Asymptotic Estimates for the Sequences of the Operators on the Space of Multivariate Periodic Functions
Year: 2026
Type: article
Abstract: In the paper we prove asymptotic estimates for sequences of linear positive operators on the space $$C_{2\pi }\left( \mathbb {R}^{k}\right) $$ C 2 π R k . A sample result: Let $$K_{n}:\mathbb {R}\rightarrow \left[ 0,\infty \right) $$ K n : R → 0 , ∞ be a sequence of even continuous $$2\pi $$ 2 π -periodic functions such that $$\frac{1}{ 2\pi }\int _{-\pi }^{\pi }K_{n}\left( t\right) dt=1$$ 1 2 π ∫ - π π K n t d t = 1 for all $$n\in \mathbb {N}$$ n ∈ N and let $$V_{n}:C_{2\pi }\left( \mathbb {R}^{k}\right) \rightarrow C_{2\pi }\left( \mathbb {R}^{k}\right) $$ V n : C 2 π R k → C 2 π R k be the operator defined by $$\begin{aligned} & V_{n}\left( f\right) \left( x_{1},...,x_{k}\right) =\frac{1}{\left( 2\pi \right) ^{k}}\int _{\left[ -\pi ,\pi \right] ^{k}}f\left( x_{1}-t_{1},...,x_{k}-t_{k}\right) \\ & \quad K_{n}\left( t_{1}\right) \cdot \cdot \cdot K_{n}\left( t_{k}\right) dt_{1}\cdot \cdot \cdot dt_{k}. \end{aligned}$$ V n f x 1 , . . . , x k = 1 2 π k
Source: Results in Mathematics
Author: Dumitru Popa
Institution: Ovidius University
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W7147311078