Efficient and effective algorithms are designed to compute the coordinates of nearly optimal points for multivariate polynomial interpolation on a general geometry. “Nearly optimal” refers to the property that the set of points has a Lebesgue constant near to the minimal Lebesgue constant with respect to multivariate polynomial interpolation on a finite region. The proposed algorithms range fro...

• The exterior measure of any set A ⊂ R is μ∗(A) = inf E⊂A μ(E) where the infimum is taken over all elementary sets. • A is Lebesgue measurable if for all > 0 there exists an open set O ⊃ A such that μ∗(O \A) ≤ . • The (Lebesgue) measure of a measurable set is μ(A) = μ∗(A). A function f is measurable if the sets {t | f(t) ≤ a} are measurable for all a ∈ R. Definition 2 (Lebesgue integral) • For...

The �-dimensional Lebesgue space is the measurable space (E���(E�))— where E = [0 � 1) or E = R—endowed with the Lebesgue measure, and the “calculus of functions” on Lebesgue space is just “real and harmonic analysis.” The �-dimensional Gauss space is the same measure space (R���(R�)) as in the previous paragraph, but now we endow that space with the Gauss measure P� in place of the Lebesgue me...

Preface In these notes we explain the measure theoretic foundations of modern probability. The notes are used during a course that had as one of its principal aims a swift introduction to measure theory as far as it is needed in modern probability, e.g. to define concepts as conditional expectation and to prove limit theorems for martingales. Everyone with a basic notion of mathematics and prob...

C.F. Gauss [Gaus1] [Gaus2] showed that the the number of lattice points in circles is the area, πt , up to a tolerance of order at most the circumference, i.e. the square root, O( √ t) . Sierpinski in 1904, following work of Voronoi [Vor] on the Dirichlet Divisor Problem, lowered the exponent in Gauss’s result to 1/3 [Sierp1] [Sierp2]. Further improvements that have been obtained will be descri...

This paper studies Lebesgue integral of a fuzzy closed set-valued stochastic process with respect to the time t. Firstly, a progressively measurable fuzzy closed set-valued stochastic process is discussed and an almost everywhere problem in the former Aumann type Lebesgue integral of the level-set process is pointed out. Secondly, a new definition of the Lebesgue integral by decomposable closur...

double system of exponents with complex-valued coefficients is considered. under some conditions on the coefficients, we prove that if this system forms a basis for the morrey-lebesgue type space on $left[-pi , pi right]$, then it is isomorphic to the classical system of exponents in this space.