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Optimal cubature formulas related to tomography for certain classes of functions defined on a cube. (English) Zbl 1262.41018
Let \(\omega\) be a modulus of continuity, and let \([0,1]^d\subset \mathbb{R}^d\). Let \[ H_p^\omega([0,1]^d):=\{f\in C([0,1]^d): |f(x)-f(y)|\leq \omega(\|x-y\|_p),\;x,y\in [0,1]^d \}. \] The paper considers the optimization of cubature formulas when \(p=\infty\). Let \(I=\{1,\dotsc,d\}\). If \(J\subset I\) then \(L(J):=\{\mathbf{z}\in \mathbb{R}^d : \forall i\in J \Rightarrow z_i=0\}\). Also put \(L(\mathbf{x};J)=\mathbf{x}+L(J)\). The set of cubature formulas \(q:C([0,1]^d)\to \mathbb{R}\) is denoted by \(Q_{n,k}^d\), where \(q\) is of the form
\[ q(f)=\sum_{i=1}^n a_i \int_{L(\mathbf{x}^{i};J_i)}f(\mathbf{t})d\mathbf{t},\tag{1} \] where \(\{a_i\}_{i=1}^n\subset \mathbb{R}\), \(\{\mathbf{x}^{i}\}_{i=1}^n\subset [0,1]^d\), \(J_1,\dotsc,J_n \subset I\), \(|J_1|=\dotsb=|J_n|=k\). In particular, if \(k=d\), then the expression on the right side of (1) is the typical linear combination of values of \(f\).
If \(R(f;q)\) is the cubature error, the error on the class \(\mathcal{M}\) is \(R(\mathcal{M};q)\). The optimal cubature formula is \(\mathcal{E}(\mathcal{M};Q_{n,k}^d):=\inf\{R(\mathcal{M};q);\; q\in Q_{n,k}^d\}\).
The main results of the paper are two theorems. The first one finds formulas which are \(Q_{n,k}^d\)-optimal for \(\mathcal{M}=H_\infty^\omega([0,1]^d)\), and proves that \[ \mathcal{E}(H_\infty^\omega([0,1]^d);Q_{n,k}^d)=(2N)^k k\int_0^{1/2N}t^{k-1}\omega(t)\,dt. \] The second result establishes an asymptotic formula for \(\mathcal{E}(H_\infty^\omega([0,1]^d);Q_{n,k}^d)\) as \(n\to \infty\).

41A55 Approximate quadratures
41A63 Multidimensional problems
44A12 Radon transform
65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas