Geodesics and Lebesgue area

The Additivity of the Lebesgue Area

A triple of continuous functions T: x(ut u ), i = l, 2, 3, defined on a closed square Q [ O g w x ^ l , 0 ^ w 2 ^ l ] represents a surface © (1.6, 1.17, 1.21). If r is any closed rectangle in Q then we may speak of the triple Tr consisting of the above triple T with its range of definition restricted to r. This triple generates a surface @(r). If r\ and r2 have no interior points in common, and...

The Isoperimetric Inequality and the Lebesgue Definition of Surface Area

The literature of these classical isoperimetric inequalities is very extensive (comprehensive presentations may be found in Blaschke [l] and Bonnesen [l](1)). In most instances, only convex curves and convex surfaces are considered, or else it is assumed that the curves and surfaces involved are sufficiently regular to permit the use of the classical formulas for the quantities a(C), 1(C), A(S)...

Topological Methods in the Theory of Lebesgue Area

Lebesgue functions and Lebesgue constants in polynomial interpolation

The Lebesgue constant is a valuable numerical instrument for linear interpolation because it provides a measure of how close the interpolant of a function is to the best polynomial approximant of the function. Moreover, if the interpolant is computed by using the Lagrange basis, then the Lebesgue constant also expresses the conditioning of the interpolation problem. In addition, many publicatio...

Lebesgue Measure

How do we measure the ”size” of a set in IR? Let’s start with the simplest ones: intervals. Obviously, the natural candidate for a measure of an interval is its length, which is used frequently in differentiation and integration. For any bounded interval I (open, closed, half-open) with endpoints a and b (a ≤ b), the length of I is defined by `(I) = b − a. Of course, the length of any unbounded...