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Some Estimates of Fourier Transforms


Kovrijkine, Oleg E. (2000) Some Estimates of Fourier Transforms. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/0p2k-ah86.


This work consists of two independent parts. In the first part we prove several results related to the theorem of Logvinenko and Sereda on determining sets for functions with Fourier transforms supported in a parallelepiped. We obtain a polynomial instead of exponential bound in this theorem, and we extend it to the case of functions with Fourier transforms supported in the union of a bounded number of parallelepipeds. When dimension d = 1 we also consider the case of infinitely many lacunary intervals. We generalize the Zygmund theorem for lacunary series whose Fourier coefficients are replaced with polynomials of uniformly bounded degree. We give also a necessary condition for the support of Fourier transforms for which the Logvinenko-Sereda theorem still holds.

In the second part we prove that the L²([0,1]d x SO(d)) norm of periodizations of a function from L¹(ℝd) is equivalent to the L²(ℝd) norm of the function itself in higher dimensions. We generalize the statement for functions from Lp(ℝd) where 1 ≤ p < (2d)/(d + 2) spirit of the Stein-Tomas theorem. We also show that the following theorem due to M. Kolountzakis and T. Wolff does not hold if dimension d = 2: if periodizations of a function from L¹(ℝd) are constants, then the function is continuous and bounded provided that the dimension d is at least three.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Wolff, Thomas H.
Thesis Committee:
  • Wolff, Thomas H. (chair)
  • Keel, Markus
  • Luxemburg, W. A. J.
  • Makarov, Nikolai G.
Defense Date:12 May 2000
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Funding AgencyGrant Number
Record Number:CaltechTHESIS:11212019-172159302
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:13594
Deposited By: Mel Ray
Deposited On:22 Nov 2019 01:41
Last Modified:16 Apr 2021 16:42

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