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Lower Bounds on the Continuation of Holomorphic Functions

2008
*
Electronical Notes in Theoretical Computer Science
*

We show that

doi:10.1016/j.entcs.2008.12.018
fatcat:aib67ouof5fvlklo5atrrze3fq
*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*...##
###
Some properties of squeezing functions on bounded domains

2012
*
Pacific Journal of Mathematics
*

This paper introduces

doi:10.2140/pjm.2012.257.319
fatcat:nlhvgutqizhhldsgjr6t3warl4
*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. ...##
###
On Some Properties of Squeezing Functions of Bounded Domains
[article]

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*. ...

##
###
A note on composition operators acting on holomorphic Sobolev spaces

2011
*
Proceedings of the American Mathematical Society
*

A

doi:10.1090/s0002-9939-2011-10944-4
fatcat:zai543sslfe4vgup72v6ntwbfe
*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. ...##
###
Norm Convergence in the Space of Hyper-functions

2019
*
Asian Journal of Engineering and Applied Technology
*

Hyperfunctions

doi:10.51983/ajeat-2019.8.2.1145
fatcat:ea2cr4c45ralxafd54syxup7cu
*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 ...##
###
Random walks are determined by their trace on the positive half-line
[article]

2019
*
arXiv
*
pre-print

We prove that

arXiv:1902.08481v1
fatcat:ztmpxyrmc5dqnaiopgyfwopplq
*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. ...##
###
Random walks are determined by their trace on the positive half-line

2020
*
Annales Henri Lebesgue
*

We prove that

doi:10.5802/ahl.64
fatcat:ro5x4b2e4ve67chibsc5fpeoj4
*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 ...##
###
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

2020
*
Ufimskii Matematicheskii Zhurnal
*

We consider a problem

doi:10.13108/2020-12-2-72
fatcat:jrhltxwzivb4jhrpc2nrz3lbhm
*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*. ...##
###
A representation theorem for functions holomorphic off the real axis

1972
*
Transactions of the American Mathematical Society
*

Let/be

doi:10.1090/s0002-9947-1972-0293111-2
fatcat:embvuwowyvfexpjrtgacyns5fy
*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. ...##
###
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]

2004
*
arXiv
*
pre-print

These notes are concerned with Abel sums and connections with analytic extensions

arXiv:math/0403290v1
fatcat:pzkdlnexqrbwhilft2r2u7dgjy
*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 ...##
###
Norm Estimates for Weighted Composition Operators on Spaces of Holomorphic Functions

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. ...

##
###
A Hilbert bundle characterization of Hilbert C*-modules

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 ...

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