Lederman, Roy R; Steinerberger, Stefan Lower Bounds for Truncated Fourier and Laplace Transforms Journal Article Integral Equations and Operator Theory, 87 (4), pp. 529–543, 2017, ISSN: 0378-620X, 1420-8989. Links | BibTeX | Tags: Fourier Transform, Laplace Transform @article{lederman_lower_2017, title = {Lower Bounds for Truncated Fourier and Laplace Transforms}, author = {Roy R Lederman and Stefan Steinerberger}, url = {http://link.springer.com/10.1007/s00020-017-2364-z}, doi = {10.1007/s00020-017-2364-z}, issn = {0378-620X, 1420-8989}, year = {2017}, date = {2017-04-01}, urldate = {2020-08-13}, journal = {Integral Equations and Operator Theory}, volume = {87}, number = {4}, pages = {529--543}, keywords = {Fourier Transform, Laplace Transform}, pubstate = {published}, tppubtype = {article} } |
Lederman, Roy R; Steinerberger, Stefan Stability Estimates for Truncated Fourier and Laplace Transforms Technical Report (arXiv:1605.03866), 2016. Abstract | Links | BibTeX | Tags: Laplace Transform @techreport{lederman_stability_2016, title = {Stability Estimates for Truncated Fourier and Laplace Transforms}, author = {Roy R Lederman and Stefan Steinerberger}, url = {https://arxiv.org/abs/1605.03866v1}, year = {2016}, date = {2016-05-01}, urldate = {2020-08-13}, number = {arXiv:1605.03866}, abstract = {We prove sharp stability estimates for the Truncated Laplace Transform and Truncated Fourier Transform. The argument combines an approach recently introduced by Alaifari, Pierce and the second author for the truncated Hilbert transform with classical results of Bertero, Grünbaum, Landau, Pollak and Slepian. In particular, we prove there is a universal constant $c textgreater0$ such that for all $f textbackslashin Ltextasciicircum2(textbackslashmathbbR)$ with compact support in $[-1,1]$ normalized to $textbackslashtextbarftextbackslashtextbar_Ltextasciicircum2[-1,1] = 1$ $$ textbackslashint_-1textasciicircum1textbartextbackslashwidehatf(ξ)textbartextasciicircum2dξ textbackslashgtrsim textbackslashleft(ctextbackslashlefttextbackslashtextbarf_x textbackslashrighttextbackslashtextbar_Ltextasciicircum2[-1,1] textbackslashright)textasciicircumvphantom- ctextbackslashlefttextbackslashtextbarf_x textbackslashrighttextbackslashtextbar_Ltextasciicircum2[-1,1]vphantom$$ The inequality is sharp in the sense that there is an infinite sequence of orthonormal counterexamples if $c$ is chosen too small. The question whether and to which extent similar inequalities hold for generic families of integral operators remains open.}, keywords = {Laplace Transform}, pubstate = {published}, tppubtype = {techreport} } We prove sharp stability estimates for the Truncated Laplace Transform and Truncated Fourier Transform. The argument combines an approach recently introduced by Alaifari, Pierce and the second author for the truncated Hilbert transform with classical results of Bertero, Grünbaum, Landau, Pollak and Slepian. In particular, we prove there is a universal constant $c textgreater0$ such that for all $f textbackslashin Ltextasciicircum2(textbackslashmathbbR)$ with compact support in $[-1,1]$ normalized to $textbackslashtextbarftextbackslashtextbar_Ltextasciicircum2[-1,1] = 1$ $$ textbackslashint_-1textasciicircum1textbartextbackslashwidehatf(ξ)textbartextasciicircum2dξ textbackslashgtrsim textbackslashleft(ctextbackslashlefttextbackslashtextbarf_x textbackslashrighttextbackslashtextbar_Ltextasciicircum2[-1,1] textbackslashright)textasciicircumvphantom- ctextbackslashlefttextbackslashtextbarf_x textbackslashrighttextbackslashtextbar_Ltextasciicircum2[-1,1]vphantom$$ The inequality is sharp in the sense that there is an infinite sequence of orthonormal counterexamples if $c$ is chosen too small. The question whether and to which extent similar inequalities hold for generic families of integral operators remains open. |
Lederman, Roy R; Rokhlin, V On the Analytical and Numerical Properties of the Truncated Laplace Transform. Part II Journal Article SIAM Journal on Numerical Analysis, 54 (2), pp. 665–687, 2016, ISSN: 0036-1429, 1095-7170. Links | BibTeX | Tags: Algorithms, Laplace Transform, Numerical Analysis @article{lederman_analytical_2016, title = {On the Analytical and Numerical Properties of the Truncated Laplace Transform. Part II}, author = {Roy R Lederman and V Rokhlin}, url = {http://epubs.siam.org/doi/10.1137/15M1028583}, doi = {10.1137/15M1028583}, issn = {0036-1429, 1095-7170}, year = {2016}, date = {2016-01-01}, urldate = {2020-08-13}, journal = {SIAM Journal on Numerical Analysis}, volume = {54}, number = {2}, pages = {665--687}, keywords = {Algorithms, Laplace Transform, Numerical Analysis}, pubstate = {published}, tppubtype = {article} } |
Lederman, Roy R; Rokhlin, V On the Analytical and Numerical Properties of the Truncated Laplace Transform I. Journal Article SIAM Journal on Numerical Analysis, 53 (3), pp. 1214–1235, 2015, ISSN: 0036-1429, 1095-7170. Links | BibTeX | Tags: Algorithms, Laplace Transform, Numerical Analysis @article{lederman_analytical_2015, title = {On the Analytical and Numerical Properties of the Truncated Laplace Transform I.}, author = {Roy R Lederman and V Rokhlin}, url = {http://epubs.siam.org/doi/10.1137/140990681}, doi = {10.1137/140990681}, issn = {0036-1429, 1095-7170}, year = {2015}, date = {2015-01-01}, urldate = {2020-08-13}, journal = {SIAM Journal on Numerical Analysis}, volume = {53}, number = {3}, pages = {1214--1235}, keywords = {Algorithms, Laplace Transform, Numerical Analysis}, pubstate = {published}, tppubtype = {article} } |
Lederman, Roy R On the Analytical and Numerical Properties of the Truncated Laplace Transform Technical Report Yale CS (YALEU/DCS/TR-1490), 2014. BibTeX | Tags: Algorithms, Laplace Transform, Numerical Analysis @techreport{lederman_analytical_2014, title = {On the Analytical and Numerical Properties of the Truncated Laplace Transform}, author = {Roy R Lederman}, year = {2014}, date = {2014-01-01}, number = {YALEU/DCS/TR-1490}, pages = {82}, institution = {Yale CS}, keywords = {Algorithms, Laplace Transform, Numerical Analysis}, pubstate = {published}, tppubtype = {techreport} } |