Recent zbMATH articles in MSC 43https://zbmath.org/atom/cc/432022-11-17T18:59:28.764376ZUnknown authorWerkzeugDiscrepancy of minimal Riesz energy pointshttps://zbmath.org/1496.111022022-11-17T18:59:28.764376Z"Marzo, Jordi"https://zbmath.org/authors/?q=ai:marzo.jordi"Mas, Albert"https://zbmath.org/authors/?q=ai:mas.albertThis paper is concerned with upper bounds for the spherical cap discrepancy of the set of minimizers of the Riesz \(s\)-energy and the logarithmic energy on the sphere \({\mathbb S}^d\).
Recall, that a spherical cap \(D_r(x)\) is the set \( \{y \in{\mathbb S}^d\ :\ |x-y|<r\}, \) where \(x\in {\mathbb S}^d \), \(r\in[0,2]\), and the spherical cap discrepancy of a point set \(X\subset {\mathbb S}^d\) is
\[
\sup_{r,x}\Big|\frac{\#\big(X\cap D_r(x)\big)}{\# X}-\tilde{\sigma}(D_r(x))\Big|,
\]
where \(\tilde{\sigma}\) is the uniform probability measure on \({\mathbb S}^d\). The Riesz/logarithmic energy of a set \(V\subset {\mathbb S}^d\) is given by
\[
E_s(V)=\sum_{\substack{x,y\in V \\
x\neq y}}R_s(x,y),\hspace{0.4cm}\mbox{ where }\hspace{0.4cm} R_s(x,y)=\begin{cases} |x-y|^{-s}, &\mbox{for }0<s<d\\
-\log(|x-y|),&\mbox{for }s=0.\end{cases}
\]
Let \(V_N\) denote all subsets of \({\mathbb S}^d\) with \(N\)-elements, where \(N\in {\mathbb N}\). Since the kernel \(R_s(x,y)\) is lower semi-continuous and \({\mathbb S}^d\) is compact, there exists a minimizing configuration \(\omega^s_N=\{x_1,\ldots,x_N\}\in V_N\) for the energy, for every \(N\in {\mathbb N}\). Let \(\chi_A\) denote the characteristic function of the set \(A\), i.e., \(\chi_A(x)=1\) if \(x\in A\) and \(\chi_A(x)=0\) else. The authors show the following upper bound:
Theorem 1.1. Let \(\omega^s_N=\{x_1,\ldots,x_N\}\) be the \(N\)-point minimizer of the Riesz or logarithmic energy, then
\[
c_{s,d}\cdot\sup_{r,x}\Big|\frac{\#\big(\omega^s_N\cap D_r(x)\big)}{N}-\tilde{\sigma}(D_r(x))\Big|\ \leq\ \chi_{[0,d-2]}(s)\cdot N^{-\frac{2}{d(d-s+1)}}+\chi_{(d-2,d)}(s)\cdot N^{-\frac{2(d-s)}{d(d-s+4)}},
\]
with a constant \(c_{s,d}\) that depends on \(d,s\) only.
This theorem is derived by relating the spherical cap discrepancy with a notion of discrepancy involving Sobolev norms (Theorem 1.5 and Proposition 5.2).
The behavior of following integral operator on \(L^2\big({\mathbb S}^d\big)\) is investigated (Section 2 and Section 3):
\[
R_s(f)=\int_{{\mathbb S}^d}R_s(x,y)f(y)d\tilde{\sigma}(y),
\]
where, among other things, it is shown (in Proposition 2.2) that \(R_s\) diagonalizes in the standard basis of spherical harmonics, and its eigenvalues are computed as well as their asymptotic behavior. Further, some formal identities were derived (in Remark 2.3) for \(R_s(x,y)\) in terms of Gegenbauer, also known as ultraspherical polynomials, which might be of independent interest.
The authors derive a differential equation to relate \(R_s(x,x_0)\) to \(R_{s+2}(x,x_0)\) for \(x\neq x_0\) (in Lemma 2.5) via the Laplace-Beltrami operator, which is later used to show superharmonicity of \(R_s(x,x_0)\) near \(x_0\) (in Lemma 3.1), and to obtain an upper bound (in Corollary 3.7) of
\[
\frac{1}{N^2}\sum_{j\neq k}\int_{D_j}\int_{D_k}R_s(x,y)d\tilde{\sigma}(x)d\tilde{\sigma}(y),
\]
where each \(D_j\) is a small disc around \(x_j\in\omega^s_N \). This result is then applied to derive the upper bound for the Sobolev discrepancy (in Theorem 1.5).
