Harmonic Analysishttp://hdl.handle.net/20.500.11824/32024-02-24T13:31:48Z2024-02-24T13:31:48ZWalter Rudin meets Elias M. SteinBakas, O.Ciccone, V.Wright, J.http://hdl.handle.net/20.500.11824/17342024-01-15T23:19:54Z2023-01-01T00:00:00ZWalter Rudin meets Elias M. Stein
Bakas, O.; Ciccone, V.; Wright, J.
Walter Rudin and Elias M. Stein were giants in the world of mathemat-
ics. They were loved and admired from students and researchers to teachers
and academics, both young and old. They touched many of us through their
inspiring books at the undergraduate and postgraduate level. Although they
were leading researchers in both harmonic analysis and several complex vari-
ables, we are not aware whether they interacted and discussed mathematics. In
this article, Rudin and Stein meet mathematically through a reformulation of
the beautiful theory of Fourier series with gaps that Rudin developed in the
1950s as an equivalent Fourier restriction problem from the 1970s, a problem
Stein proposed and which remains a fundamental, central problem in Euclidean
harmonic analysis today.
Walter Rudin was born in Vienna on 2 May, 1921 and emigrated to the US
in 1945, completing his PhD at Duke University in 1949. While a C. L. E.
Moore Instructor at MIT in the early 1950s, Walter was asked to teach a real
analysis course but he could not find a textbook that he liked so he decided to
write Principles of Mathematical Analysis which despite its age, has remained
the paragon of high quality. After a stint of teaching at the University of
Rochester, he took up a position at the University of Wisconsin, Madison in
1959 where he remained until his retirement as Vilas Professor in 1991. He died
at his home in Madison on 20 May, 2010.
Elias M. Stein (known to friends and colleagues as Eli) was born in Antwerp
on 13 January, 1931 and emigrated with his family to the US in 1941, settling in
New York where Eli attended high school. He went to the University of Chicago,
received his PhD in 1955, and then went to MIT as a C.L.E. Moore Instructor
before Antoni Zygmund told Eli “it’s time to return to Chicago.” In 1963, Stein
moved to Princeton University as a full professor where he remained until he
died on 23 December, 2018.
Between 2003 and 2011, Eli expanded the presentation of Walter’s Principles
and published a series of four books aimed at advanced undergraduates. This
series is quickly becoming an important part of any young analyst’s education.However the majority of books written by Rudin and Stein are postgraduate
textbooks and research monographs (too many to list here), mainly in the areas
of harmonic analysis and several complex variables where both men were central
figures.
In this article, these two luminaries meet in the world of mathematical anal-
ysis. We look back at some important work Rudin did in the 1950s and recast
it in terms of a far-reaching problem from the 1970s that Stein gave us.
2023-01-01T00:00:00ZOn the doubling condition in the infinite-dimensional settingKosz, D.http://hdl.handle.net/20.500.11824/17302024-01-09T23:20:34Z2023-12-01T00:00:00ZOn the doubling condition in the infinite-dimensional setting
Kosz, D.
We present a systematic approach to the problem whether a topologically infinite-dimensional space can be made homogeneous in the Coifman–Weiss sense. The answer to the question is negative, as expected. Our leading representative of spaces with this property is $\mathbb T^{\omega} = \mathbb T \times \mathbb T \times \cdots$ with the natural product topology.
2023-12-01T00:00:00ZSharp estimates for Jacobi heat kernels in conic domainsHanrahan, D.Kosz, D.http://hdl.handle.net/20.500.11824/17292024-01-09T23:20:32Z2023-01-01T00:00:00ZSharp estimates for Jacobi heat kernels in conic domains
Hanrahan, D.; Kosz, D.
We prove genuinely sharp estimates for the Jacobi heat kernels introduced in the context of the multidimensional cone $\mathbb V^{d+1}$and its surface $\mathbb V^{d+1}_0$. To do so, we combine the theory of Jacobi polynomials on the cone explored by Xu with the recent techniques by Nowak, Sjögren, and Szarek, developed to find genuinely sharp estimates for the spherical heat kernel.
2023-01-01T00:00:00ZSharp constants in inequalities admitting the Calderón transference principleKosz, D.http://hdl.handle.net/20.500.11824/17282024-01-09T23:20:36Z2023-01-01T00:00:00ZSharp constants in inequalities admitting the Calderón transference principle
Kosz, D.
The aim of this note is twofold. First, we prove an abstract version of the Calderón transference principle for inequalities of admissible type in the general commutative multilinear and multiparameter setting. Such an operation does not increase the constants in the transferred inequalities. Second, we use the last information to study a certain dichotomy arising in problems of finding the best constants in the weak type $(1,1)$ and strong type $(p,p)$ inequalities for one-parameter ergodic maximal operators.
2023-01-01T00:00:00Z