Timeline for The Cauchy Transform, and the convergence of the Fourier-Stieltjes transforms of a sequence of measures
Current License: CC BY-SA 4.0
16 events
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| S Jan 20, 2021 at 0:07 | history | bounty ended | CommunityBot | ||
| S Jan 20, 2021 at 0:07 | history | notice removed | CommunityBot | ||
| Jan 16, 2021 at 21:47 | history | edited | MCS | CC BY-SA 4.0 |
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| Jan 16, 2021 at 0:57 | comment | added | Mateusz Kwaśnicki | Finally, you may like to add further tags to attract more attention. In particular, why not add a top-level tag (perhaps cv.complex-variables)? | |
| Jan 16, 2021 at 0:56 | comment | added | Mateusz Kwaśnicki | Also, convergence of Fourier coefficients implies weak convergence of measures, I suppose? Or do I misunderstand the question completely? | |
| Jan 16, 2021 at 0:51 | comment | added | Mateusz Kwaśnicki | I have seen quite a few related results for generalised Stieltjes transforms, which seems to be a similar concept on a half-line $(0,\infty)$ rather than on a circle. If I am not mistaken, in this case locally uniform convergence of transforms implies vague convergence of representing measures (I suppose, by virtue of appropriate inversion formulae). I expect the same is true for the circle, but I must admit I have no references. | |
| Jan 15, 2021 at 21:00 | history | edited | MCS | CC BY-SA 4.0 |
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| Jan 13, 2021 at 16:18 | comment | added | Tom Copeland | A multi-dimensional neighborhood--an extension/interpolation of the Hilbert transform, the Stieltjes-Cauchy transform in free probability theory, even perhaps the Todd operator--so a number of possible approaches. Unfortunately, I have little time to spend on it, but I'm certainly interested in any results. | |
| Jan 13, 2021 at 0:33 | comment | added | MCS | Mmm... when you put it that way, I'd say that it is related—in that the formulas involved are in an open neighborhood about those topics. The sequences of measures in question are, in fact, the (partial) time averages of a dirac delta under the adjoint of a linear operator on functions on the disk, one which admits a representation as a contour integral against a kernel—a rational function of two complex variables. I could write a small paper on the questions I currently have. The present question is my effort to reduce it to its essentials, to increase my chance of getting a response. | |
| Jan 12, 2021 at 22:19 | comment | added | Tom Copeland | Then not related to the Euler integral for the beta function, various methods of analytic continuation for it, Mellin transform extension, Pochhammer contour integral rep., convolution rep. | |
| Jan 12, 2021 at 21:33 | history | edited | MCS | CC BY-SA 4.0 |
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| Jan 12, 2021 at 21:31 | comment | added | MCS | No. This is a question of the representability of holomorphic functions in terms of their boundary behavior, and the stability of this representability with respect to sequences convergent in various topologies. | |
| Jan 12, 2021 at 0:07 | comment | added | Tom Copeland | Is this not related to fractional calculus? | |
| S Jan 11, 2021 at 22:43 | history | bounty started | MCS | ||
| S Jan 11, 2021 at 22:43 | history | notice added | MCS | Draw attention | |
| Jan 9, 2021 at 21:42 | history | asked | MCS | CC BY-SA 4.0 |