A minimax problem admitting the equioscillation characterization of Bernstein and Erdös
The paper under review deals with the following minimax problem. Let \(a<b\), \(n\geq 1\), and \(X:=\{{\mathbf x}= (x_1, \dots, x_n): x_0:= a<x_1 <\cdots <x_{n+1}: =b\}\). Let \(f_i( {\mathbf x}) \geq 0\), \(i=1,2, \dots, n+1\), be continuously differentiable functions on \(X\) and \(f( {\mathbf x}): =\max_{1\leq i\leq n+1} f_i({\mathbf x})\). The minimax problem is concerned with finding a vector \({\mathbf y}\in X\) such that \(f({\mathbf y}) =\inf_{{\mathbf x} \in X} f({\mathbf x})\). The results of the paper are equioscillation characterizations of Bernstein and Erdős for the minimax solution. One of the main results is: let the functions \(f_i\) satisfy the conditions \[ \lim_{\min_{0\leq j\leq n} (x_{j+1} -x_j)\to 0} \max_{1\leq i\leq n} \bigl| f_{i+1} ({\mathbf x})-f_i({\mathbf x}) \bigr| =\infty, \tag{1} \] and \[ D_k({\mathbf x}): =\text{det} \left({\partial f_i({\mathbf x}) \over \partial x_j} \right)^{nn+ 1}_{j=1, i=1,i\neq k} \neq 0,\quad {\mathbf x} \in X,\;k=1,2, \dots, n+1, \tag{2} \] then there exists a unique vector \({\mathbf y}\in X\) such that (3) \(f({\mathbf y}) =\min_{{\mathbf x} \in X} f({\mathbf x})\). The following statements are also valid under the assumptions (1) and (2). Equation (3) holds if and only if \(f_1({\mathbf y}) =f_2({\mathbf y}) \dots= f_{n+1} ({\mathbf y})\); and for any vector \({\mathbf x} \in X \setminus \{{\mathbf y}\} \min_{1\leq i\leq n+1} f_i({\mathbf x}) <f({\mathbf y}) <\max_{1\leq n\leq n+1} f_i({ \mathbf x})\).
scientific article; zbMATH DE number 1197041