Given a function b, holomorphic on the disc and bounded by 1, one can construct an associated reproducing kernel Hilbert space called the de Branges–Rovnyak space H(b). We explore representations of such spaces via descriptions of the corresponding families of orthogonal polynomials. We find relevant structures in the linear systems involved in a diversity of cases when b is rational. We also establish a form of invariance under some composition operators on H(b) spaces.
A new alternative numerical procedure to the Szegő quadrature formulas for the estimation of integrals with respect to a positive Borel measure μ supported on the unit circle is presented. As in many practical situations, we assume that the values of the integrand F are only known at a finite number of points, which we will assume to be equispaced on the unit circle. Our technique consists of obtaining an approximating Laurent polynomial L to F by interpolation in the Hermite sense in a collection of these points that mimic the zeros of a para-orthogonal polynomial with respect to μ, and to use the values of F at the remaining nodes to improve the accuracy of the approximation by a process of simultaneous complex regression. Some numerical examples are carried out.
In the paper we introduce the concept of the Kuratowski distance between nonempty and bounded subsets of a Banach space. We show that the Kuratowski measure of noncompactness of a nonempty and bounded subset of a Banach space is equal to the Kuratowski distance of that subset to the family of all nonempty and relatively compact subsets of the mentioned Banach space.
The origin of obscuration in active galactic nuclei (AGN) is still a matter of contention. It is unclear whether obscured AGN are primarily due to line-of-sight effects (Orientation model), a transitory, dust-enshrouded phase in galaxy evolution (Evolution models), or a combination of both. The role of an inner torus around the central supermassive black hole also remains unclear in pure Evolution models. We use cosmological semi-analytic models and semi-empirical prescriptions to explore obscuration effects in AGN at cosmic noon, in the range 1 < z < 3. We consider a realistic object-by-object modelling of AGN evolution including different AGN light curves (LCs) composed of phases of varying levels of obscuration, usually (but not uniquely) with a larger degree of obscuration before the peak of AGN activity, mimicking the possible clearing effects of strong AGN feedback. Evolution models characterized by AGN LCs with relatively short pre-peak obscured phases followed by more extended optical/ultraviolet (UV) visible post-peak phases, struggle to reproduce the high fraction of obscured AGN at z similar to 2-3 inferred from X-ray surveys. Evolution models characterized by AGN LCs with sharp post-peak declines or persistent or multiple obscuration phases are more successful, although they still face challenges in reproducing the steady drop in the fractions of obscured AGN with increasing luminosity measured by some groups. Invoking a fine-tuning in the input LCs, with more luminous AGN defined by longer optical/UV visible windows, can improve the match to the decreasing fractions of obscured AGN with luminosity. Alternatively, a long-lived central torus-like component, with thickness decreasing with increasing AGN power, naturally boosts the luminosity-dependent fractions of obscured AGN, suggesting that small-scale orientation effects may still represent a key component even in Evolution models. We also find that in our models major mergers and starbursts, when considered in isolation, fall short in accounting for the large fractions of highly obscured faint AGN detected at cosmic noon.
Aim: Identifying climate refugia is a pressing priority for conservation planning under global change, particularly in oceanic archipelagos with high levels of endemicity and topographic complexity. Location: Canary Islands. Methods: Here, we developed a spatial framework to identify and recursively map refugia probability in the Canary Islands across a broad set of indicators, including multivariate climate analogues, topographic complexity, poleward aspect, wetness, and forestry cover. We integrated these into continuous refugia probability maps and assessed their spatial patterns across the islands. Results: Our results reveal clear east-west patterns, with highest refugia potential values concentrated in poleward-oriented, rugged and forested areas in the western part of the archipelago. The additive integration of the different blocks enables the identification of refugia probability. The tree-cover block highlights the role of vegetation and forest patches, promoting refugia. In drier areas, however, topography and poleward-facing aspects become more influential, improving refugia detection on islands with strong environmental contrasts. Critically, many of the most resilient refugia remain outside existing protected areas, especially on biodiversity-rich and heterogeneous islands, highlighting how our approach can help identify areas of high conservation potential. Main Conclusions: Our proposed framework is reproducible, data-efficient, and transferable to other oceanic regions, providing a flexible tool for conservation decisions and adaptive design of Protected Areas systems in vulnerable landscapes. By identifying areas likely to host biodiversity under climate change, our approach supports adaptive conservation planning, protected area expansion, and climate-informed prioritisation for managers.
