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Lower Bounds on the Continuation of Holomorphic Functions

Robert Rettinger
2008 Electronical Notes in Theoretical Computer Science  
We show that continuation of holomorphic functions needs time at least Ω(n 2 ) in the uniform case, which is optimal up to a polylogarithmic factor.  ...  In the non-uniform case, i.e. assuming f to be computable in time O(t) on an open subset of its domain, we show, that continuation of f cannot be robustly done in time o(n · t(n)).  ...  Uniform and Computational Bounds In this paragraph we will apply the above combinatorical lemma to achieve lower bounds on the complexity needed to continue a holomorphic function in the situation of  ... 
doi:10.1016/j.entcs.2008.12.018 fatcat:aib67ouof5fvlklo5atrrze3fq

Some properties of squeezing functions on bounded domains

Fusheng Deng, Qian Guan, Liyou Zhang
2012 Pacific Journal of Mathematics  
This paper introduces the notion of squeezing functions on bounded domains and studies some of their properties.  ...  The relation to geometric and analytic structures of bounded domains will be investigated. Existence of related extremal maps and continuity of squeezing functions are proved.  ...  Acknowledgements The authors thank Xiangyu Zhou, the PhD advisor of the first two authors, for invaluable instruction and discussions.  ... 
doi:10.2140/pjm.2012.257.319 fatcat:nlhvgutqizhhldsgjr6t3warl4

On Some Properties of Squeezing Functions of Bounded Domains [article]

Fusheng Deng, Qian Guan, Liyou Zhang
2011 arXiv   pre-print
Holomorphic homogeneous regular domains are exactly domains whose squeezing functions have positive lower bounds.  ...  The relation to geometric and analytic structures of bounded domains will be investigated. Existence of related extremal maps and continuity of squeezing functions are proved.  ...  Continuity of squeezing functions In this section, we will prove that the squeezing function on any bounded domain is continuous.  ... 
arXiv:1109.3920v1 fatcat:4oekn3tpqnbajeym4jx2f46hhi

A note on composition operators acting on holomorphic Sobolev spaces

Boo Rim Choe, Hyungwoon Koo, Wayne Smith
2011 Proceedings of the American Mathematical Society  
A holomorphic self-map ϕ of the unit disk is constructed such that the composition operator induced by ϕ is bounded on the Hardy-Sobolev space H 1 2 of order 2 as well as on the ordinary holomorphic Lipschitz  ...  So, the main points here are that our construction provides a composition operator which is bounded on smaller spaces, but not on a larger space and that all the function spaces involved are standard ones  ...  Let g be the holomorphic function on D determined by the conditions that g(0) = 0 and g = ψ + 1. Note that g is continuously differentiable on D.  ... 
doi:10.1090/s0002-9939-2011-10944-4 fatcat:zai543sslfe4vgup72v6ntwbfe

Norm Convergence in the Space of Hyper-functions

A. N. Deepthi, N. R. Mangalambal
2019 Asian Journal of Engineering and Applied Technology  
Hyperfunctions of bounded exponential growth with compact support are mainly considering here.We have developed the results using the defining function of the hyperfunction.  ...  Classical Mathematics is not sufficient to justify some functions like Dirac's delta function, Heaviside's unit step function in the mathematical modelling of some physical problems.  ...  Definition 10[1]: A hyper-function is called holomorphic at , if the lower and upper component of the defining function can analytically be continued to a full(two-dimensional) neighborhood of the real  ... 
doi:10.51983/ajeat-2019.8.2.1145 fatcat:ea2cr4c45ralxafd54syxup7cu

Random walks are determined by their trace on the positive half-line [article]

Mateusz Kwaśnicki
2019 arXiv   pre-print
We prove that the law of a random walk X_n is determined by the one-dimensional distributions of (X_n, 0) for n = 1, 2, ..., as conjectured recently by Loïc Chaumont and Ron Doney.  ...  Equivalently, the law of X_n is determined by its upward space-time Wiener-Hopf factor. Our methods are complex-analytic.  ...  I thank Alexander Ulanovskiȋ for discussions on references [6, 8] . The main part of this article was written during the 39th Conference on Stochastic Processes and their Applications in Moscow.  ... 
arXiv:1902.08481v1 fatcat:ztmpxyrmc5dqnaiopgyfwopplq

Random walks are determined by their trace on the positive half-line

Mateusz Kwaśnicki
2020 Annales Henri Lebesgue  
We prove that the law of a random walk X n is determined by the onedimensional distributions of max(X n , 0) for n = 1, 2, . . . , as conjectured recently by Loïc Chaumont and Ron Doney.  ...  Equivalently, the law of X n is determined by its upward spacetime Wiener-Hopf factor. Our methods are complex-analytic. Résumé.  ...  The main part of this article was written during the 39 th Conference on Stochastic Processes and their Applications in Moscow. BIBLIOGRAPHY  ... 
doi:10.5802/ahl.64 fatcat:ro5x4b2e4ve67chibsc5fpeoj4

Page 1628 of Mathematical Reviews Vol. 46, Issue 6 [page]

1973 Mathematical Reviews  
Let H,, denote the space of germs at w of continuous functions on D that are holomorphic in D, a bounded strictly pseudoconvex domain in (*.  ...  The singular set of an inner function is the set of points on 7” over which the function has no analytic continuation.  ... 

