Monotonicity and the Approximate Symmetric Derivative

1986 ◽  
Vol 12 (1) ◽  
pp. 121
Author(s):  
Larson
Keyword(s):  
Symmetric Derivative

The First Symmetric Derivative

1967 ◽  
Vol 74 (6) ◽  
pp. 708 ◽  
Author(s):  
C. E. Aull
Keyword(s):  
Symmetric Derivative

scholarly journals On The K-Pseudo Symmetric and Ordinary Differentiation

2016 ◽  
Vol 49 (2) ◽  
Author(s):  
E. Łazarow ◽  
M. Turowska
Keyword(s):  
Continuous Function ◽  
Lebesgue Measure ◽  
Measure Zero ◽  
Lebesgue Measure Zero ◽  
Definition Of ◽  
Ordinary Derivative ◽  
Symmetric Derivative

AbstractIn 1972, S. Valenti introduced the definition of k-pseudo symmetric derivative and has shown that the set of all points of a continuous function, at which there exists a finite k-pseudo symmetric derivative but the finite ordinary derivative does not exist, is of Lebesgue measure zero. In 1993, L. Zajícek has shown that for a continuous function f, the set of all points, at which f is symmetrically differentiable but no differentiable, is σ-(1 - ε) symmetrically porous for every ε > 0. The question arises: can we transferred the Zajícek’s result to the case of the k-pseudo symmetric derivative?In this paper, we shall show that for each 0 < ε < 1 the set of all points of a continuous function, at which there exists a finite k-pseudo symmetric derivative but the finite ordinary derivative does not exist, is σ-(1 - ε)-porous.

ON THE SYMMETRIC DERIVATIVE

1980 ◽  
Vol 6 (2) ◽  
pp. 235
Author(s):  
Larson
Keyword(s):  
Symmetric Derivative

Symmetric derivative of nowhere monotone functions, I

1970 ◽  
Vol 23 (3) ◽  
pp. 247-253 ◽  
Author(s):  
N. K. Kundu ◽  
S. N. Mukhopadhyay
Keyword(s):  
Monotone Functions ◽  
Symmetric Derivative

scholarly journals Differentiable positive definite kernels and Lipschitz continuity

1988 ◽  
Vol 104 (2) ◽  
pp. 361-369 ◽  
Author(s):  
James A. Cochran ◽  
Mark A. Lukas
Keyword(s):  
Approximation Theory ◽  
Finite Rank ◽  
Asymptotic Estimate ◽  
Lipschitz Continuity ◽  
Piecewise Linear ◽  
Positive Definite ◽  
Basis Functions ◽  
Bounded Region ◽  
Positive Definite Kernels ◽  
Symmetric Derivative

AbstractReade[11] has shown that positive definite kernels K(x, t) which satisfy a Lipschitz condition of order α on a bounded region have eigenvalues which are asymptotically O(1/n1+α). In this paper we extend this result to positive definite kernels whose symmetric derivative Krr(x, t) = ∂2rK(x, t)/∂xτ ∂tτ is in Lipα and establish λn(K) = O(1/n2r+1+α). If ∂Krr/∂t is in Lipα, the anticipated asymptotic estimate is also derived.The proofs use a well-known result of Chang [2], recently rederived by Ha [5], and estimates based upon finite rank approximations to the kernels in question. In these latter estimates we employ the familiar piecewise linear ‘hat’ basis functions of approximation theory.

Sign in / Sign up

Export Citation Format

Share Document