This paper generalizes an unpublished result due to \textit{T. Wolff} [``Fekete points on spheres'', Preprint]. The conclusion of Theorem 1.1 (partially) improves upon results, obtained by Kleiner, Sjögren, Götz and Brauchart.
Some typos were found on page 479, where the average of a function \(f\) over a disc \(D\) should be integrated over \(D\); on page 499 we find \(R_s(x-y)\) instead of \(R_s(x,y)\).
Reviewer: Damir Ferizović (Leuven)Hypergeometry, integrability and Lie theory. Virtual conference, Lorentz Center, Leiden, the Netherlands, December 7--11, 2020https://zbmath.org/1496.170012022-11-17T18:59:28.764376ZPublisher's description: This volume contains the proceedings of the virtual conference on Hypergeometry, Integrability and Lie Theory, held from December 7--11, 2020, which was dedicated to the 50th birthday of Jasper Stokman.
The papers represent recent developments in the areas of representation theory, quantum integrable systems and special functions of hypergeometric type.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Etingof, Pavel; Kazhdan, David}, Characteristic functions of \(p\)-adic integral operators, 1-27 [Zbl 07602313]
\textit{Garbali, Alexandr; Zinn-Justin, Paul}, Shuffle algebras, lattice paths and the commuting scheme, 29-68 [Zbl 07602314]
\textit{Kolb, Stefan}, The bar involution for quantum symmetric pairs -- hidden in plain sight, 69-77 [Zbl 07602315]
\textit{Koornwinder, Tom H.}, Charting the \(q\)-Askey scheme, 79-94 [Zbl 07602316]
\textit{Rains, Eric M.}, Filtered deformations of elliptic algebras, 95-154 [Zbl 07602317]
\textit{Regelskis, Vidas; Vlaar, Bart}, Pseudo-symmetric pairs for Kac-Moody algebras, 155-203 [Zbl 07602318]
\textit{Reshetikhin, N.; Stokman, J. V.}, Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains, 205-241 [Zbl 07602319]
\textit{Rösler, Margit; Voit, Michael}, Elementary symmetric polynomials and martingales for Heckman-Opdam processes, 243-262 [Zbl 07602320]
\textit{Schomerus, Volker}, Conformal hypergeometry and integrability, 263-285 [Zbl 07602321]
\textit{Varchenko, Alexander}, Determinant of \(\mathbb{F}_p\)-hypergeometric solutions under ample reduction, 287-307 [Zbl 07602322]
\textit{Varchenko, Alexander}, Notes on solutions of KZ equations modulo \(p^s\) and \(p\)-adic limit \(s\to\infty\), 309-347 [Zbl 07602323]Limiting Sobolev and Hardy inequalities on stratified homogeneous groupshttps://zbmath.org/1496.350232022-11-17T18:59:28.764376Z"van Schaftingen, Jean"https://zbmath.org/authors/?q=ai:van-schaftingen.jean"Yung, Po-Lam"https://zbmath.org/authors/?q=ai:yung.polamSummary: We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn-Sobolev and Korn-Hardy inequalities on stratified homogeneous groups.AP-frames and stationary random processeshttps://zbmath.org/1496.420072022-11-17T18:59:28.764376Z"Centeno, Hernán D."https://zbmath.org/authors/?q=ai:centeno.hernan-d"Medina, Juan M."https://zbmath.org/authors/?q=ai:medina.juan-miguelSummary: It is known that, in general, an AP-frame is an \(L^2 (\mathbb{R})\)-frame and conversely. Here, in part as a consequence of the Ergodic Theorem, we prove a necessary and sufficient condition for a Gabor system \(\{ g(t - k) e^{il(t-k)}, l \in \mathbb{L} = \omega_0 \mathbb{Z}, k \in \mathbb{K} = t_0 \mathbb{Z}\}\) to be an \(L^2 (\mathbb{R})\)-Frame in terms of Gaussian stationary random processes. In addition, if \(X = (X(t))_{t \in \mathbb{R}}\) is a wide sense stationary random process, we study density conditions for the associated