The approximate integration of functions of two variables
A formula is considered for the approximate integration \[ (1)\quad \int^{1}_{0}\int^{1}_{0}f(x,y)dx dy=\sum^{m,n}_{i,j=1}\lambda_{ij}Z_{ij}+R(f) \] of functions f from the class \(W_ 0^{rs}L_ 2\), for which \(f^{(r,s)}\) belongs to the class \(L_ 2\) over the unit square; the partial derivatives exist and are piecewise-continuous. Moreover, \(f^{(k,0)}_{(0,\ell)}=f^{(0,\ell)}(x,0)=0\), \(K=\overline{0,r-1}\); \(\ell =\overline{0,S-1}\), \(Z_{ij}\) are approximate values of the function f in the given nodes. The unique optimal formula of (1) type is obtained; \(\lambda_{ij}\) are derived from the system of linear equations. For the function f from the other class \(W^{rs}L_ 2\) the unique best cubature formula is obtained with the integrals over each of the variables. Applying the result achieved to a subset of the class \(W^{rr}L_ 2\) yields a cubature formula with the number of nodes reduced in comparison with the known formula obtained earlier (the precision order remains the same).
scientific article; zbMATH DE number 4013242