Bounded Point Evaluation in C n

1995 ◽  
Vol 347 (6) ◽  
pp. 2169
Author(s):  
R. M. Range ◽  
M. I. Stessin
Keyword(s):  
Point Evaluation ◽  
Bounded Point Evaluation

scholarly journals Some remarks on the spaceR2(E)

1983 ◽  
Vol 6 (3) ◽  
pp. 459-466
Author(s):  
Claes Fernström
Keyword(s):  
Compact Subset ◽  
Complex Plane ◽  
Rational Functions ◽  
Compact Set ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Point Evaluation ◽  
Bounded Point Evaluation ◽  
Point Evaluations ◽  
Necessary And Sufficient

LetEbe a compact subset of the complex plane. We denote byR(E)the algebra consisting of the rational functions with poles offE. The closure ofR(E)inLp(E),1≤p<∞, is denoted byRp(E). In this paper we consider the casep=2. In section 2 we introduce the notion of weak bounded point evaluation of orderβand identify the existence of a weak bounded point evaluation of orderβ,β>1, as a necessary and sufficient condition forR2(E)≠L2(E). We also construct a compact setEsuch thatR2(E)has an isolated bounded point evaluation. In section 3 we examine the smoothness properties of functions inR2(E)at those points which admit bounded point evaluations.

scholarly journals Smoothness properties of functions inR2(x)at certain boundary points

1979 ◽  
Vol 2 (3) ◽  
pp. 415-426
Author(s):  
Edwin Wolf
Keyword(s):  
Lebesgue Measure ◽  
Compact Subset ◽  
Complex Plane ◽  
Rational Functions ◽  
Point Evaluation ◽  
Boundary Points ◽  
Bounded Point Evaluation ◽  
Dimensional Lebesgue Measure

LetXbe a compact subset of the complex planeℂ. We denote byR0(X)the algebra consisting of the (restrictions toXof) rational functions with poles offX. Letmdenote2-dimensional Lebesgue measure. Forp≥1, letRp(X)be the closure ofR0(X)inLp(X,dm).In this paper, we consider the casep=2. Letxϵ∂Xbe both a bounded point evaluation forR2(X)and the vertex of a sector contained inIntX. LetLbe a line which passes throughxand bisects the sector. For thoseyϵL∩Xthat are sufficiently nearxwe prove statements about|f(y)−f(x)|for allfϵR2(X).

scholarly journals Functions in the spaceR2(E)at boundary points of the interior

1983 ◽  
Vol 6 (2) ◽  
pp. 363-370
Author(s):  
Edwin Wolf
Keyword(s):  
Lebesgue Measure ◽  
Compact Subset ◽  
Complex Plane ◽  
Limit Point ◽  
Rational Functions ◽  
Point Evaluation ◽  
Boundary Points ◽  
Bounded Point Evaluation ◽  
Dimensional Lebesgue Measure

LetEbe a compact subset of the complex planeℂ. We denote byR(E)the algebra consisting of (the restrictions toEof) rational functions with poles offE. Letmdenote2-dimensional Lebesgue measure. Forp≥1, letRp(E)be the closure ofR(E)inLp(E,dm).In this paper we consider the casep=2. Letx ϵ ∂Ebe a bounded point evaluation forR2(E). Suppose there is aC>0such thatxis a limit point of the sets={y|y ϵ Int E,Dist(y,∂E)≥C|y−x|}. For thosey ϵ Ssufficiently nearxwe prove statements about|f(y)−f(x)|for allf ϵ R(E).

scholarly journals Matrix integrals & finite holography

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Dionysios Anninos ◽  
Beatrix Mühlmann
Keyword(s):  
Critical Exponents ◽  
Continuum Theory ◽  
Point Of View ◽  
Leading Order ◽  
Minimal Models ◽  
Point Evaluation ◽  
Brst Cohomology ◽  
Matrix Integrals ◽  
The Matrix ◽  
The Continuum

Abstract We explore the conjectured duality between a class of large N matrix integrals, known as multicritical matrix integrals (MMI), and the series (2m − 1, 2) of non-unitary minimal models on a fluctuating background. We match the critical exponents of the leading order planar expansion of MMI, to those of the continuum theory on an S2 topology. From the MMI perspective this is done both through a multi-vertex diagrammatic expansion, thereby revealing novel combinatorial expressions, as well as through a systematic saddle point evaluation of the matrix integral as a function of its parameters. From the continuum point of view the corresponding critical exponents are obtained upon computing the partition function in the presence of a given conformal primary. Further to this, we elaborate on a Hilbert space of the continuum theory, and the putative finiteness thereof, on both an S2 and a T2 topology using BRST cohomology considerations. Matrix integrals support this finiteness.

Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions

1999 ◽  
Vol 42 (2) ◽  
pp. 139-148 ◽  
Author(s):  
José Bonet ◽  
Paweł Dománski ◽  
Mikael Lindström
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Essential Norm ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Point Evaluation ◽  
Weighted Banach Spaces ◽  
Spaces Of Analytic Functions

AbstractEvery weakly compact composition operator between weighted Banach spaces of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

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