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The paper proposes proof of the possibility of an integral representation of a positive definite kernel of two pairs of variables. Using this kernel, we use the technique of constructing a new Hilbert space in which symmetric differential operators formally commute. In this case, the kernel satisfie...
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| Date: | 2023 |
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| Main Author: | |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2023
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| Subjects: | |
| Online Access: | http://journal.iasa.kpi.ua/article/view/279777 |
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| Journal Title: | System research and information technologies |
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System research and information technologies| Summary: | The paper proposes proof of the possibility of an integral representation of a positive definite kernel of two pairs of variables. Using this kernel, we use the technique of constructing a new Hilbert space in which symmetric differential operators formally commute. In this case, the kernel satisfies a system of differential equations with partial derivatives. It is known that a kernel given in a subdomain of the real plane, generally speaking, does not always imply an extension to the entire plane. This possibility is related to the problem of the existence of a commuting self-adjoint extension of symmetric operators. The author applies his own results related to a commuting self-adjoint extension in a wider Hilbert space. The resulting representation in the form of an integral of elementary positive-definite kernels with respect to the spectral measure generated by the resolution of the identity of the operators allows us to extend the positive-definite kernel to the entire plane. |
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