Sharir and Welzl introduced an abstract framework for optimization problems, called LP-type problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we believe, simpler and more natural framework: violator spaces, which constitute a proper generalization of LP-type problems. We show that Clarkson's randomized algorithms for low-d...

Sharir and Welzl introduced an abstract framework for optimization problems, called LP-type problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we believe, simpler and more natural framework: violator spaces, which constitute a proper generalization of LP-type problems. We show that Clarkson’s randomized algorithms for low-d...

We establish two inequalities of Stein-Weiss type for the Riesz potential operator Iα,γ (B−Riesz potential operator) generated by the Laplace-Bessel differential operator ΔB in the weighted Lebesgue spaces Lp,|x|β ,γ . We obtain necessary and sufficient conditions on the parameters for the boundedness of Iα,γ from the spaces Lp,|x|β ,γ to Lq,|x|−λ ,γ , and from the spaces L1,|x|β ,γ to the weak...

The purpose of this paper is threefold: first to mainly review several recent results concerning the fine spectrum of the operator ∆v over the sequence spaces c and lp, where 1 < p < ∞; second to provide some new results concerning the residual spectrum and the continuous spectrum of the operator ∆v over the sequence spaces c and lp; and third to modify the definition of the operator ∆v and to ...

A Banach space X has the approximation property if and only if every compact set in X is in the range of a one-to-one bounded linear operator from a space that has a Schauder basis. Characterizations are given for Lp spaces and quotients of Lp spaces in terms of covering compact sets in X by operator ranges from Lp spaces. A Banach space X is a L1 space if and only if every compact set in X is ...