Timeline for What does Mellin inversion "really mean"?
Current License: CC BY-SA 4.0
39 events
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| Oct 29, 2022 at 23:41 | comment | added | Tom Copeland | I disagree with the brief remarks, oft repeated, on the history of the Mellin transform in the Bertrand et al. paper. Before Riemann, Euler used the Mellin transform (at least for real arguments) in integral reps of the Euler gamma and beta functions, and Mellin gave much credit to Pincherle for inspiring his development of the transform pair. | |
| Oct 29, 2022 at 23:38 | comment | added | Tom Copeland | A good overview of the mechanics of using the Mellin transform pair--encoding info on a function $f(x)$ into singularities of its Mellin transform $F(s)$--while also covering the topics in this stream is "The Mellin Transform" by Jacqueline Bertrand, Pierre Bertrand, and Jean-Philippe Ovarlez (hal.archives-ouvertes.fr/hal-03152634/document). | |
| Oct 28, 2022 at 5:41 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| Oct 27, 2022 at 18:45 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Refined links
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| Feb 5, 2021 at 21:23 | comment | added | Tom Copeland | Other examples of use of RMF at mathoverflow.net/questions/379428/… | |
| May 6, 2020 at 2:27 | comment | added | Tom Copeland | See also the Mellin interpolation of the symmetric power sums of eigenvalues of a matrix in "On Functional Determinants of Laplacians in Polygons and Simplicial Complexes" by Erik Aurell and Per Salomonson. | |
| Nov 25, 2019 at 16:27 | comment | added | Tom Copeland | Recent application of this Mellin interpolation in "Zeta functions connecting multiple zeta values and poly-Bernoulli numbers" (Defs. 4.4 and 4.5, pg. 9) by Kanejko and Tsumura arxiv.org/abs/1811.07736 | |
| Jul 30, 2018 at 22:41 | comment | added | Tom Copeland | For more operator interpolations, see "Mellin Interpolation of Differential Ops and Associated Infinigens and Appell Polynomials: The Ordered, Laguerre, and Scherk-Witt-Lie Diff Ops" at tcjpn.wordpress.com/2015/09/16/… | |
| Jul 30, 2018 at 20:58 | comment | added | Tom Copeland | Continuing applications: "Simplicity in AdS Perturbative Dynamics" by Ellis Ye Yuan arxiv.org/abs/1801.07283 p. 104 | |
| Jun 26, 2017 at 16:12 | comment | added | Tom Copeland | Related: pg. 26 of "Multiple polylogarithms and mixed Tate motives" by Goncharov arxiv.org/abs/math/0103059. | |
| Sep 15, 2015 at 1:13 | comment | added | Tom Copeland | For $ y < 0$, the Heaviside step function in the last integral is to be interpreted as setting the upper limit of the integral as the first zero from the origin of $ (1-|y|tx^{-|y|})^{\alpha/|y|}$, i.e., the upper limit should be $ t = x^{|y|}/|y|$ . | |
| Sep 14, 2015 at 21:32 | history | edited | Tom Copeland | CC BY-SA 3.0 |
Related to Witt Lie algebra
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| Sep 13, 2015 at 19:58 | history | edited | Tom Copeland | CC BY-SA 3.0 |
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| Aug 7, 2015 at 20:46 | comment | added | Tom Copeland | See also tcjpn.wordpress.com/2015/08/05/… | |
| Sep 27, 2012 at 22:21 | comment | added | Tom Copeland | The Ramanujan divergent series relation to the Riemann zeta function can be generalized to more general Dirichlet series through $f(x)= \sum_{n=0}^{\infty } \varphi (n)e^{-nx} \rightarrow \sum_{j=0}^{\infty}\left ( \sum_{n=1}^{\infty }\varphi (n)n^j \right )\frac{\left ( -x \right )^j}{j!}\rightarrow \sum_{j=0}^{\infty} D_{\varphi}\left ( -j \right ) \frac{\left ( -x \right )^j}{j!}$ | |
| Sep 12, 2012 at 23:56 | comment | added | Tom Copeland | In a similar vein, look at $f(x)=H(1-x)(1-x)^\alpha=H(1-x)\sum_{n=0}^{\infty }\frac{\alpha!}{(\alpha-n)!}\frac{(-x)^n}{n!}$ with $H(x)$ the Heaviside step function, giving a fundamental result for fractional calculus, and for the iconic Ramanujan divergent series look at $f(x)=\frac{1}{e^x-1}=\sum_{n=1}^{\infty }e^{-{nx}}\rightarrow \sum_{j=0}^{\infty }\left ( \sum_{n=1}^{\infty }n^j \right )\frac{(-x)^j}{j!}\rightarrow \sum_{j=0}^{\infty }\zeta (-j)\frac{(-x)^j}{j!}$ . | |
| May 15, 2012 at 8:46 | history | edited | Tom Copeland | CC BY-SA 3.0 |
Reformatted, added important example of transform pair
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| Mar 22, 2012 at 4:06 | history | edited | Tom Copeland | CC BY-SA 3.0 |
Fixed index
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| Mar 22, 2012 at 3:49 | history | edited | Tom Copeland | CC BY-SA 3.0 |
Fixed link
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| Mar 22, 2012 at 3:30 | history | edited | Tom Copeland | CC BY-SA 3.0 |
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| Mar 22, 2012 at 3:14 | history | edited | Tom Copeland | CC BY-SA 3.0 |
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| Nov 3, 2011 at 13:38 | history | answered | Tom Copeland | CC BY-SA 3.0 |