Geodesic
On approximation of periodic functions by modified Steklov averages in~$L_2$
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 29, Tome 429 (2014), pp. 20-33

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In the space $L_2$ of periodic functions, sharp (in the sence of constants) estimates from below for the deviation of the modified Steklov functions of the first and second order in terms of the modulus of continuity are established. Similar results are also obtained for even continuous periodic functions with nonnegative Fourier coefficients in the space $C$. 
@article{ZNSL_2014_429_a2,
     author = {V. O. Dron and V. V. Zhuk},
     title = {On approximation of periodic functions by modified {Steklov} averages in~$L_2$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {20--33},
     publisher = {mathdoc},
     volume = {429},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_429_a2/}
}
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V. O. Dron; V. V. Zhuk. On approximation of periodic functions by modified Steklov averages in~$L_2$. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 29, Tome 429 (2014), pp. 20-33. http://geodesic.mathdoc.fr/item/ZNSL_2014_429_a2/