These notes are based on a Conference Board of the Mathematical Sciences
mini-course given in August of 2013 at Clemson, South Carolina. I am deeply
grateful to Constanze Liaw, Mishko Mitkovski and Brett Wick for inviting me to be
a CBMS speaker and for all the hard work they put into organizing the conference.
More than 40 researchers attended the conference, several of whom contributed
outstanding lectures. The lectures were given by Michael Lacey, Doron Lubinsky,
Shahaf Nitzan, Bill Ross, Carl Sundberg and Sergei Treil.
On behalf of the organizers and participants of the conference, I wish to thank
the National Science Foundation and Conference Board of the Mathematical Sci-
ences for their generous support.
Much credit is due to Nikolai Makarov, whose ideas initiated the Toeplitz ap-
proach in the area of the Uncertainty Principle. The last two chapters of the
notes are devoted to our joint work, where these methods were developed. I also
would like to thank Misha Sodin for convincing me to apply this approach to the
Gap and Type problems and for numerous invaluable discussions. My interest in
the P´ olya-Levinson problem and Bernstein’s weighted uniform approximation also
stems from communications with Nikolai and Misha. I am grateful to Alexander
Eremenko for bringing the problem on oscillations of Fourier integrals to my atten-
tion. Thought-provoking collaboration with my former student Mishko Mitkovski
produced chapters 3 and 4.
Original results discussed in these notes were obtained with support of the NSF
through grants DMS-1101278, DMS-0800300 and DMS-0500852.
I am especially indebted to my wife Svetlana for helping me find the classical
texts mentioned in the notes and for constant moral support.