scholarly journals Differential Operators of Infinite Order and the Distribution of Zeros of Entire Functions

1994 ◽  
Vol 186 (3) ◽  
pp. 799-820 ◽  
Author(s):  
T. Craven ◽  
G. Csordas
Keyword(s):  
Infinite Order ◽  
Differential Operators ◽  
Entire Functions ◽  
Distribution Of Zeros ◽  
Zeros Of Entire Functions

The phase problem in scattering phenomena: the zeros of entire functions and their significance

1978 ◽  
Vol 360 (1700) ◽  
pp. 25-45 ◽  
Keyword(s):  
Entire Functions ◽  
Signal To Noise Ratio ◽  
Phase Problem ◽  
Distribution Of Zeros ◽  
Real Zeros ◽  
Zeros Of Entire Functions ◽  
Functions Of Exponential Type ◽  
Zeros Of Functions ◽  
The Fourier Transform ◽  
Complex Zeros

The paper outlines an approach to the calculation of the phase from intensity data based on the properties of the distribution of zeros of functions of exponential type. This leads to a reinterpretation of such phenomena as Gibbs’ or speckle, which underlines their intrinsic unity. The phase problem is solved for functions which present complex zeros by apodization, i.e. by creating a sufficiently large zero-free area. The method is based on a compromise between signal to noise ratio and resolution and is meaningful provided the apodization required is not too severe. Real zeros, for which the phase problem is trivial, occur only for the special case of eigenfunctions of the Fourier transform

Zeros of entire functions of infinite order

1992 ◽  
Vol 58 (4) ◽  
pp. 371-384
Author(s):  
A. M. Russakovskii
Keyword(s):  
Infinite Order ◽  
Entire Functions ◽  
Zeros Of Entire Functions

scholarly journals Infinite order differential operators in spaces of entire functions

2003 ◽  
Vol 277 (2) ◽  
pp. 423-437 ◽  
Author(s):  
Yuri Kozitsky ◽  
Piotr Oleszczuk ◽  
Lech Wołowski
Keyword(s):  
Infinite Order ◽  
Differential Operators ◽  
Entire Functions

scholarly journals Holomorphic functions, relativistic sum, Blaschke products and superoscillations

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Daniel Alpay ◽  
Fabrizio Colombo ◽  
Stefano Pinton ◽  
Irene Sabadini
Keyword(s):  
Infinite Order ◽  
Complex Analysis ◽  
Differential Operators ◽  
Entire Functions ◽  
Growth Conditions ◽  
Weak Values ◽  
Blaschke Products ◽  
Independent Field ◽  
Superoscillating Functions ◽  
Band Limited

AbstractSuperoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. The notion of superoscillation is a particular case of that one of supershift. In the recent years, superoscillating functions, that appear for example in weak values in quantum mechanics, have become an interesting and independent field of research in complex analysis and in the theory of infinite order differential operators. The aim of this paper is to study some infinite order differential operators acting on entire functions which naturally arise in the study of superoscillating functions. Such operators are of particular interest because they are associated with the relativistic sum of the velocities and with the Blaschke products. To show that some sequences of functions preserve the superoscillatory behavior it is of crucial importance to prove that their associated infinite order differential operators act continuously on some spaces of entire functions with growth conditions.

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