Pointwise convergence of Fourier series – Carleson’s theorem

Postgraduate Thesis uoadl:1319731 424 Read counter

Unit:
Κατεύθυνση Θεωρητικά Μαθηματικά
Library of the School of Science
Deposit date:
2016-09-21
Year:
2016
Author:
Σκαρμόγιαννης Νικόλαος
Supervisors info:
Απόστολος Γιαννόπουλος Καθηγητής
Original Title:
Σημειαική σύγκλιση της σειράς Fourier-Το θεώρημα του Carleson
Languages:
Greek
Translated title:
Pointwise convergence of Fourier series – Carleson’s theorem
Summary:
The aim of this thesis is to provide a complete a self-contained proof of the
Carleson-Hunt theorem: if f is a p-integrable 2π-periodic function, where p>1,
then the Fourier series S[f](x) converges to f(x) almost everywhere.
In the first chapter we present some interpolation theorems for sublinear
operators that will be needed later on: among them are the theorem of
Marcinkiewicz and the interpolation theorem of Stein-Weiss. Then we formulate a
basic lemma-theorem and show that accepting this we can easily deduce the
Carleson-Hunt theorem.
In the second chapter we introduce two linear operators that help us to prove
that the Hilbert transform and the modified Hilbert transform are of type p for
every p >1, as well as some exponential estimates for these two
transforms.καθώς
In the third chapter we introduce the notion of dyadic intervals that are
needed in the sequel, and some modified Hilbert transforms which correspond to
them. Then, we introduce the modified Fourier coefficients and reduce the
question to the problem to show that a suitable operator M* is of type p on the
indicator functions of measurable sets.
In the last more technical chapter we define some exceptional sets whose
measure is of "type p" and using them we show that M* has the desired property.
This completes the proof.
Keywords:
Series, Fourier, Pointwise, Convegence, p-integrable
Index:
No
Number of index pages:
0
Contains images:
No
Number of references:
11
Number of pages:
141
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