TY - JOUR
T1 - The chain rule as a functional equation
AU - Artstein-Avidan, Shiri
AU - König, Hermann
AU - Milman, Vitali
N1 - Funding Information:
* Corresponding author. E-mail address: [email protected] (H. König). 1 The author is supported by ISF grant 865-07 and by BSF grant 2006079. 2 The author is supported by the Alexander von Humboldt Foundation, by ISF grant 387/09 and by BSF grant 2006079.
PY - 2010/12
Y1 - 2010/12
N2 - We consider operators T from C1(R{double-struck}) to C(R{double-struck}) satisfying the "chain rule". T(f o g)=(Tf) o g ·Tg,f,gεC1(R{double-struck}), and study under which conditions this functional equation admits only the derivative or its powers as solutions. We also consider T operating on other domains like Ck(R{double-struck}) for kεN{double-struck}0 or k=∞ and study the more general equation T(f o g)=(Tf) o g ·Ag, f,gεC1(R{double-struck}) where both T and A map C1(R{double-struck}) to C(R{double-struck}).
AB - We consider operators T from C1(R{double-struck}) to C(R{double-struck}) satisfying the "chain rule". T(f o g)=(Tf) o g ·Tg,f,gεC1(R{double-struck}), and study under which conditions this functional equation admits only the derivative or its powers as solutions. We also consider T operating on other domains like Ck(R{double-struck}) for kεN{double-struck}0 or k=∞ and study the more general equation T(f o g)=(Tf) o g ·Ag, f,gεC1(R{double-struck}) where both T and A map C1(R{double-struck}) to C(R{double-struck}).
KW - Chain rule
KW - Functional equation
UR - https://www.scopus.com/pages/publications/77956184143
U2 - 10.1016/j.jfa.2010.07.002
DO - 10.1016/j.jfa.2010.07.002
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AN - SCOPUS:77956184143
SN - 0022-1236
VL - 259
SP - 2999
EP - 3024
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 11
ER -