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This is an advanced school for young researchers featuring five minicourses in modern analysis, such as wavelets, discrete and continuous duality with respect to Dirac and Schrödinger operators, singular integral operators, Padé approximations, orthogonal polynomials. The target audience includes graduate, master and senior bachelor students of any mathematical specialty.
Institutions participating in the organization of the event:
Please register to participate: link.
Wavelets (Maria Skopina)
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To the spectral theory of 1-D Schrödinger and Dirac operators with point interactions and quantum graphs (Mark Malamud)
The lectures will be devoted to the duality of certain spectral properties of the operators mentioned above and their discrete counterparts . Among others we will discuss self-adjointness, semiboundedness, discreteness and absolutely continuous properties, compactness and finiteness of negative parts of these operators, etc.
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Entropy function in the theory of orthogonal polynomials (Roman Bessonov)
This minicourse is dedicated to the recent results in the theory of orthogonal polynomials on the unit circle which employ the method of the entropy function of the respective probability measure. The main goal of the lectures is to present a detailed description of this method and discuss related open problems. The basics of the theory of orthogonal polynomials required for the course will be introduced along the way.
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Estimates of differential operators in L^1 and related questions (Dmitriy Stolyarov)
In 1938 S.L. Sobolev proved his famous embedding theorem: the Sobolev space W_p^1 embeds continuously into the L^q space, provided 1/p — 1/q = 1/d, d being the dimension of the underlying space, and p > 1. This result was extended to the case p=1 twenty years later by E. Gagliardo and L. Nirenberg. In early 2000’s J. Bourgain and H. Brezis observed that there exist similar estimates, which are relatively easy to obtain for p > 1, but are much more involved (if they are even valid) for p = 1. One of the key aspects in these matters is attributed to the vectorial nature of the differential operators in question (as, for instance, the fact that the gradient is a vector function, and not a scalar one). Nowadays, there is a certain change of perspective regarding these topics: such inequalities are believed to be interesting not only in a hermetic sense — as a challenge to one’s analytical prowess, — but also as a way to establish deep connections to the geometric measure theory. We will describe the (by now) classical Bourgain—Brezis theory, highlight the aforementioned geometric connections, and explain how simple tricks from harmonic analysis help in this business.
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Padé approximations, their generalizations and related problems (Aleksandr Komlov)
Padé approximants are the best rational approximations of a given power series. We show that Padé approximants are closely related to orthogonal polynomials, Chebyshev (functional) continued fractions, and Jacobi operators. This mini-course will cover the basics of the classical Stahl theory of convergence of Padé approximants of multivalued analytic functions. To do this, we will touch on the potential theory on the complex plane and Riemann sphere. We also consider such generalizations of Padé polynomials as Hermite-Padé polynomials of types I and II. For them, there is no general convergence theory analogous to the Stahl theory. We discuss the cases in which it is possible to describe their asymptotic behavior, how to use them for asymptotic recovery of the values of multivalued analytic functions, and formulate new problems arising here.
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Anton Baranov, Yurii Belov, Pavel Mozolyako, Saint Petersburg State University