Geodesic
An optimal lattice cubature formula in Banach spaces of periodic functions
Matematičeskie zametki, Tome 17 (1975) no. 1, pp. 67-69.

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We establish the optimality of a formula of rectangles among cubature formulas on a given lattice taken in one of the equivalent normalizations on the Banach space of the periodic functions,which is compactly embedded in the space of continuous functions with a norm which is invariant with respect to translations of the arguments of the functions. 
@article{MZM_1975_17_1_a7,
     author = {M. D. Ramazanov},
     title = {An optimal lattice cubature formula in {Banach} spaces of periodic functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {67--69},
     publisher = {mathdoc},
     volume = {17},
     number = {1},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_17_1_a7/}
}
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M. D. Ramazanov. An optimal lattice cubature formula in Banach spaces of periodic functions. Matematičeskie zametki, Tome 17 (1975) no. 1, pp. 67-69. http://geodesic.mathdoc.fr/item/MZM_1975_17_1_a7/