Geodesic
Estimates for mixed moduli of smoothness in $L_q$ metric via mixed moduli of smoothness in $L_1$ metric
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2018), pp. 12-26

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Interrelations between mixed fractional moduli of smoothness considered in metrics of $L_1$ and $L_q$ are studied in the paper. 
@article{VMUMM_2018_2_a1,
     author = {M. K. Potapov and B. V. Simonov},
     title = {Estimates for mixed moduli of smoothness in $L_q$ metric via mixed moduli of smoothness in $L_1$ metric},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {12--26},
     publisher = {mathdoc},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2018_2_a1/}
}
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M. K. Potapov; B. V. Simonov. Estimates for mixed moduli of smoothness in $L_q$ metric via mixed moduli of smoothness in $L_1$ metric. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2018), pp. 12-26. http://geodesic.mathdoc.fr/item/VMUMM_2018_2_a1/