On a theorem of Denjoy and on approximate derivative

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

scholarly journals On the PropertyN-1

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Stanisław Kowalczyk ◽  
Małgorzata Turowska
Keyword(s):  
Continuous Function ◽  
Lebesgue Measure ◽  
Continuous Functions ◽  
Approximate Derivative ◽  
Banach's Theorem ◽  
Full Lebesgue Measure

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.

Approximate Derivative Calculated by Using Continuous Wavelet Transform

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

scholarly journals Solving the Cauchy problem for the Helmholtz equation using cubic smoothing splines

2021 ◽  
Author(s):  
Mary Nanfuka ◽  
Fredrik Berntsson ◽  
John Mango
Keyword(s):  
Cauchy Problem ◽  
Helmholtz Equation ◽  
Smoothing Splines ◽  
Cauchy Data ◽  
Rectangular Domain ◽  
Stability Estimates ◽  
Second Derivative ◽  
Numerical Tests ◽  
The Cauchy Problem ◽  
Approximate Derivative

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.

scholarly journals On ω-approximately continuous Perron-Stieltjes and Denjoy-Stieltjes integral

1974 ◽  
Vol 18 (2) ◽  
pp. 129-152 ◽  
Author(s):  
D. N. Sarkhel
Keyword(s):  
General Type ◽  
Monotone Function ◽  
Nondecreasing Function ◽  
Stieltjes Integral ◽  
Approximate Derivative ◽  
Perron Integral ◽  
Definition Of

The aim of the present paper is to introduce a definition of the Perron-Stieltjes integral employing the notion of approximate derivative with respect to a nondecreasing function ω and to study some of the properties of the integral. Various authors have studied the Perron integral and Perron-Stieltjes integral in different ways, most of which can be found in the references appended in the list of the bibliography. Among them Ridder [10] uses the concept of approximate co-derivative but he assumes that the monotone function a associated with co is continuous. Finally we consider a more general type of integral, the co-approximately continuous Denjoy-Stieltjes integral, defined descriptively by the method of Saks [11].

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