Geodesic
Approximation of Classes $B^r_{p,\theta}$ of Periodic Functions of One and Several Variables
Matematičeskie zametki, Tome 87 (2010) no. 3, pp. 429-442

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We obtain order-sharp estimates of best approximations to the classes $B^r_{p,\theta}$ of periodic functions of several variables in the space $L_q$, $1\le p,q\le\infty$, by trigonometric polynomials with “numbers” of harmonics from step hyperbolic crosses. In the one-dimensional case, we establish the order of deviation of Fourier partial sums of functions from the classes $B^{r_1}_{1,\theta}$ in the space $L_1$.
Keywords: class $B^r_{p,\theta}$ of periodic functions, trigonometric polynomial, hyperbolic cross, Fourier hyperbolic sum
Mots-clés : Bernoulli kernel, Valée-Poussin kernel, Fejér kernel. 
@article{MZM_2010_87_3_a10,
     author = {A. S. Romanyuk},
     title = {Approximation of {Classes} $B^r_{p,\theta}$ of {Periodic} {Functions} of {One} and {Several} {Variables}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {429--442},
     publisher = {mathdoc},
     volume = {87},
     number = {3},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_87_3_a10/}
}
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A. S. Romanyuk. Approximation of Classes $B^r_{p,\theta}$ of Periodic Functions of One and Several Variables. Matematičeskie zametki, Tome 87 (2010) no. 3, pp. 429-442. http://geodesic.mathdoc.fr/item/MZM_2010_87_3_a10/