Symbolic Computation Applied to the Study of the Kernel of a Singular Integral Operator with Non-Carleman Shift and Conjugation
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- 作者：Ana C. Conceição ; Rui C. Marreiros ; José C. Pereira
- 关键词：Singular integral operators ; Non ; Carleman shift ; Conjugation ; Kernel dimension ; Factorization algorithms ; Symbolic computation ; Wolfram Mathematica
- 刊名：Mathematics in Computer Science
- 出版年：2016
- 出版时间：September 2016
- 年：2016
- 卷：10
- 期：3
- 页码：365-386
- 全文大小：1,508 KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Mathematics
Computer Science, general - 出版者：Springer Basel
- ISSN：1661-8289
- 卷排序：10
文摘
On the Hilbert space \(\widetilde{L}_{2}(\mathbb {T})\) the singular integral operator with non-Carleman shift and conjugation \(K=P_{+}+(aI+AC)P_{-}\) is considered, where \(P_{\pm }\) are the Cauchy projectors, \(A=\sum \nolimits _{j=0}^{m}a_{j}U^{j}\), \(a,a_{j}\), \(j=\overline{1,m}\), are continuous functions on the unit circle \(\mathbb {T}\), U is the shift operator and C is the operator of complex conjugation. We show how the symbolic computation capabilities of the computer algebra system Mathematica can be used to explore the dimension of the kernel of the operator K. The analytical algorithm [ADimKer-NonCarleman] is presented; several nontrivial examples are given.