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 |