Year 2023/2024

No 156  October 4th, 2023   Organizational meeting

No 157  October 11th, 2023   Jerzy K±kol (UAM, Poznań)
About Banach-Mazur's problem from the 1930s
Abstract. One of the famous unsolved problems in Functional Analysis asks (Banach-Mazur's problem (1932)) if every infinite-dimensional Banach space can be mapped by a continuous linear operator onto an infinite-dimensional separable Banach space. This problem is known under the name Separable quotient problem. For many concrete Banach spaces the answer is positive, for example, reflexive Banach spaces, or even weakly compactly generated Banach spaces. Argyros, Dodos and Kanellopoulos, proved that every dual Banach space has a separable quotient. On the other hand, Rosenthal showed that all Banach spaces C(X) of continuous (real-valued) functions on X have a quotient isomorphic to c_0 or l_2. We provide several useful methods to examine which Banach spaces admit a separable quotient, and the same problem will be studied for spaces C_p(X) with the pointwise topology. The talk gathers also quite new results. A connection with Efimov compact spaces X will be also discussed.
No 158  October 25th, 2023   Marta Kosek
Non-autonomous Julia sets for sequences of polynomials used in the approximation of the complex Green function
Abstract.We consider sequences of polynomials satisfying the conditions of the well known Kalmar-Walsh theorem. We investigate the non-autonomous filled Julia sets of such sequences. Our toy example is given by the Chebyshev polynomials on the interval.
No 159  November 8th, 2023   Przemysław Sprus
Some remarks on filled-in Julia sets
Abstract: We will study some disks contained in the filled-in Julia sets associated with complex polynomials of the form f(z)=z^n+c for some values c \in [-2, \frac{1}{4}] and n \geq 2. References: D. Dumitru, R. Ioan, Some remarks on filled-in Julia sets, An. Univ. Oradea Fasc. Mat.
No 160  November 15th, 2023   Tomasz Beberok
On L_p Bernstein-type inequalities
Abstract: In this talk we verify sharp Bernstein-type inequalities for multivariate algebraic polynomials in L_p norm on the simplex and certain two-dimensional cuspidal domains. The talk will be based partly on the article "Sharp Bernstein inequalities on simplex" by Yan Ge and Yuan Xu.
No 161  November 29th, 2023   Michał Kudra
Solution to the problem of Hadamard powers of stable polynomials using Hermite-Biehler Theorem
No 162  December 6th, 2023   Grzegorz Lewicki
On two-strongly unique best approximation in the complex case
Abstract. During my talk the following theorem will be proved. Let $X$ be a Banach space over $\mathbb{C}$ and $Y \subset X$ be a finite-dimensional subspace. Let $x \in X \setminus Y.$ Assume that $ P_Y(x) = \{0\}$ where $$P_Y(x) = \{ y \in Y: \|x-y\|= dist(x,Y)\}.$$ Let $$E(x) = \{ f \in ext(S(X^*)): f(x) = \|x\| \}$$be a finite set consisting of $n$ elements. Assume that$$M= sup\{|f(x)|: f \in Ext(S(X^*))\setminus |E|(x) \} < \|x\|,$$ if $Ext(S(X^*))\setminus |E|(x) \neq \emptyset,$ where $$|E|(x) = \{ f \in ext(S(X^*)): |f(x)| = \|x\| \}$$ Then there exists $ r >0$ such that for any $ y \in Y,$ $$ \|x-y\|^2 \geq \|x\|^2 + r \|y\|^2.$
No 163  December 13th, 2023   Dimitri Jordan Kenne
Numerical computation of pseudo-Leja points
No 164  December 20th, 2023   Grzegorz Lewicki
On two-strongly unique best approximation in the complex case (continued)
Abstract. During my talk the following theorem will be proved. Let $X$ be a Banach space over $\mathbb{C}$ and $Y \subset X$ be a finite-dimensional subspace. Let $x \in X \setminus Y.$ Assume that $ P_Y(x) = \{0\}$ where $$P_Y(x) = \{ y \in Y: \|x-y\|= dist(x,Y)\}.$$ Let $$E(x) = \{ f \in ext(S(X^*)): f(x) = \|x\| \}$$be a finite set consisting of $n$ elements. Assume that$$M= sup\{|f(x)|: f \in Ext(S(X^*))\setminus |E|(x) \} < \|x\|,$$ if $Ext(S(X^*))\setminus |E|(x) \neq \emptyset,$ where $$|E|(x) = \{ f \in ext(S(X^*)): |f(x)| = \|x\| \}$$ Then there exists $ r >0$ such that for any $ y \in Y,$ $$ \|x-y\|^2 \geq \|x\|^2 + r \|y\|^2.$
No 165  January 10th, 2024   Tomasz Kobos
Spaces with maximal projection constants revisited
Abstract. The absolute projection constant is an important numerical invariant of the normed space, related to some other well-known notions of the theory of Banach spaces. Projection constants have been extensively studied for several decades. This field has a long history of remarkable results, but interestingly, also of some incorrect proofs. The first correct proof of the fact that 4/3 is the maximal possible value of an absolute projection constant of a real two-dimensional normed spaces was published by Chalmers and Lewicki in 2010. Only recently, DerÄ gowska and Lewandowska determined the maximal possible value of projection constant in every dimension n, for which there exists an equiangular tight frame of the maximal cardinality. These include the dimensions n=2, 3, 7, 23 in the real case and up to dimension n=17 in the complex case. Based on the reasoning of DerÄ gowska and Lewandowska, we are able to characterize all spaces with the maximal projection constant in the aferomentioned dimensions. During the process, we shall discuss the history of the existing literature on this subject, including some incorrect and incomplete reasonings. As a result, we obtain that the two-dimensional real normed space with the regular hexagon as a unit ball is the unique two-dimensional real normed space with the maximal projection constant. In every other dimension a space with a maximal projection constant is not unique.
No 166  January 17th, 2024   Leokadia Białas-Cież
Dubiner's distance and admissible meshes
No 167  February 28th, 2024  
Organizational meeting
No 168  March 6th, 2024   Grzegorz Lewicki
Minimal projections onto the lines with respect to L_p norm
No 169  March 20th, 2024   Marta Kosek
A few remarks on the complex Green function and its application
No 170  March 27th, 2024   Przemysław Sprus
Convex hulls of polynomial Julia sets
Abstract: We will prove that for every complex polynomial $p$ of degree $d\geq 2$ the convex hull $H_p$ of the Julia set $J_p$ of $p$ satisfies $p^{-1}(H_p) \subset H_p$. We will also characterise polynomials for which the equality $p^{-1}(H_p)=H_p$ is achieved.
References: M. Stawiska, Convex hulls of polynomial Julia sets, Proceedings of the AMS, Vol. 149, No. 1, January 2021, pp. 245-250
No 171  April 10th, 2024   Grzegorz Lewicki
On 2-strongly unique best approximation