stationary sequences \(\{\langle X, g_{k, l} \rangle, l \in \mathbb{L}, k \in \mathbb{K}\}\).Multi-dimensional \(c\)-almost periodic type functions and applicationshttps://zbmath.org/1496.420092022-11-17T18:59:28.764376Z"Kostic, Marko"https://zbmath.org/authors/?q=ai:kostic.markoSummary: In this article, we analyze multi-dimensional Bohr \((\mathcal{B}, c)\)-almost periodic type functions. The main structural characterizations for the introduced classes of Bohr \((\mathcal{B}, c)\)-almost periodic type functions are established. Several applications of our abstract theoretical results to the abstract Volterra integro-differential equations in Banach spaces are provided, as well.The circular maximal operator on Heisenberg radial functionshttps://zbmath.org/1496.420222022-11-17T18:59:28.764376Z"Beltran, David"https://zbmath.org/authors/?q=ai:beltran.david"Guo, Shaoming"https://zbmath.org/authors/?q=ai:guo.shaoming"Hickman, Jonathan"https://zbmath.org/authors/?q=ai:hickman.jonathan"Seeger, Andreas"https://zbmath.org/authors/?q=ai:seeger.andreasDenote by \(\mathbb{H}^n\) the Heisenberg group, the set \(\mathbb{R}\times\mathbb{R}^{2n}\) equipped with the following non-commutative group operation: for all \((u,x), (v,y)\in\mathbb{H}^n\),
\[
(u,x)\cdot (v,y):=(u+v+x^TBy,x+y).
\]
Here, \(B:=\begin{pmatrix}0&-bI_n\\
bI_n&0\end{pmatrix}\) for some \(b\neq 0\) (one typically choose \(b=1/2\)). For \(\mu_1\equiv\mu\), the normalized surface measure on \(\{0\}\times\mathbb{S}^{2n-1}\), let \(\mu_t\) denote its dilation supported on \(t\mathbb{S}^{2n-1}\). For a function \(f:\mathbb{H}^n\rightarrow\mathbb{C}\), one may formally define its spherical means as \[ f\ast\mu_t(u,x):=\int_{\mathbb{S}^{2n-1}}\!f(u-tx^TBy,x-ty)\,d\mu(y) \] and its spherical maximal function as
\[
Mf(u,x):=\sup_{t>0}|f\ast\mu_t(u,x)|.
\]
In this paper, the authors complement known \(L^p\)-boundedness results for \(M\) on \(\mathbb{H}^n\), \(n\geq{2}\), by initiating the study of the case \(n=1\), where currently nothing is known for any \(p<\infty\). For \(2<p\leq\infty\), they show the existence of a constant \(C_p\), depending only on \(p\), such that
\[
\|Mf\|_{L^p(\mathbb{H}^1)}\leq C_p\|f\|_{L^p(\mathbb{H}^1)}
\]
for all \(\mathbb{H}\)-radial functions \(f\) on \(\mathbb{H}^1\). A function \(f:\mathbb{H}^1\rightarrow\mathbb{C}\) is said to be \(\mathbb{H}\)-radial if \(f(u,Rx)=f(u,x)\) for all \((u,x)\in\mathbb{H}^1\) and all \(R\) belonging to the special orthogonal group, \(SO(2)\). Equivalently, \(f\) is \(\mathbb{H}\)-radial if and only if there exists some function \(f_0:\mathbb{R}\times[0,\infty)\rightarrow\mathbb{C}\) such that \(f(u,x)=f_0(u,|x|)\) for all \((u,x)\in\mathbb{H}^1\).
The authors accomplish this by reducing the problem to studying the boundedness of a maximal function given by \(\sup_{t>0} |A_tf|\), where \(\{A_t\}\) are non-convolution averaging operators on \(\mathbb{R}^2\). While the reduction is not difficult, the associated curve distribution has vanishing rotational and cinematic curvatures, precluding the straightforward application of the standard techniques used to study the Euclidean spherical maximal function. A significant portion of this paper is spent overcoming these challenges, along the way performing an \(L^2\) analysis of two-parameter oscillatory integrals with two-sided fold singularities.
The appendices contain, among other things, a discussion of the use of repeated integration by parts often seen when studying oscillatory integrals.
