In this talk we will introduce histopolation, an approximation technique that replaces nodal evaluations with integral over compact sets or, equivalently, the corresponding averages. Despite being frequently employed in many applications, such as finite volumes and weights-based finite elements, the classical "uniform" approximation associated with histopolation is rather unexplored. In this seminar, we give an overview of the most impactful techniques in the one dimensional framework, introduce an appropriate Lebesgue constant, and analyze the conditioning of the resulting linear system, as well as efficient techniques for the resolution. We offer a parallelism with usual nodal interpolation and see where the two methods move away from each other. To conclude, possible applications are discussed.

From the Garloff–Wagner theorem it follows that the set of stable polynomials \(\mathcal{H}_n\), i.e. those whose zeros all lie in the open left half of the complex plane, together with the Hadamard product*, forms an abelian semigroup contained in the abelian group of polynomials with positive real coefficients \(\mathbb{R}_n^+\). By the idealizer of the set \(\mathcal{H}_n\) we mean the largest subsemigroup of \(\mathbb{R}_n^+\) in which \(\mathcal{H}_n\) is an ideal. During the talk, new results providing a partial characterization of this idealizer will be presented.

Based on the work of A. Kroó, "Optimal meshes for weighted multivariate polynomials in \(\mathbb{R}^{d}\)".

For so-called ‘guided polynomial sequences’, the guide (from which their name is derived) turns out to be the Green function of any regular compact set. Equivalent conditions to the ‘guided’ property will be presented, together with examples of such sequences and their applications in the theory of Julia sets.

Let \(T=\{t_n,\; n\ge 1\}\) be a sequence of points in the interval \([0,1]\), dense in \([0,1]\). Let \(1 < p < \infty\). For the sequence \(T\) we consider two dictionaries: \(\mathcal{C}_T\), consisting of characteristic functions of intervals created by consecutively adding the points \(\{t_n, n\ge 1\}\), and \(\mathcal{H}_T\), consisting of general Haar functions corresponding to the sequence of points \(T\). We study the approximation spaces \(A_q^\alpha(L^p,\mathcal{C}_T)\) and \(A_q^\alpha(L^p,\mathcal{H}_T)\), defined by the orders of the best \(n\)-term approximation in the \(L^p[0,1]\) norm, with respect to the elements of the dictionaries \(\mathcal{C}_T\) and \(\mathcal{H}_T\), respectively. Of course, one always has \( A_q^\alpha(L^p,\mathcal{H}_T) \subset A_q^\alpha(L^p,\mathcal{C}_T). \) From the paper by P. Petrushev, “Multivariate \(n\)-term rational and piecewise polynomial approximation”, J. Approx. Theory 121 (2003), 158–197, it follows that for \(1< p < \infty\) and \(D\) - a sequence of dyadic points, the equality \( A_q^\alpha(L^p,\mathcal{H}_D) = A_q^\alpha(L^p,\mathcal{C}_D) \) holds. We ask whether this property also holds for an arbitrary sequence of points \(T\). In the talk I intend to present:

• A Bernstein-type inequality \( BI(\mathcal{H}_T, \beta, p, \tau) \), on which the embedding \( A_q^\alpha(L^p,\mathcal{C}_T) \subset A_q^\alpha(L^p,\mathcal{H}_T) \) for \( \alpha < \beta \) depends.
• A geometric characterization of the Bernstein inequality \( BI(\mathcal{H}_T, \beta, p, \tau) \).
• A further discussion of the space \( A_q^\alpha(L^p,\mathcal{C}_T) \).

In recent years, various nonlinear approximation methods have been extensively studied. One such method is greedy approximation with respect to bases. In the talk, I plan to address the following points:

• Greedy bases (and related ones: almost greedy, quasi-greedy) in Banach spaces and their basic properties.
• Spaces of best n-term approximation for greedy bases and their characterization in terms of the coefficients of the basis expansion.
• The existence/non-existence of greedy (and related) bases in specific function spaces (\(L^p\), Sobolev, Besov, and Lizorkin–Triebel spaces), along with examples.

During the talk, I will present a certain class of non-autonomous Julia sets that possess the HCP (Hölder Continuity Property). I will provide a proof of this property in a specific case and mention possible generalizations. Additionally, I will present a number of non-trivial examples of non-autonomous Julia sets.