TY - JOUR
T1 - A note on operator equations describing the integral
AU - König, H.
AU - Milman, V.
PY - 2013
Y1 - 2013
N2 - We study operator equations generalizing the chain rule and the substitution rule for the integral and the derivative of the type fog + c=I (Tfog.Tg), f,g ∈ 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(fog)=Tfog.Tg, f,g ∈ 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.
AB - We study operator equations generalizing the chain rule and the substitution rule for the integral and the derivative of the type fog + c=I (Tfog.Tg), f,g ∈ 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(fog)=Tfog.Tg, f,g ∈ 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.
KW - Chain rule
KW - Integral
KW - Leibniz rule
KW - Operator equation
UR - http://www.scopus.com/inward/record.url?scp=84883099083&partnerID=8YFLogxK
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AN - SCOPUS:84883099083
SN - 1812-9471
VL - 9
SP - 51
EP - 58
JO - Journal of Mathematical Physics, Analysis, Geometry
JF - Journal of Mathematical Physics, Analysis, Geometry
IS - 1
ER -