Reviewer: Ryan Gibara (Cincinnati)On a (no longer) new Segal algebra: a review of the Feichtinger algebrahttps://zbmath.org/1496.430012022-11-17T18:59:28.764376Z"Jakobsen, Mads S."https://zbmath.org/authors/?q=ai:jakobsen.mads-sielemannSummary: Since its invention in 1979 the Feichtinger algebra has become a useful Banach space of functions with applications in time-frequency analysis, the theory of pseudo-differential operators and several other topics. It is easily defined on locally compact Abelian groups and, in comparison with the Schwartz(-Bruhat) space, the Feichtinger algebra allows for more general results with easier proofs. This review paper develops the theory of Feichtinger's algebra in a linear and comprehensive way. The material gives an entry point into the subject and it will also bring new insight to the expert. A further goal of this paper is to show the equivalence of the many different characterizations of the Feichtinger algebra known in the literature. This task naturally guides the paper through basic properties of functions that belong to this space, over operators on it, and to aspects of its dual space. Additional results include a seemingly forgotten theorem by Reiter on Banach space isomorphisms of the Feichtinger algebra, a new identification of Feichtinger's algebra as the unique Banach space in \(L^{1}\) with certain properties, and the kernel theorem for the Feichtinger algebra. A historical description of the development of the theory, its applications, and a list of related function space constructions is included.Convergence and almost sure properties in Hardy spaces of Dirichlet serieshttps://zbmath.org/1496.430022022-11-17T18:59:28.764376Z"Bayart, Frédéric"https://zbmath.org/authors/?q=ai:bayart.fredericThis is a remarkable continuation of a series of recent articles trying to establish a modern theory of general Dirichlet series \(D = \sum_n a_n e^{-\lambda_n s}\); here \(s \in \mathbb{C}\) is a complex variable, the \(a_n\)'s form the coefficients, and \(\lambda = (\lambda_n)\) is the frequency (i.e., a sequence of non-negative, strictly increasing real numbers tending to \(\infty\)).
The article is divided into three sections. The first section isolates a new condition on \(\lambda\) (denoted by ($NC$)), which ensures that a somewhere convergent \(\lambda\)-Dirichlet series defining a bounded function on the right half-plane, converges uniformly on every smaller half-plane. Conditions of this type are fundamental for the understanding of general Dirichlet series, and \((NC)\) indeed extends famous work of H.~Bohr (condition \((BC)\)) and E.~Landau (condition \((LC)\)). Using alternative techniques (as e.g. Saksman's vertical convolution formula), Bayart's new condition \((NC)\) is applied to improve recent maximal inequalities from [\textit{I. Schoolmann}, Math. Nachr. 293, No. 8, 1591--1612 (2020; Zbl 07261807)].
For \(1 \leq p \leq \infty \) the Hardy space \(\mathcal{H}_p(\lambda)\) of \(\lambda\)-Dirichlet series is defined as the completion of all finite \(\lambda\)-Dirichlet polynomials \(D = \sum_{n=1}^N a_n e^{-\lambda_n s}\) under the \(\mathcal{H}_p\)-norm \[\|D\|_p = \lim_{T \to \infty} \frac{1}{2T} \Big(\int_{-T}^{T} |D(it)|^p dt \Big)^{1/p}\,.\] However, this internal description is often not sufficient to understand the structure of these Banach spaces. Following earlier work of Bayart for ordinary Dirichlet series (\(\lambda = (\log n)\)), a~group approach is suggested in [\textit{A. Defant} and \textit{I. Schoolmann}, J. Fourier Anal. Appl. 25, No.~6, 3220--3258 (2019; Zbl 1429.43004)], and among others carefully studied in [\textit{A. Defant} and \textit{I. Schoolmann}, Math. Ann. 378, No.~1--2, 57--96 (2020; Zbl 1483.43006)] and [\textit{A. Defant} and \textit{I. Schoolmann}, J. Funct. Anal. 279, No.~5, Article ID 108569, 36~p. (2020; Zbl 1470.43006)].
The second section of this article answers non-trivial questions arising from these works -- mainly extending a famous theorem of Helson to Dirichlet series in \(\mathcal{H}_1(\lambda).\) It shows that if \(\lambda\) satisfies the new condition \((NC)\) and \(D = \sum_n a_n e^{-\lambda_n s} \in \mathcal{H}_1(\lambda),\) then for almost all homomorphisms \(\omega: \mathbb{R} \to \mathbb{T}\) the Dirichlet series \(D = \sum_n \omega(\lambda_n)a_n e^{-\lambda_n s}\) converges pointwise on \([\Re s>0]\). Moreover a maximal inequality is added, and a non-trivial counterexample showing that such a result is false for arbitrary \(\mathcal{H}_1(\lambda)\)-Dirichlet series (we note that for \(1 < p < \infty\) Helson's theorem in the above sense holds for any \(\mathcal{H}_p(\lambda)\)-Dirichlet series without any further assumption on~\(\lambda\)). These results indicate that the Hardy spaces \(\mathcal{H}_p(\lambda)\) seem to behave well whenever we consider their `almost everywhere properties'.
