On the approximate derivative of superposition

2017 ◽  
Vol 9 (2) ◽  
Author(s):  
K. Krzyżewski
2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Stanisław Kowalczyk ◽  
Małgorzata Turowska

We construct a continuous functionf:[0,1]→Rsuch thatfpossessesN-1-property, butfdoes not have approximate derivative on a set of full Lebesgue measure. This shows that Banach’s Theorem concerning differentiability of continuous functions with Lusin’s property(N)does not hold forN-1-property. Some relevant properties are presented.


2002 ◽  
Vol 42 (2) ◽  
pp. 274-283 ◽  
Author(s):  
Lei Nie ◽  
Shouguo Wu ◽  
Xiangqin Lin ◽  
Longzhen Zheng ◽  
Lei Rui

Author(s):  
Mary Nanfuka ◽  
Fredrik Berntsson ◽  
John Mango

AbstractWe consider the Cauchy problem for the Helmholtz equation defined in a rectangular domain. The Cauchy data are prescribed on a part of the boundary and the aim is to find the solution in the entire domain. The problem occurs in applications related to acoustics and is illposed in the sense of Hadamard. In our work we consider regularizing the problem by introducing a bounded approximation of the second derivative by using Cubic smoothing splines. We derive a bound for the approximate derivative and show how to obtain stability estimates for the method. Numerical tests show that the method works well and can produce accurate results. We also demonstrate that the method can be extended to more complicated domains.


1962 ◽  
Vol 66 (5) ◽  
pp. 435-440 ◽  
Author(s):  
Solomon Marcus

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