Estimates for entropy numbers of sets of smooth functions on the torus Td

Estimates for n-widths of multiplier operators of multiple Walsh series

“…In [2,9,10,18], the asymptotic behavior of entropy numbers of multiplier operators was studied, using different techniques. Estimates were obtained for entropy numbers of sets of finitely and infinitely differentiable functions and sets of analytic functions, on homogeneous spaces and on the torus.…”

“…Estimates for entropy numbers of different operators and embeddings between function spaces of the types Besov and Sobolev with mixed smoothness into Lebesgue have been studied [8,11,12,19,21]. In the papers [10,18] sharp estimates were obtained for entropy numbers of sets of infinitely differentiable functions and of analytic functions, on two-points homogeneous spaces and on the torus, in several cases. We do not know other papers where studies of this type have been carried out.…”

“…In the present paper, we study estimates for entropy numbers of multiplier operators of type Λ and Λ * . The multiplier operators of types Λ and Λ * were studied in [4,18]. In particular, we consider orthonormal systems {φ m } m∈N d 0 of the type φ m (x) = φ (1)…”

“…(2) * U p are sets of infinitely differentiable functions if 0 < r < 1 and of analytic functions if r ≥ 1 (see [18]). If {φ m } is the multiple Walsh system on [0, 1] d , Λ (1) U p and Λ…”

“…In this paper, the trigonometric system considered on the d-dimensional torus T d is constructed by making the product of the usual real trigonometric system on the unit circle (see Proposition 3.1 and Remark 5.4). In [18], estimates for entropy numbers of the operator Λ (2) were studied considering another trigonometric system on the torus. The upper and lower estimates were obtained for all real number r, 0 < r ≤ 1 and for any dimension d. We note that the estimates in [18] when d = 2 are true only if 0 < r ≤ 76/208 and for d = 3 only if 0 < r ≤ 11/16.…”