Abstract
In this paper we study the approximation properties of measurable and square-integrable functions. In particular we show that any $L^2$-bounded function can be approximated in $L^2$-norm by smooth functions defined on a highly oscillating boundary of thick multi-structures in ${\mathbb{R}}^n$. We derive the norm estimates for the approximating functions and study their asymptotic behaviour.
Highlights
The main object of our consideration in this paper is a class of functions defined on a domain Q, < R”, whose boundary 6Q, contains the very highly oscillating part with respect to a small parameter ¢,as ¢e—>0
We say that Q, is a thick multi-structure in R”, if Q, consists of somefixed domain Q* and a large numberof cylinders with axesparallel to Ox, and ¢ -periodically distributed along some manifold ХУ, on the boundary of Q*.This manifold is called the joint zone and the domain Q” is called the junction’s body
Let ¢ be a thick multi-structure in R", which consists ofsome domain Q* and a large number of thin cylinders G, with a small cross section of the size eC and єperiodically distributed along some manifold X on the boundary of °
Summary
Изучается проблема аппроксимации Г’ - ограниченных функций, заданных на густых сингулярных соединениях. Что любая функция из указаного класса может быть аппраксимирована в норме пространства і гладкой функцией, которая задана только на боковой поверхности быстро осцилирующеге сингулярного соединения. Получены оценки таких приближений и изучено их асимптотическое поведение. Ключовые слова: ‘густое сингулярное’ соединение, ‘аппроксимационные свойства, сингулярные меры, сходимость в переменных просторанствах. In this paper we study the approximation properties of measurable and square-integrable. In particular we show that any І. Bounded functions can be approximated in 27 -погт by smooth functions defined on a highly oscillating boundary of thick muiti-structures in R". We derive the norm estimates for the approximating functions and study their asymptotic behavior
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
