Symmetric and quantum symmetric derivatives of Lipschitz functions

[摘要] The symmetric derivative of a real valued function f at the real number x is defined to be {GRAPHICS} when that limit exists, and if additionally x not equal 0, the quantum symmetric derivative is defined to be {GRAPHICS} when that limit exists. An increasing function phi : R+ --> R satisfying {GRAPHICS} defines by {f: \f (x + h) - f (x)\ less than or equal toC-f phi(h)} a class of continuous functions which we call a Lipschitz class of functions smoother than Lip 1/2. The symmetric derivative and the quantum symmetric derivative are equivalent pointwise everywhere for functions that are in any Lipschitz class smoother than Lip 1/2, but not necessarily for functions that are Lipschitz of order 1/2. (C) 2003 Elsevier Inc. All rights reserved.

[发布日期] 2003-12-15

[关键词] quantum derivative;q-derivative;quantum symmetric derivative;symmetric derivative;Lipschitz function