Geodesic
Estimates for functionals with a~known finite set of moments in terms of deviations of operators constructed with the use of the Steklov averages and finite differences
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 26, Tome 392 (2011), pp. 32-66

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The paper presents a new technique for deriving the estimates mentioned in the title. In particular, this technique allows one to establish Jackson-type inequalities for moduli of continuity of arbitrary order with relatively small constants. The results obtained are valid in different spaces of periodic and nonperiodic functions. 
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     author = {O. L. Vinogradov and V. V. Zhuk},
     title = {Estimates for functionals with a~known finite set of moments in terms of deviations of operators constructed with the use of the {Steklov} averages and finite differences},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {32--66},
     publisher = {mathdoc},
     volume = {392},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_392_a1/}
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O. L. Vinogradov; V. V. Zhuk. Estimates for functionals with a~known finite set of moments in terms of deviations of operators constructed with the use of the Steklov averages and finite differences. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 26, Tome 392 (2011), pp. 32-66. http://geodesic.mathdoc.fr/item/ZNSL_2011_392_a1/