On Functions Whose Mean Value Abscissas Are Midpoints, with Connections to Harmonic Functions
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arXiv
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| Format: | Preprint |
| Published: |
2016
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| _version_ | 1866915408432332800 |
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| author | Carter, Paul Lowry-Duda, David |
| author_facet | Carter, Paul Lowry-Duda, David |
| contents | We investigate functions with the property that for every interval, the slope at the midpoint of the interval is the same as the average slope. More generally, we find functions whose average slopes over intervals are given by the slope at a weighted average of the endpoints of those intervals. This is equivalent to finding functions satisfying a weighted mean value property. In the course of our exploration, we find connections to harmonic functions that prompt us to explore multivariable analogues and the existence of "weighted harmonic functions." |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1608_02558 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | On Functions Whose Mean Value Abscissas Are Midpoints, with Connections to Harmonic Functions Carter, Paul Lowry-Duda, David Classical Analysis and ODEs 26A06 We investigate functions with the property that for every interval, the slope at the midpoint of the interval is the same as the average slope. More generally, we find functions whose average slopes over intervals are given by the slope at a weighted average of the endpoints of those intervals. This is equivalent to finding functions satisfying a weighted mean value property. In the course of our exploration, we find connections to harmonic functions that prompt us to explore multivariable analogues and the existence of "weighted harmonic functions." |
| title | On Functions Whose Mean Value Abscissas Are Midpoints, with Connections to Harmonic Functions |
| topic | Classical Analysis and ODEs 26A06 |
| url | https://arxiv.org/abs/1608.02558 |