Bernstein inequality for multivariate functions with smooth Fourier images
UDC 517.5 Let $K$ be a compact set in ${\Bbb R}^n$ with $(O)$-property and let $1\leq p\leq \infty$. Then there exists a constant $C_K< \infty $ independent of $f$ and $\alpha$ such that $$\|D^{\alpha } f \|_p \leq C_K \sup\limits_{\xi \in K } |\xi ^{\alpha} |\, \|f\|_{\mathcal{H}_{p,K,3}...
Gespeichert in:
| Datum: | 2026 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2026
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/6386 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: | |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512342424420352 |
|---|---|
| author | Bang, Ha Huy Huy, Vu Nhat Bang, Ha Huy Huy, Vu Nhat |
| author_facet | Bang, Ha Huy Huy, Vu Nhat Bang, Ha Huy Huy, Vu Nhat |
| author_sort | Bang, Ha Huy |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-02-23T13:26:31Z |
| description | UDC 517.5
Let $K$ be a compact set in ${\Bbb R}^n$ with $(O)$-property and let $1\leq p\leq \infty$. Then there exists a constant $C_K< \infty $ independent of $f$ and $\alpha$ such that $$\|D^{\alpha } f \|_p \leq C_K \sup\limits_{\xi \in K } |\xi ^{\alpha} |\, \|f\|_{\mathcal{H}_{p,K,3}}$$ for all $\alpha \in \mathbb{Z}_+^n$ and $f\in \mathcal{H}_{p,K,3},$ where $\mathcal{H}_{p,K,3}=\big\{f \in L^p({\Bbb R}^n)\colon {\rm supp\,} \widehat f \subset K ,D^{(3,3,\ldots,3)} \widehat{f} \in C({\Bbb R}^n) \big\},$ $\|f\|_{\mathcal{H}_{p,K,3}} = \big\|D^{(3,3,\ldots,3)} \widehat{f}\,\big\|_\infty$, and $\widehat{f}$ is the Fourier transform of $f$. Note that $K$ is said to have the $(O)$-property if there exists a constant $C>0$ such that $$\sup\limits_{{\bf x} \in K} |{\bf x}^{\alpha + e_j} | \geq C\sup\limits_{{\bf x} \in K } |{\bf x}^{\alpha} |$$ for all $\alpha \in \mathbb{Z}_+^n$ and $j=1,2, \ldots ,n$. |
| doi_str_mv | 10.37863/umzh.v74i11.6386 |
| first_indexed | 2026-03-24T03:27:16Z |
| format | Article |
| fulltext |
Skip to main content
Skip to main navigation menu
Skip to site footer
Open Menu
Ukrains’kyi Matematychnyi Zhurnal
Current
Archives
Submissions
Major topics of interest
About
About Journal
Editorial Team
Ethics & Disclosures
Contacts
Search
Register
Login
Home
/
Login
Login
Required fields are marked with an asterisk: *
Subscription required to access item. To verify subscription, log in to journal.
Login
Username or Email
*
Required
Password
*
Required
Forgot your password?
Keep me logged in
Login
Register
Language
English
Українська
Information
For Readers
For Authors
For Librarians
subscribe
Subscribe
Latest publications
Make a Submission
Make a Submission
STM88 menghadirkan Link Gacor dengan RTP tinggi untuk peluang menang yang lebih sering! Bergabunglah sekarang dan buktikan keberuntungan Anda!
Pilih STM88 sebagai agen toto terpercaya Anda dan nikmati kenyamanan bermain dengan sistem betting cepat, result resmi, dan bonus cashback harian.
