On optimization of ``interval'' and ``pointwise'' quadrature formulas for classes of monotone functions (Q2759329)

The author considers two problems of optimization of quadrature formulas for monotone functions. Let \(F_d,d=1,2,\ldots,\) be classes of nondecreasing functions in every variable \(f:[0,1]^d\to[0,1].\) The functional NEWLINE\[NEWLINE \Phi(f;h,x)= \begin{cases} \frac{1}{2h}\int_{x-h}^{x+h}f(t) dt,& h > 0; \\ f(x),& h=0; \end{cases} NEWLINE\]NEWLINE is defined by a function of the class \(F_1.\) The set \(Q_n=Q_n(H_n, X_n) = \{q:F_1\to{\mathbb R},q(f)=q(f;H_n,X_n,\varphi_n)= \varphi_n(\Phi(f;x_1,h_1),\ldots,\Phi(f;x_n,h_n))\}\) is defined for a fixed collection \(H_n\) and \(X_n.\) The problem of optimization for functions of the class \(F_1\) is reduced to find the functional \(q\in Q_n\) which realizes the infimum NEWLINE\[NEWLINER(F_1, Q_n) = \inf_{q\in Q_n} \sup_{f\in F_1}\Biggl|\int_0^1 f(t) dt - q(f)\Biggr|.NEWLINE\]NEWLINE The author also considers the sets \( Q = Q(H_n), \) \( Q = Q_n^{\delta}, \) \( \delta \in (0, 1), \) where \( 2\sum_{i=1}^n h_i = \delta . \) NEWLINENEWLINENEWLINEThe problem of optimization of cubature formulae for the classes \( F_d, \) \(d=2,3,\ldots,\) is considered for fixed knots \(Y_n = \{y_1,y_2,\ldots,y_n\}.\) The set \(Y_n\) and arbitrary functions \(\varphi_n:{\mathbb R}^n\to{\mathbb R} \) generate the cubature formula \(S(f;Y_n,\varphi_n)=\varphi_n(f(y_1),\ldots,f(y_n)). \) The problem of optimization is reduced to finding a solution \(\varphi_n \) to the problem \(r(F_d; Y_n) = \inf_{\varphi_n} \sup_{f \in F_d} |\int_{[0,1]^d} f(t) dt - S(f; Y_n, \varphi_n)|\).