Uniform estimation of a segment function by a polynomial strip of fixed width

The best uniform approximation of a segment function on an interval by a polynomial strip of fixed width (in ordinate) with respect to the Hausdorff measure at each point of the interval is considered. Ranges of strip widths are indicated for which this problem gives outer and inner estimates for the graph of the segment function in terms of the polynomial strip, and a range of strip widths is given for which the problem has an independent value. A necessary and sufficient condition for the existence of a solution and uniqueness conditions are obtained in a form comparable to the Chebyshev alternance. A range of strip widths is indicated for which the solution of the problem is always unique. Certain variational properties of the solution are examined.