Universal Series on a Riemann Surface

2011 ◽  
Vol 63 (5) ◽  
pp. 1025-1037
Author(s):  
Raphäel Clouâtre
Keyword(s):  
Riemann Surface ◽  
Holomorphic Function ◽  
Compact Subset ◽  
Fundamental Solutions ◽  
Partial Sums ◽  
Universal Series ◽  
Cauchy Riemann ◽  
Series Of Functions

Abstract Every holomorphic function on a compact subset of a Riemann surface can be uniformly approximated by partial sums of a given series of functions. Those functions behave locally like the classical fundamental solutions of the Cauchy–Riemann operator in the plane.

Universal Power Series in ℂN

2011 ◽  
Vol 54 (2) ◽  
pp. 230-236 ◽  
Author(s):  
Raphaël Clouâtre
Keyword(s):  
Power Series ◽  
Holomorphic Function ◽  
Compact Subset ◽  
Partial Sums

AbstractWe establish the existence of power series in ℂN with the property that the subsequences of the sequence of partial sums uniformly approach any holomorphic function on any well chosen compact subset outside the set of convergence of the series. We also show that, in a certain sense, most series enjoy this property.

scholarly journals Analytic Functions on Some Riemann Surfaces

1963 ◽  
Vol 22 ◽  
pp. 211-217 ◽  
Author(s):  
Nobushige Toda ◽  
Kikuji Matsumoto
Keyword(s):  
Riemann Surface ◽  
Compact Subset ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Interesting Theorem ◽  
Hyperbolic Riemann Surface

Some years ago, Kuramochi gave in his paper [5] a very interesting theorem, which can be stated as follows.THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class Of OHR(OHD,resp.). Then, for any compact subset K of R such that R—K is connected, R—K as an open Riemann surface belongs to the class 0AB(OAD resp.).

Uniform summability of double Walsh-Fourier series of functions of bounded partial Λ-variation

2014 ◽  
Vol 64 (6) ◽  
Author(s):  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Necessary Conditions ◽  
Partial Sums ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Uniformly Convergence ◽  
Negative Order ◽  
Cesáro Means ◽  
Sufficient And Necessary Conditions ◽  
Series Of Functions

AbstractThe sufficient and necessary conditions on the sequence Λ = {λn} are found for the uniformly convergence of Cesàro means of negative order of cubic partial sums of double Walsh-Fourier series of functions of bounded partial Λ-variation.

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