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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.17018 |
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| _version_ | 1866916866113404928 |
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| author | Sun, Jiawei Zu, Chao Lu, Yufeng |
| author_facet | Sun, Jiawei Zu, Chao Lu, Yufeng |
| contents | In 1958, Helson and Lowdenslager extended the theory of analytic functions to a general class of groups with ordered duals. In this context, analytic functions on such a group $G$ are defined as the integrable functions whose Fourier coefficients lie in the positive semigroup of the dual of $G$. In this paper, we found some applications of their theory to infinite-dimensional complex analysis. Specifically, we considered a special order on $\mathbb{Z}^\infty$ and corresponding analytic continuous functions on $\mathbb{T}^ω$, which serves as the counterpart of the disk algebra in infinitely many variables setting. By characterizing its maximal ideals, we have generalized the following theorem to the infinite-dimensional case: For a positive function $w$ that is integrable and log-integrable on $\mathbb{T}^d$, there exists an outer function $g$ such that $w=|g|^2$ if and only if the support of $\hat{\log w}$ is a subset of $\mathbb{N}^d\cap (-\mathbb{N})^d$. Furthermore, we have found the counterpart of the above function algebra in the closed right half-plane, and the representing measures of each point in the right half-plane for this algebra. As an application of the order, we provided a new proof of the infinite-dimensional Szegö's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17018 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Notes on a special order on $\mathbb{Z}^\infty$ Sun, Jiawei Zu, Chao Lu, Yufeng Functional Analysis In 1958, Helson and Lowdenslager extended the theory of analytic functions to a general class of groups with ordered duals. In this context, analytic functions on such a group $G$ are defined as the integrable functions whose Fourier coefficients lie in the positive semigroup of the dual of $G$. In this paper, we found some applications of their theory to infinite-dimensional complex analysis. Specifically, we considered a special order on $\mathbb{Z}^\infty$ and corresponding analytic continuous functions on $\mathbb{T}^ω$, which serves as the counterpart of the disk algebra in infinitely many variables setting. By characterizing its maximal ideals, we have generalized the following theorem to the infinite-dimensional case: For a positive function $w$ that is integrable and log-integrable on $\mathbb{T}^d$, there exists an outer function $g$ such that $w=|g|^2$ if and only if the support of $\hat{\log w}$ is a subset of $\mathbb{N}^d\cap (-\mathbb{N})^d$. Furthermore, we have found the counterpart of the above function algebra in the closed right half-plane, and the representing measures of each point in the right half-plane for this algebra. As an application of the order, we provided a new proof of the infinite-dimensional Szegö's theorem. |
| title | Notes on a special order on $\mathbb{Z}^\infty$ |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2502.17018 |