A Note on Operator Equations Describing the Integral
We study operator equations generalizing the chain rule and the substitution rule for the integral and the derivative of the type f ○ g + c = I (Tf ○ g ∙ Tg), f, g є C¹(R), (1) where T : C¹ (R) → C(R) and where I is defined on C(R). We consider suitable conditions on I and T such that (1) is w...
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| Datum: | 2013 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2013
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| Schriftenreihe: | Журнал математической физики, анализа, геометрии |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/106736 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A Note on Operator Equations Describing the Integral / H. König, V. Milman // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 1. — С. 51-58. — Бібліогр.: 4 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Zusammenfassung: | We study operator equations generalizing the chain rule and the substitution rule for the integral and the derivative of the type f ○ g + c = I (Tf ○ g ∙ Tg), f, g є C¹(R), (1) where T : C¹ (R) → C(R) and where I is defined on C(R). We consider suitable conditions on I and T such that (1) is well-defined and, after reformulating (1) as V (f ○ g) = Tf ○ g ∙ Tg, f, g є C¹(R) (2) with V : C¹ (R) → C(R), give the general form of T, V and I. Simple initial conditions then guarantee that the derivative and the integral are the only solutions for T and I. We also consider an analogue of the Leibniz rule and study surjectivity properties there. |
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