Estimates of the approximation characteristics of the classes B p,θ Ω of periodic functions of several variables with given majorant of mixed moduli of continuity

Abstract

We obtain order-sharp estimates of the orthogonal projection widths of the classes B Ω p,θ of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers.

Approximation of classes of periodic functions of several variables with given majorant of mixed moduli of continuity

“…First, we establish the upper bounds in (10). According to (8), it is sufficient to obtain the upper bound for the orthoprojective width d ⊥ M (B Ω p,θ , L q ). For this purpose, we consider an approximation of the functions f ∈ B Ω p,θ by trigonometric polynomials t Q(N) of the form t Q(N) (x) = ∑ s∈χ(N) δ s ( f , x).…”

“…We note that the approximation of certain classes of periodic functions of many variables with mixed generalized smoothness by trigonometric polynomials with "numbers" of harmonics from the sets that are analogs of Q(N) was started in work [16]. Later, the approximations by trigonometric polynomials with "numbers" of harmonics from the sets Q(N) were studied in works [8,19,20] and other ones.…”

Estimates of approximative characteristics of the classes $B^{\Omega}_{p,\theta}$ of periodic functions of several variables with given majorant of mixed moduli of continuity in the space $L_{q}$

“…We now give several known assertions, which are used in the subsequent considerations. As was noted above, Ω(t) is a function of the form (5). For a natural N, we set…”

“…In addition, we assume that b j ∈ R, j = 1, d, and 0 < r < l. Hence, properties 1-4 and the conditions (S) and (S l ) are satisfied for the function Ω(t) of the form (5).…”

“…At present, a lot of works are known, in which the quantities d ⊥ M (F, L q ) and d B M (F, L q ) were studied for various classes of functions. We mention works [14,16,17,22,24], where the quantities ( 6) and (7) were considered for the classes of functions of many variables W r p,α , H r p , B r p,θ , and H Ω p (see also numerous references therein). The quantities d ⊥ M (B Ω p,θ , L q ) and d B M (B Ω p,θ , L q ) for the classes of functions of many variables with a given function Ω(t) of the form (5) under the condition b j < r, j = 1, d, were considered in works [4][5][6][7].…”