Singular integral operators. The case of an unlimited contour
Let \(\Gamma\)be a closed or unclosed unlimited contour, a shift \(\alpha(t)\) maps homeomorphicly the contour \(\Gamma\) onto itself with preserving or reversing the direction on \(\Gamma\) and also satisfies the conditions: for some natural \(n\geq2\), \(\alpha_n(t)\equiv t\), and \(\alpha_j(t)\n...
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| Format: | Article |
| Language: | English |
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Publishing House of the Romanian Academy
2005-08-01
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| Series: | Journal of Numerical Analysis and Approximation Theory |
| Subjects: | |
| Online Access: | https://www.ictp.acad.ro/jnaat/journal/article/view/802 |
| _version_ | 1818197187737157632 |
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| author | V. Neaga |
| author_facet | V. Neaga |
| author_sort | V. Neaga |
| collection | DOAJ |
| description | Let \(\Gamma\)be a closed or unclosed unlimited contour, a shift \(\alpha(t)\) maps homeomorphicly the contour \(\Gamma\) onto itself with preserving or reversing the direction on \(\Gamma\) and also satisfies the conditions: for some natural \(n\geq2\), \(\alpha_n(t)\equiv t\), and \(\alpha_j(t)\not\equiv t\) for \(1\leq j<n\). In this work we study subalgebra \(\Sigma\) of algebra\(L(L_p(\Gamma,\rho))\), which contains all operators of the form\[\left (M \varphi \right) (t) = \sum_{k=0}^{n-1} \bigg \{a_k (t) \varphi (\alpha_k (t)) + \tfrac{b_k(t)}{\pi {\rm i} } \int_{\Gamma} \tfrac{\varphi ( \tau )}{\tau - \alpha_k (t)} d \tau \bigg \}\]with piecewise-continuous coefficients. The existence of such an isomorphism between \(\Sigma\) and some algebra \(\frak A\) of singular operators with Cauchy kernel that an arbitrary operator from \(\Sigma\) and its image are Noetherian or not Noetherian simultaneously is proved. It allows to introduce the concept of a symbol for all operators from \( \Sigma \) and, using the known results for algebra \( \frak A \), in terms of a symbol to receive conditions of Noetherian property. |
| first_indexed | 2024-12-12T01:46:00Z |
| format | Article |
| id | doaj.art-b18ef2b234e04042bb1b1d0dafdcac6c |
| institution | Directory Open Access Journal |
| issn | 2457-6794 2501-059X |
| language | English |
| last_indexed | 2024-12-12T01:46:00Z |
| publishDate | 2005-08-01 |
| publisher | Publishing House of the Romanian Academy |
| record_format | Article |
| series | Journal of Numerical Analysis and Approximation Theory |
| spelling | doaj.art-b18ef2b234e04042bb1b1d0dafdcac6c2022-12-22T00:42:35ZengPublishing House of the Romanian AcademyJournal of Numerical Analysis and Approximation Theory2457-67942501-059X2005-08-01342Singular integral operators. The case of an unlimited contourV. Neaga0State University of Moldova, Chisinau, Moldova, Republic ofLet \(\Gamma\)be a closed or unclosed unlimited contour, a shift \(\alpha(t)\) maps homeomorphicly the contour \(\Gamma\) onto itself with preserving or reversing the direction on \(\Gamma\) and also satisfies the conditions: for some natural \(n\geq2\), \(\alpha_n(t)\equiv t\), and \(\alpha_j(t)\not\equiv t\) for \(1\leq j<n\). In this work we study subalgebra \(\Sigma\) of algebra\(L(L_p(\Gamma,\rho))\), which contains all operators of the form\[\left (M \varphi \right) (t) = \sum_{k=0}^{n-1} \bigg \{a_k (t) \varphi (\alpha_k (t)) + \tfrac{b_k(t)}{\pi {\rm i} } \int_{\Gamma} \tfrac{\varphi ( \tau )}{\tau - \alpha_k (t)} d \tau \bigg \}\]with piecewise-continuous coefficients. The existence of such an isomorphism between \(\Sigma\) and some algebra \(\frak A\) of singular operators with Cauchy kernel that an arbitrary operator from \(\Sigma\) and its image are Noetherian or not Noetherian simultaneously is proved. It allows to introduce the concept of a symbol for all operators from \( \Sigma \) and, using the known results for algebra \( \frak A \), in terms of a symbol to receive conditions of Noetherian property.https://www.ictp.acad.ro/jnaat/journal/article/view/802Lyapunov closed curvesNoetherian singular integralpiecewise Lyapunov contoursingular integral equationssingular operators with Cauchy kernel singular operators with shift |
| spellingShingle | V. Neaga Singular integral operators. The case of an unlimited contour Journal of Numerical Analysis and Approximation Theory Lyapunov closed curves Noetherian singular integral piecewise Lyapunov contour singular integral equations singular operators with Cauchy kernel singular operators with shift |
| title | Singular integral operators. The case of an unlimited contour |
| title_full | Singular integral operators. The case of an unlimited contour |
| title_fullStr | Singular integral operators. The case of an unlimited contour |
| title_full_unstemmed | Singular integral operators. The case of an unlimited contour |
| title_short | Singular integral operators. The case of an unlimited contour |
| title_sort | singular integral operators the case of an unlimited contour |
| topic | Lyapunov closed curves Noetherian singular integral piecewise Lyapunov contour singular integral equations singular operators with Cauchy kernel singular operators with shift |
| url | https://www.ictp.acad.ro/jnaat/journal/article/view/802 |
| work_keys_str_mv | AT vneaga singularintegraloperatorsthecaseofanunlimitedcontour |