Linear Isometries of Spaces of Absolutely Continuous Functions

1982 ◽  
Vol 34 (2) ◽  
pp. 298-306 ◽  
Author(s):  
V. D. Pathak
Keyword(s):  
Real Line ◽  
Upper Bound ◽  
Continuous Functions ◽  
Finite Complex ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
End Points ◽  
Arbitrary Compact Subset ◽  
Image Position ◽  
Complex Valued

Let X be an arbitrary compact subset of the real line R which has at least two points. For each finite complex valued function f on X we denote by V(f; X) (and call it the weak variation of f on X) the least upper bound of the numbers ∑i|f(bi) – f(ai)| where {[ai, bi]} is any sequence of non-overlapping intervals whose end points belong to X. A function f is said to be of bounded variation (BV) on X if V(f; X) < ∞. A function f is said to be absolutely continuous (AC) on X, if given any ∈ > 0 there exists an n > 0 such that for every sequence of non-overlapping intervals {[au bi]} whose end points belong to X, the inequalityimplies that([7], p. 221, 223).

scholarly journals Ostrowski Via a Two Functions Pompeiu's Inequality

2016 ◽  
Vol 24 (3) ◽  
pp. 123-139
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Schwarz Inequality ◽  
Continuous Functions ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Complex Valued

Abstract In this paper, some generalizations of Pompeiu's inequality for two complex-valued absolutely continuous functions are provided. They are applied to obtain some new Ostrowski type results. Reverses for the integral Cauchy-Bunyakovsky-Schwarz inequality are provided as well.

scholarly journals Separating Maps between Spaces of Vector-Valued Absolutely Continuous Functions

2010 ◽  
Vol 53 (3) ◽  
pp. 466-474 ◽  
Author(s):  
Luis Dubarbie
Keyword(s):  
Real Line ◽  
Continuous Functions ◽  
Dimensional Case ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Disjointness Preserving ◽  
Vector Valued

AbstractIn this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied.

scholarly journals On the structure of the set of solutions of the Darboux problem for hyperbolic equations

1986 ◽  
Vol 29 (1) ◽  
pp. 7-14 ◽  
Author(s):  
F. S. de Blasi ◽  
J. Myjak
Keyword(s):  
Integrable Function ◽  
Hyperbolic Equations ◽  
Continuous Functions ◽  
Darboux Problem ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Image Position ◽  
Almost All

Consider the Darboux problemwhere φ,ψ:I→Rd (I=[0,1]) are given absolutely continuous functions with φ(0)=ψ(0), and the mapping f : Q × Rd→Rd (Q = I × I) satisfies the following hypotheses:(A1) f(.,.,z) is measurable for every z ∈ Rd;(A2) f(x, y,.) is continuous for a.a. (almost all) (x, y) ∈ Q;(A3) there exists an integrable function α:Q →[0, + ∞) such that |f(x, y, z)|≦α(x, y) for every (x, y, z)∈ Q × Rd.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

scholarly journals The spectral theorem for well-bounded operators

1993 ◽  
Vol 54 (3) ◽  
pp. 334-351 ◽  
Author(s):  
Ian Doust ◽  
Qiu Bozhou
Keyword(s):  
Functional Calculus ◽  
Continuous Functions ◽  
Weak Compactness ◽  
Compact Interval ◽  
Spectral Theorem ◽  
Type B ◽  
Bounded Operators ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous

AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

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