|
| id | umjimathkievua-article-6386 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:27:16Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2e/7ffba3e1169679d8c5532e21fe45e72e |
| spelling | umjimathkievua-article-63862026-02-23T13:26:31Z Bernstein inequality for multivariate functions with smooth Fourier images Bernstein inequality for multivariate functions with smooth Fourier images Bang, Ha Huy Huy, Vu Nhat Bang, Ha Huy Huy, Vu Nhat $L^p$- spaces, Bernstein inequality, generalized functions 2010 AMS Subject Classification: 26D10, 46E30 UDC 517.5 Let $K$ be a compact set in ${\Bbb R}^n$ with $(O)$-property and let $1\leq p\leq \infty$. Then there exists a constant $C_K< \infty $ independent of $f$ and $\alpha$ such that $$\|D^{\alpha } f \|_p \leq C_K \sup\limits_{\xi \in K } |\xi ^{\alpha} |\, \|f\|_{\mathcal{H}_{p,K,3}}$$ for all $\alpha \in \mathbb{Z}_+^n$ and $f\in \mathcal{H}_{p,K,3},$ where $\mathcal{H}_{p,K,3}=\big\{f \in L^p({\Bbb R}^n)\colon {\rm supp\,} \widehat f \subset K ,D^{(3,3,\ldots,3)} \widehat{f} \in C({\Bbb R}^n) \big\},$ $\|f\|_{\mathcal{H}_{p,K,3}} = \big\|D^{(3,3,\ldots,3)} \widehat{f}\,\big\|_\infty$, and $\widehat{f}$ is the Fourier transform of $f$. Note that $K$ is said to have the $(O)$-property if there exists a constant $C>0$ such that $$\sup\limits_{{\bf x} \in K} |{\bf x}^{\alpha + e_j} | \geq C\sup\limits_{{\bf x} \in K } |{\bf x}^{\alpha} |$$ for all $\alpha \in \mathbb{Z}_+^n$ and $j=1,2, \ldots ,n$. УДК 517.5 Нерівність Бернштейна для функцій багатьох змінних з гладкими зображеннями Фур’є  Нехай $K$ – компактна множина в ${\Bbb R}^n$, що має $(O)$-властивість і $1\leq p\leq \infty$.  Тоді існує стала $C_K< \infty $, незалежна від $f$ та  $\alpha$, така, що $$ \|D^{\alpha } f \|_p \leq C_K \sup\limits_{\xi \in K } |\xi ^{\alpha} |\, \|f\|_{\mathcal{H}_{p,K,3}}$$ для всіх $\alpha \in \mathbb{Z}_+^n$ і $f\in \mathcal{H}_{p,K,3},$ де $\mathcal{H}_{p,K,3}=\big\{f \in L^p({\Bbb R}^n)\colon {\rm supp\,} \widehat f \subset K ,D^{(3,3,\ldots ,3)} \widehat{f} \in C({\Bbb R}^n) \big\},$ $\|f\|_{\mathcal{H}_{p,K,3}}  = \big\| D^{(3,3,\ldots,3)} \widehat{f}\,\big\|_\infty$ і $\widehat{f}$ є перетворенням Фур'є $f$.  Зауважимо, що $K$ має $(O)$-властивість, якщо існує стала $C>0$ така, що $$\sup\limits_{{\bf x} \in K} |{\bf x}^{\alpha + e_j} | \geq C\sup\limits_{{\bf x} \in  K } |{\bf x}^{\alpha} |$$ для всіх $\alpha \in \mathbb{Z}_+^n$ і $j=1,2, \ldots ,n$.  Institute of Mathematics, NAS of Ukraine 2026-02-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6386 10.37863/umzh.v74i11.6386 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 11 (2022); 1558 - 1570 Український математичний журнал; Том 74 № 11 (2022); 1558 - 1570 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6386/9334 Copyright (c) 2022 Ha Huy Bang, Vu Nhat Huy |
| spellingShingle | Bang, Ha Huy Huy, Vu Nhat Bang, Ha Huy Huy, Vu Nhat Bernstein inequality for multivariate functions with smooth Fourier images |
| title | Bernstein inequality for multivariate functions with smooth Fourier images |
| title_alt | Bernstein inequality for multivariate functions with smooth Fourier images |
| title_full | Bernstein inequality for multivariate functions with smooth Fourier images |
| title_fullStr | Bernstein inequality for multivariate functions with smooth Fourier images |
| title_full_unstemmed | Bernstein inequality for multivariate functions with smooth Fourier images |
| title_short | Bernstein inequality for multivariate functions with smooth Fourier images |
| title_sort | bernstein inequality for multivariate functions with smooth fourier images |
| topic_facet | $L^p$- spaces Bernstein inequality generalized functions 2010 AMS Subject Classification: 26D10 46E30 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6386 |
| work_keys_str_mv | AT banghahuy bernsteininequalityformultivariatefunctionswithsmoothfourierimages AT huyvunhat bernsteininequalityformultivariatefunctionswithsmoothfourierimages AT banghahuy bernsteininequalityformultivariatefunctionswithsmoothfourierimages AT huyvunhat bernsteininequalityformultivariatefunctionswithsmoothfourierimages |