A further non-trivial problem is to determine the optimal half-plane \([\Re s > \sigma_{\mathcal{H}_p(\lambda)} ]\), where \(\sigma_{\mathcal{H}_p(\lambda)}\) is defined to be the best \(\sigma \in \mathbb{R}\) such that the convergence abscissa \(\sigma_{c}(D)\) of all \(D \in \mathcal{H}_p(\lambda)\) is \(\leq \sigma\). Previous work of Bayart shows that, based on the multiplicativity of \(\lambda = (\log n)\) and the hypercontractivity of the Poisson kernel acting on the Hardy space \(H_1(\mathbb{T})\), we have \(\sigma_{\mathcal{H}_p((\log n))} = \frac{1}{2}\) for all \(1 \leq p < \infty\). A natural guess would be that \(\sigma_{\mathcal{H}_p(\lambda)} = \frac{L(\lambda)}{2} \) for all \(1 \leq p < \infty\), where \(L(\lambda) = \limsup_n \frac{\log n}{\lambda_n} \) (following Bohr, this number equals the width of the largest possible strip on which a \(\lambda\)-Dirichlet series converges but does not converge absolutely). For \(p=2\) this is indeed true, but in contrast to the ordinary case, there is no hope to get a similar result for \(p\neq 2\). Among others, it is proved that \(\sigma_{\mathcal{H}_1(\lambda)} \leq 2 \sigma_{\mathcal{H}_2(\lambda)}\) for all frequencies, and that there exists a frequency with Bohr's condition \((BC)\) for which we here even have equality.
Reviewer: Andreas Defant (Oldenburg)Fourier transforms on finite group actions and bent functionshttps://zbmath.org/1496.430032022-11-17T18:59:28.764376Z"Fan, Yun"https://zbmath.org/authors/?q=ai:fan.yun"Xu, Bangteng"https://zbmath.org/authors/?q=ai:xu.bangtengSummary: Highly nonlinear functions (bent functions, perfect nonlinear functions, etc.) on finite fields and finite (abelian or nonabelian) groups have been studied in numerous papers. As a generalization of bent functions on finite groups, bent functions on group actions have been studied in quite a few papers. In this paper, we study Fourier transforms and bent functions on finite nonabelian group actions. Let \(G\) be a finite (nonabelian) group acting on a finite set \(X\). Our goal is to develop a Fourier analysis on \(X\) and study the bentness of complex valued functions on \(X\) via their Fourier transforms. We will first construct a \(G\)-dual set \(\widehat{X} \), which is a special basis of the \(G\)-space of all complex valued functions on \(X\). This \(G\)-dual set \({\widehat{X}}\) plays a role similar to that of \({\widehat{G}} \). Then, for a complex-valued function \(f\) on \(X\), we define its Fourier transform \({\widehat{f}}\) as a function on \({\widehat{X}}\) and characterize the bentness of \(f\) by its Fourier transform \({\widehat{f}} \). Some known and new results about bent functions on finite groups will be obtained as direct consequences. We will also discuss the constructions of bent functions and construct examples of bent functions on group actions of the dihedral group of order 6.Improved critical Hardy inequalities on 2-dimensional quasi-ballshttps://zbmath.org/1496.430042022-11-17T18:59:28.764376Z"Sabitbek, Bolys"https://zbmath.org/authors/?q=ai:sabitbek.bolys"Suragan, Durvudkhan"https://zbmath.org/authors/?q=ai:suragan.durvudkhan"Yessirkegenov, Nurgissa"https://zbmath.org/authors/?q=ai:yessirkegenov.nurgissa-aSummary: In this note we obtain a remainder estimate for improved critical Hardy inequalities on a 2-dimensional quasi-ball on homogeneous Lie groups. These results are new even in the Abelian case of \(\mathbb{R}^2\) in terms of choosing any choice of homogeneous quasi-norm as well as replacing the full gradient by the radial derivative.
