On the convergence of Newton series and the asymptotics of finite differences
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913487643475968 |
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| author | Bruda, Glenn |
| author_facet | Bruda, Glenn |
| contents | Suppose a complex function $f$ has a Lebesgue measurable inverse Laplace transform. We show that the $n$th order forward and backward differences of $f$ at $z_0\in\mathbb{C}$ tend to zero as $n\to\infty$ whenever $z_0$ lies in the region of absolute convergence of $f$. Under the same hypothesis, we show that the Newton series of $f$ centered at $z_0$ exists and converges in the half-plane $\Re(z)>\Re(z_0)$. Assuming instead that $f$ has a Lebesgue measurable inverse Fourier transform, we show that the $n$th order forward, backward, and central differences of $f$ at any $y\in\mathbb{R}$ are $o(2^n)$. Consequently, we show that the binomial sum $\sum_{k\geq0}{n\choose k}f(k)$ is $o(2^n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04288 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the convergence of Newton series and the asymptotics of finite differences Bruda, Glenn Complex Variables Classical Analysis and ODEs 40A30 Suppose a complex function $f$ has a Lebesgue measurable inverse Laplace transform. We show that the $n$th order forward and backward differences of $f$ at $z_0\in\mathbb{C}$ tend to zero as $n\to\infty$ whenever $z_0$ lies in the region of absolute convergence of $f$. Under the same hypothesis, we show that the Newton series of $f$ centered at $z_0$ exists and converges in the half-plane $\Re(z)>\Re(z_0)$. Assuming instead that $f$ has a Lebesgue measurable inverse Fourier transform, we show that the $n$th order forward, backward, and central differences of $f$ at any $y\in\mathbb{R}$ are $o(2^n)$. Consequently, we show that the binomial sum $\sum_{k\geq0}{n\choose k}f(k)$ is $o(2^n)$. |
| title | On the convergence of Newton series and the asymptotics of finite differences |
| topic | Complex Variables Classical Analysis and ODEs 40A30 |
| url | https://arxiv.org/abs/2404.04288 |