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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 $$n\to \infty$$.

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