Geodesic
On best quadrature formulas in the reproducing kernel Hilbert space
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 1 (1998) no. 4, pp. 313-320

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The best quadrature formulas with free nodes in some Hilbert space with a reproducing kernel are considered. It is shown that they are at the same time optimal among the formulas with the same nodes and having directional derivatives. To prove this, the error norm gradient of the formula with the optimal weights is determined as a function of nodes. 
@article{SJVM_1998_1_4_a2,
     author = {A. V. Gavrilov},
     title = {On best quadrature formulas in the reproducing kernel {Hilbert} space},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {313--320},
     publisher = {mathdoc},
     volume = {1},
     number = {4},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_1998_1_4_a2/}
}
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A. V. Gavrilov. On best quadrature formulas in the reproducing kernel Hilbert space. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 1 (1998) no. 4, pp. 313-320. http://geodesic.mathdoc.fr/item/SJVM_1998_1_4_a2/