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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2602.10373 |
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| _version_ | 1866917266678874112 |
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| author | Heinävaara, Otte |
| author_facet | Heinävaara, Otte |
| contents | We give a precise functional comparison between classical and free convolutions. If $μ$ and $ν$ are compactly supported probability measures, we show that the expectation of $f$ over the classical convolution $μ* ν$ is at least the expectation of $f$ over the free convolution $μ\boxplus ν$, as long as the fourth derivative of $f$ is non-negative. Conversely, the non-negativity of the fourth derivative is necessary for such a comparison.
This comparison is based on the positivity of a related measure on $\mathbb{R}^{2}$, which we dub the convolution comparison measure. We give an expression for this measure using a curious identity involving Hermitian matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10373 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convolution comparison measures Heinävaara, Otte Functional Analysis Operator Algebras Probability 46L54 We give a precise functional comparison between classical and free convolutions. If $μ$ and $ν$ are compactly supported probability measures, we show that the expectation of $f$ over the classical convolution $μ* ν$ is at least the expectation of $f$ over the free convolution $μ\boxplus ν$, as long as the fourth derivative of $f$ is non-negative. Conversely, the non-negativity of the fourth derivative is necessary for such a comparison. This comparison is based on the positivity of a related measure on $\mathbb{R}^{2}$, which we dub the convolution comparison measure. We give an expression for this measure using a curious identity involving Hermitian matrices. |
| title | Convolution comparison measures |
| topic | Functional Analysis Operator Algebras Probability 46L54 |
| url | https://arxiv.org/abs/2602.10373 |