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ON T H E MODULE OF STRONG D E R I V A T I V E S OVER T H E RING OF CONTINUOUS FUNCTIONS OF TWO V A R I A B L E S

ON T H E MODULE OF STRONG D E R I V A T I V E S OVER T H E RING OF CONTINUOUS FUNCTIONS OF TWO V... DEMONSTRATIO MATHEMATICAVol. XVIINo 41984Wtodzimierz £l$zakON T H E MODULE OF STRONG DERIVATIVES OVER THE RINGOF CONTINUOUS FUNCTIONS OF TWO VARIABLESIn [12] a simple characterization of the system of a l lfunctions f : R — s u c h that f * g i s a derivative for eachcontinuous function g:R —»R was proved. In this note, applyingthe theory of double Denjoy-Celidze integral ( c f , [4]-[7],[9]» [ n ] » [13]> [14]), we obtain the two-dimensional analogueof J . Marik r e s u l t .Let—»R be a r e a l function of two variables and for2x . y e R , l e t I(x;y) - < a c R 2 j x < » < y } 3 fo .3^] * [»2*^2]be a rectangle contained in H 2 . The order relation x $ y f o r* » { x 1 l ( x 2 ) , y m ( y ^ y g i s B * «»ana here thatfor i - 1 , 2 .The increment of a function F on I(x{y) will be defined by theformula(1)A[pjl{xjy)] =- F(y.,,x 2 ) - F(x 1 F Y 2 ) + F http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

ON T H E MODULE OF STRONG D E R I V A T I V E S OVER T H E RING OF CONTINUOUS FUNCTIONS OF TWO V A R I A B L E S

Demonstratio Mathematica , Volume 17 (4): 16 – Oct 1, 1984

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Publisher
de Gruyter
Copyright
© by Wtodzimierz Ślęzak
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1984-0407
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XVIINo 41984Wtodzimierz £l$zakON T H E MODULE OF STRONG DERIVATIVES OVER THE RINGOF CONTINUOUS FUNCTIONS OF TWO VARIABLESIn [12] a simple characterization of the system of a l lfunctions f : R — s u c h that f * g i s a derivative for eachcontinuous function g:R —»R was proved. In this note, applyingthe theory of double Denjoy-Celidze integral ( c f , [4]-[7],[9]» [ n ] » [13]> [14]), we obtain the two-dimensional analogueof J . Marik r e s u l t .Let—»R be a r e a l function of two variables and for2x . y e R , l e t I(x;y) - < a c R 2 j x < » < y } 3 fo .3^] * [»2*^2]be a rectangle contained in H 2 . The order relation x $ y f o r* » { x 1 l ( x 2 ) , y m ( y ^ y g i s B * «»ana here thatfor i - 1 , 2 .The increment of a function F on I(x{y) will be defined by theformula(1)A[pjl{xjy)] =- F(y.,,x 2 ) - F(x 1 F Y 2 ) + F

Journal

Demonstratio Mathematicade Gruyter

Published: Oct 1, 1984

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