On Quality of Approximation by Families of Generalized Sampling Series

Properties of moduli of smoothness inLp(Rd)

“…In the case 1 ≤ p ≤ ∞, the first equivalence (1.12) is well known, see, e.g., [11,Ch. 6,§ 5] or [30,Appendix A]; see also [13]. For periodic functions f ∈ L p (T), 0 < p < 1, and α ∈ (1/p − 1) + , ∞ , equivalence (1.14) was derived in [33].…”

“…For Jackson's inequality (1.29) in the case 1 ≤ p ≤ ∞ see, e.g., [64, p. 279]. In the case 0 < p < 1, α ∈ N, and d = 1, this inequality was derived in [61] (see also [5]). Sharp Jackson inequality (1.30) was obtained in [10].…”

On moduli of smoothness and Fourier multipliers in L p , 0 < p< 1

“…In the case 1 ≤ p ≤ ∞, the first equivalence (1.11) is well known, see, e.g., [11,Ch. 6,§ 5] or [27,Appendix A]; see also [13]. For periodic functions f ∈ L p (T), 0 < p < 1, and α ∈ (1/p − 1) + , ∞ , equivalence (1.12) was derived in [30].…”

“…For Jackson's inequality (1.25) in the case 1 ≤ p ≤ ∞ see, e.g., [58, p. 279]. In the case 0 < p < 1, α ∈ N, and d = 1, this inequality was derived in [56] (see also [5]). Inequality (1.26) was obtained in [10].…”