Sectorial normalization of simplest germs of semi-hyperbolic maps in a half-neighborhood

P A Shaikhullina
2020 Ufimskii Matematicheskii Zhurnal  
We consider a problem on analytic classification of semi-hyperbolic maps on the plane for an example of the simplest class of germs, namely, the class of germs that are formally equivalent to F , which  ...  A key step in the classification is an analytic normalization of the germs on sectorial domains forming a cut neighbourhood of the origin (C 2 , 0)∖{ = 0}.  ...  a removable singular point˜− and this is why˜− can be continued to a holomorphic and bounded on C function.  ... 
doi:10.13108/2020-12-2-72 fatcat:jrhltxwzivb4jhrpc2nrz3lbhm

A representation theorem for functions holomorphic off the real axis

Albert Baernstein
1972 Transactions of the American Mathematical Society  
Let/be holomorphic in the union of the upper and lower half planes, and letp e [1, oo). We prove that there exists an entire function y and a sequence {/"} in L*(R) satisfying ||/n||J'B -+ 0 such that  ...  Thus/is in fact a pair of functions/+ and/_, independently prescribed, one holomorphic in the upper half plane, the other in the lower half plane. The singular set is now 7? u {co}.  ...  Moreover, \g\ ^M'in Q, where A/'depends only on k,p, and p, and g is continuous on the closure of Q.  ... 
doi:10.1090/s0002-9947-1972-0293111-2 fatcat:embvuwowyvfexpjrtgacyns5fy

Page 83 of Mathematical Reviews Vol. 35, Issue 1 [page]

1968 Mathematical Reviews  
Let H(U") be the algebra of holomorphic func- tions on U", and let H®(U") be the subalgebra of bounded 413 holomorphic functions on U".  ...  function on X which is bounded above is constant.  ... 

Page 2596 of Mathematical Reviews Vol. , Issue 2000d [page]

2000 Mathematical Reviews  
The authors prove an extension of results on the Hodlder- continuous extension phenomenon for proper holomorphic mappings to the case of holomorphic correspondences.  ...  (F-PROV-AT; Marseille) On the continuous extension of holomorphic correspondences. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), no. 4, 747-766 (1998).  ... 

Some topics in complex and harmonic analysis, 2 [article]

Stephen Semmes
2004 arXiv   pre-print
These notes are concerned with Abel sums and connections with analytic extensions of Fourier integrals.  ...  The Fourier transform of f is a bounded continuous function, and even uniformly continuous.  ...  Similarly, if the Fourier transform of f is supported in (−∞, 0], then the integral for the inverse Fourier transform of f defines a holomorphic function on the lower half-plane, consisting of the complex  ... 
arXiv:math/0403290v1 fatcat:pzkdlnexqrbwhilft2r2u7dgjy

Norm Estimates for Weighted Composition Operators on Spaces of Holomorphic Functions

Isabelle Chalendar, Jonathan R. Partington
2013 Complex Analysis and Operator Theory  
The methods used involve the study of lower-triangular and causal operators.  ...  This paper shows that the boundedness of a weighted composition operator on the Hardy-Hilbert space on the disc or half-plane implies its boundedness on a class of related spaces, including weighted Bergman  ...  They also thank the referee for some helpful comments. This work was partially supported by the ANR project ANR-09-BLAN-0058-01.  ... 
doi:10.1007/s11785-013-0340-4 fatcat:2nuuatjx7reidbjk3gykfzv3tu

A Hilbert bundle characterization of Hilbert C*-modules

George A. Elliott, Katsunori Kawamura
2008 Transactions of the American Mathematical Society  
of pure states of the C*-algebra with the zero functional adjoined.  ...  The category of Hilbert C*-modules over a given C*-algebra is shown to be equivalent to a certain simply described category of Hilbert bundles (i.e., continuous fields of Hilbert spaces) over the space  ...  The bounded holomorphic sections of the dual Hopf bundle associated to K consist exactly of the families of functionals on one-dimensional subspaces corresponding to fixed continuous functionals on all  ... 
doi:10.1090/s0002-9947-08-04600-x fatcat:4af56emtyjhshkyr72ncteux2e
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