Approximation by piecewise constant functions on a square (Q1889637)

The author considers the class \(S_n^r, r\in\mathbb Z_+, n\in\mathbb N,\) of piecewise constant functions \(S(x,y)\) defined on the square \(I=[0,1]^2,\) and for a set \(n_j\in\mathbb N,j\in J,\sum_{j\in J}n_j\leq n\) satisfying the following conditions: 1) for each \(x\in[0,1],\) the function \(S(x,\cdot)\) does not decrease in the variable \(y;\) 2) the set \(\bar S\) which is the union of all boundaries of the domains where \(S\) is constant can be represented as \(\bigcup\{\bar s_j\mid j\in J\}\) so that each of \(\bar s_j\) is the generalized graph of a function \(s_j\in S_{n_j}^r[0,1],\;j\in J;\) 3) the value \(S(x_0,y_0)\), \((x_0,y_0)\in \bar S,\) is equal to one of the limit values attained by \(S(x,y)\) as \((x,y)\) approaches \((x_0,y_0)\) and remains in the exterior of \(\bar S.\) Here \(S_n^r[0,1]\) is the class of piecewise polynomial functions (with no more than \(n\) pieces which are polynomials of degree not exceeding \(r\)) defined on an interval \(\Delta\subset [0,1],\) and such that they issue from a point on the boundary of the square \(I\) and come to a point of this boundary. The generalized graph of a function \(s\in S_n^r[0,1]\) is defined as its usual graph together with vertical lines connecting points of jumps (including possibly the ends of the interval \(\Delta).\) The author finds upper bounds for the rate of approximation of continuous functions on \(I\) strictly increasing with respect to \(y\) by the class \(S_n^r.\)