On the Uniform Limit of Multiple Balayage of Vector Integrals
Hubert Halkin · American Mathematical Monthly · 1966
Introduction. Letf be an integrable function from [0, 1] into an n-dimensional Euclidean space. The vector I= fofdp, represents the average of the function f on the interval [0, 1]. The vector I is also the value at t= 1 of the function g(t) = fofdl. If we consider the estimation of the average I as a continuous process then the vector g(t) may be regarded as a certain approximation of the fraction tI of the average vector I. This continuous estimation process is not very accurate if the function f fluctuates greatly. Instead of basing our estimation on a single balayage of the interval [0, 1 ] we could consider a simultaneous balayage of each of the intervals [0, -1] and [i, 1i] and introduce a function gi(t) defined over [0, 1] by the relation: