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1. Representer Theorems in Banach Spaces: Minimum Norm Interpolation, Regularized Learning and Semi-Discrete Inverse Problems

   Wang, Rui; Xu, Yuesheng (September 2021, Journal of machine learning research)

   Shawe-Taylor, John (Ed.)

   Learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering. This is often formulated as a minimum norm interpolation (MNI) problem, a regularized learning problem or, in general, a semi-discrete inverse problem (SDIP), in either Hilbert spaces or Banach spaces. The goal of this paper is to systematically study solutions of these problems in Banach spaces. We aim at obtaining explicit representer theorems for their solutions, on which convenient solution methods can then be developed. For the MNI problem, the explicit representer theorems enable us to express the infimum in terms of the norm of the linear combination of the interpolation functionals. For the purpose of developing efficient computational algorithms, we establish the fixed-point equation formulation of solutions of these problems. We reveal that unlike in a Hilbert space, in general, solutions of these problems in a Banach space may not be able to be reduced to truly finite dimensional problems (with certain infinite dimensional components hidden). We demonstrate how this obstacle can be removed, reducing the original problem to a truly finite dimensional one, in the special case when the Banach space is ℓ1(N). 

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2. Fishes of the Lower Lulua River (Kasai Basin, Central Africa): A Continental Hotspot of Ichthyofaunal Diversity under Threat

   https://doi.org/https://doi.org/10.3390/d13080341  

   José, J. Mbimbi; Melanie, L. J.; Tobit L. D. Liyandja (July 2021, Diversity)

   The ichthyofauna of the Lulua River, a large right bank tributary of the Kasai River in central Africa, is among the most poorly documented in the Kasai ecoregion. To remedy this lack of knowledge, sampling was carried out between 2007 and 2014 along the main channel and in many tributaries. A total of 201 species distributed in 81 genera, 24 families, and 12 orders are reported from the lower reaches of the Lulua. The species richness reported in this study represents a substantially improved documentation of the Lulua ichthyofauna (historically estimated at only 79 species). Here, 129 species are recorded for the first time, bringing the total number of known species to 208. Among these, five have recently been described: Raiamas brachyrhabdotos Katemo Manda, Snoeks, Choca Manda, and Vreven 2018, Distichodus kasaiensis Moelants, Snoeks, and Vreven, 2018, Distichodus polli Abwe, Snoeks, Choca Manda, and Vreven 2019, Epiplatys atractus Van Der Zee, Mbimbi, and Sonnenberg 2013, and Hypsopanchax stiassnyae Van Der Zee, Sonnenberg, and Mbimbi 2015; numerous additional taxa are currently under investigation. Recognized here as a continental hotspot of ichthyofaunal diversity, the Lulua is under extreme threat from exploitation of forest products for building materials, deforestation for shifting agriculture and charcoal exploitation, destructive fishing practices, and mining, all of which are rapidly increasing in the watershed. The present study provides baseline documentation for use in conservation planning and future devel- opmental projects in the Kasai ecoregion in general and Lulua River basin in particular. 

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3. On Harmonic Hilbert Spaces on Compact Abelian Groups

   https://doi.org/10.1007/s00041-023-09992-4  

   Das, Suddhasattwa; Giannakis, Dimitrios (February 2023, Journal of Fourier Analysis and Applications)

   Abstract Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted$$L^2$$ $L^2$spaces on the dual group. It is known that for suitably chosen subadditive weights, every such space is a Banach algebra with respect to pointwise multiplication of functions. In this paper, we study RKHSs associated with subconvolutive functions on the dual group. Sufficient conditions are established for these spaces to be symmetric Banach$$^*$$ ${}^{\ast }$-algebras with respect to pointwise multiplication and complex conjugation of functions (here referred to as RKHAs). In addition, we study aspects of the spectra and state spaces of RKHAs. Sufficient conditions are established for an RKHA on a compact abelian groupGto have the same spectrum as the$$C^*$$ $C^{\ast }$-algebra of continuous functions onG. We also consider one-parameter families of RKHSs associated with semigroups of self-adjoint Markov operators on$$L^2(G)$$ $L^2\left(G\right)$, and show that in this setting subconvolutivity is a necessary and sufficient condition for these spaces to have RKHA structure. Finally, we establish embedding relationships between RKHAs and a class of Fourier–Wermer algebras that includes spaces of dominating mixed smoothness used in high-dimensional function approximation. 

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4. Analysis of the inverse Born series: an approach through geometric function theory

   https://doi.org/10.1088/1361-6420/ac661f  

   Hoskins, Jeremy G; Schotland, John C (May 2022, Inverse Problems)

   Abstract We analyze the convergence and approximation error of the inverse Born series, obtaining results that hold under qualitatively weaker conditions than previously known. Our approach makes use of tools from geometric function theory in Banach spaces. An application to the inverse scattering problem with diffuse waves is described. 

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5. On the Norm Equivalence of Lyapunov Exponents for Regularizing Linear Evolution Equations

   https://doi.org/10.1007/s00205-023-01928-y  

   Blumenthal, Alex; Punshon-Smith, Sam (October 2023, Archive for Rational Mechanics and Analysis)

   We consider the top Lyapunov exponent associated to a dissipative linear evolution equation posed on a separable Hilbert or Banach space. In many applications in partial differential equations, such equations are often posed on a scale of nonequivalent spaces mitigating, e.g., integrability (L^p) or differentiability (W^{s,p}). In contrast to finite dimensions, the Lyapunov exponent could a priori depend on the choice of norm used. In this paper we show that under quite general conditions, the Lyapunov exponent of a cocycle of compact linear operators is independent of the norm used. We apply this result to two important problems from fluid mechanics: the enhanced dissipation rate for the advection diffusion equation with ergodic velocity field; and the Lyapunov exponent for the 2d Navier–Stokes equations with stochastic or periodic forcing. 

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