence: 99%

Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles

2020

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We consider a compact Riemann surface R of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate R into two subsets: a connected Riemann surface $$\Sigma $$ Σ , and the union $$\mathcal {O}$$ O of a finite collection of simply connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $$\mathcal {O}$$ O to the Bergman space of holomorphic forms on $$\Sigma $$ Σ is an isomorphism onto the exact one-forms, when restricted to the orthogonal complement of the set of forms on all of R. We then apply this to prove versions of the Plemelj–Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $$\Sigma $$ Σ by elements of Bergman space and Dirichlet space on fixed regions in R containing $$\Sigma $$ Σ .

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Luzin-type harmonic approximation on subsets of non-compact Riemannian manifolds

Cited by 7 publications

References 8 publications

Schiffer comparison operators and approximations on Riemann surfaces bordered by quasicircles

Schiffer comparison operators and approximations on Riemann surfaces bordered by quasicircles

Faber series for $L^2$ holomorphic one-forms on Riemann surfaces with boundary

Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles

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