Approximation of holomorphic functions by Taylor-Abel-Poisson means

Abstract

We investigate the problem of approximation of functions f holomorphic in the unit disk by means A f r ρ, ( ) as ρ → 1 -. In terms of the error of approximation by these means, a constructive characteristic of classes of holomorphic functions H p r Lip α is given. The problem of the saturation of A f r ρ, ( ) in the Hardy space H p is solved.

Approximation of functions by linear summation methods in the Orlicz-type spaces

“…This fact is related to the socalled saturation property of the approximation method, generated by the operator A ϱ,r . In particular, in [15], it was shown that the operator A ϱ,r generates the linear approximation method for holomorphic functions, which is saturated in the Hardy space H p with the saturation order (1 − ϱ) r and the saturation class H r−1 p Lip 1.…”

“…The operators A ϱ,r were first studied in [15], where the author gave a structural characteristic of the Hardy-Lipschitz classes H r p Lip α of one-variable functions, holomorphic in a unit disk on the complex plane in the terms of these operators. Approximative properties of these operators were also considered in [13,16].…”

Approximation of functions of several variables by linear methods in the space Sp

“…In general case, the operators P △ ̺,s were perhaps first considered as the aggregates of approximation of functions of one variable in [4,5]. The operators A △ ̺,r were first studied in [6], where in the terms of these operators, the author gives the structural characteristic of Hardy-Lipschitz classes H r p Lip α of one variable functions, holomorphic in the unit circle in the complex plane. In special cases, when r = s = 1, the operators A △ ̺,1 and P △ ̺,1 coincide with each other and generate the Abel-Poisson method of summation of multiple Fourier series in the triangular areas.…”

Direct and Inverse Theorems on the Approximation of 2π-Periodic Functions by Taylor–Abel–Poisson Operators

“…Assume that f ∈ L p , 1 ≤ p ≤ ∞, n, r ∈ N, n ≤ r and the function ω(t), t ∈ [0, 1], satisfies conditions 1)-4), (Z ) and (Z n ) . If relation (14) holds, then f [r−n] ∈ L p and relation (13) also holds.…”

“…then the quantity on the right-hand side of (14) decreases to zero as ̺ → 1− not faster, than the related to the so-called saturation property of the approximation method, generated by the operator A ̺,r . In particular, in [14], it was shown that the operator A ̺,r generates the linear approximation method of holomorphic functions, which is saturated in the space H p with the saturation order (1−̺) r and the saturation class H r−1 p Lip 1.…”