Peter Morris

Chebyshev subspaces of L1 with linear metric projection

Abstract In this note we consider Chebyshev subspaces (i.e., those that contain a unique nearest element to every point) of real L 1 = L 1 [0, 1]. The result we prove is a characterization of those subspaces which are Chebyshev with linear metric projections (nearest point maps). We also give an example of a Chebyshev subspace whose metric projection is not linear.

Non-Zero-Sum Games

In studying games which are not zero-sum, the distinction between cooperative and noncooperative games is crucial. There are two types of cooperation among players. The first is the making of binding agreements before play starts about how they will play (that is, coordination of strategies); the second is the making of agreements about sharing payoffs (or about “side payments” from one player to another). The aim of the first type is to increase the payoffs to the cooperating players; the aim of the second is for one player (or group of players) to induce another player to coordinate strategies.

On an approximation operator of de La Vallée Poussin

Abstract : The paper investigates the general operators of which de la Vallee Poussins is the prototype. Projections onto n-dimensional spaces which use n + 1 quanta of information concerning the projected element. The information is delivered in the form of values of n + 1 linear functionals. Families of such projections are examined to determine which are optimal in a certain sense. (Author)

On a problem of T. J. Rivlin in approximation theory

for which there exists an x E C([O, 11) such that the polynomial of best approximation of degree j to x (in the sense of CebySev) is pj (j = 0, 1, . . ., n 1). What is the answer in the particular case II = 2? In the present paper we shall consider the following more general problem: Let {Gk} be a sequence of linear subspaces of a normed linear space E. Characterize those sequences (gk) in E, with g, E Gk (k = 1,2,. . .), for which there exists an x E E such that

Stability properties of trigonometric interpolating operators

Consider the space C of all 2π-periodic continuous real functions, and the subspace π of all n-th order trigonometric polynomials. The index n is held fixed, and the spaces are endowed with the usual supremum norm. Any operator L: C → π which can be written in the form \( Lx = \sum\limits_1^m {x({s_k}){y_k}} \) with 0 ≤ sk < 2≤ and yk e π is said to be carried by the point set {s1,...,sm}. If Lx = x for all x e π, then L is a projection of C onto π. The uniform grid is defined to be the set of points tk = kπ(2n + 1)-1 for k = 0,...,2n.

Two-Person Zero-Sum Games

Let \( \vec \pi \) be a game in normal form with strategy sets X1,...,X N We say that this game is zero-sum if

N-Person Cooperative Games

\sum\limits_{i = 1}^N {{\pi _i}\,({x_1},\,.\,.\,.\,,\,{x_N})\, = \,0,}

Games in Extensive Form

for every choice of x i ∈ X i , 1 ≤ i ≤ N. The corresponding definition for a game in extensive form states that the sum of the components of \( \vec p(v) \) is zero for each terminal vertex v. This condition is certainly true for ordinary recreational games. It says that one player cannot win an amount unless the other players jointly lose the same amount. Nonrecreational games, however, tend not to be zero-sum. Competitive situations in economics and international politics are often of the type where the players can jointly do better by playing appropriately, and jointly do worse by playing stupidly. The phrase “zero-sum game” has entered the language of politics and business.