Negative Theorems on Monotone Approximation

Negative Theorems on Monotone Approximation

Negative Theorems in Approximation Theory

1. INTRODUCTION. Negative theorems have a rich tradition in mathematics. In fact mathematics seems to be unique among the sciences in that negative results are very much a part of the mathematical edifice. Most mathematical theories try to explain what is possible and also what is not. To understand the structure of a mathematical theory is also to understand its limitations. There are many dif...

On 3-monotone approximation by piecewise polynomials

Abstract. We consider 3-monotone approximation by piecewise polynomials with prescribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L1 approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone approximation by piecewise polynomials ...

Three-monotone spline approximation

For r ≥ 3, n ∈ N and each 3-monotone continuous function f on [a, b] (i.e., f is such that its third divided differences [x0, x1, x2, x3] f are nonnegative for all choices of distinct points x0, . . . , x3 in [a, b]), we construct a spline s of degree r and of minimal defect (i.e., s ∈ Cr−1[a, b]) with n −1 equidistant knots in (a, b), which is also 3-monotone and satisfies ‖ f − s‖L∞[a,b] ≤ cω...

On Egoroff's theorems on finite monotone non-additive measure space

In this note, we give four versions of Egoroff’s theorem in non-additive measure theory by using the condition (E) and the pseudo-condition (E) of set function and the duality relations between the conditions. These conditions employed are not only sufficient, but also necessary for the four kinds of Egoroff’s theorem, respectively.