For the entire collection see [Zbl 1436.46003].An uncertainty principle for spectral projections on rank one symmetric spaces of noncompact typehttps://zbmath.org/1496.430052022-11-17T18:59:28.764376Z"Ganguly, Pritam"https://zbmath.org/authors/?q=ai:ganguly.pritam"Thangavelu, Sundaram"https://zbmath.org/authors/?q=ai:thangavelu.sundaramThe authors present a weaker version of Chernoff's theorem for Bessel and Jacobi operators. This result is used to prove a refined version of Ingham's theorem for the Helgason Fourier transform on rank one Riemannian symmetric spaces of noncompact type. The authors also prove an Ingham type uncertainty principle for the generalized spectral projections associated to the Laplace-Beltrami operator. Similar Ingham type results for the generalized spectral projections associated to Dunkl Laplacian are also discussed.
Reviewer: Ashish Bansal (Delhi)A generalized polar-coordinate integration formula with applications to the study of convolution powers of complex-valued functions on \(\mathbb{Z}^d\)https://zbmath.org/1496.430062022-11-17T18:59:28.764376Z"Bui, Huan Q."https://zbmath.org/authors/?q=ai:bui.huan-q"Randles, Evan"https://zbmath.org/authors/?q=ai:randles.evanSummary: In this article, we consider a class of functions on \(\mathbb{R}^d\), called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on \(\mathbb{Z}^d\). As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function \(P\), we construct a Radon measure \(\sigma_P\) on \(S=\{\eta \in\mathbb{R}^d:P(\eta)=1\}\) which is invariant under the symmetry group of \(P\). With this measure, we prove a generalization of the classical polar-coordinate integration formula and deduce a number of corollaries in this setting. We then turn to the study of convolution powers of complex functions on \(\mathbb{Z}^d\) and certain oscillatory integrals which arise naturally in that context. Armed with our integration formula and the Van der Corput lemma, we establish sup-norm-type estimates for convolution powers; this result is new and partially extends results of Randles and Saloff-Coste
[\textit{E.~Randles} and \textit{L.~Saloff-Coste}, Rev. Mat. Iberoam. 33, No.~3, 1045--1121 (2017; Zbl 1377.42012)].The Haagerup property for twisted groupoid dynamical systemshttps://zbmath.org/1496.460512022-11-17T18:59:28.764376Z"Kwaśniewski, Bartosz K."https://zbmath.org/authors/?q=ai:kwasniewski.bartosz-kosma"Li, Kang"https://zbmath.org/authors/?q=ai:li.kang"Skalski, Adam"https://zbmath.org/authors/?q=ai:skalski.adam-gSummary: We introduce the Haagerup property for twisted groupoid \(C^\ast\)-dynamical systems in terms of naturally defined positive-definite operator-valued multipliers. By developing a version of `the Haagerup trick' we prove that this property is equivalent to the Haagerup property of the reduced crossed product \(C^\ast\)-algebra with respect to the canonical conditional expectation \(E\). This extends a theorem of Dong and Ruan [\textit{Z.~Dong} and \textit{Z.-J. Ruan}, Integral Equations Oper. Theory 73, No.~3, 431--454 (2012; Zbl 1263.46043)]
for discrete group actions, and implies that a given Cartan inclusion of separable \(C^\ast\)-algebras has the Haagerup property if and only if the associated Weyl groupoid has the Haagerup property in the sense of Tu [\textit{J.-L. Tu}, \(K\)-Theory 17, No. 3, 215--264 (1999; Zbl 0939.19001)]. We use the latter statement to prove that every separable \(C^\ast\)-algebra which has the Haagerup property with respect to some Cartan subalgebra satisfies the Universal Coefficient Theorem. This generalises a recent result of Barlak and Li [\textit{S.~Barlak} and \textit{X.~Li}, Adv. Math. 316, 748--769 (2017; Zbl 1382.46048)] on the UCT for nuclear Cartan pairs.Characterization of homological properties of \(\theta\)-Lau product of Banach algebrashttps://zbmath.org/1496.460752022-11-17T18:59:28.764376Z"Essmaili, Morteza"https://zbmath.org/authors/?q=ai:essmaili.morteza"Rejali, Ali"https://zbmath.org/authors/?q=ai:rejali.ali"Marzijarani, Azam Salehi"https://zbmath.org/authors/?q=ai:marzijarani.azam-salehiSummary: Let \(A\) and \(B\) be two Banach algebras and \(\theta\in\sigma(B)\). In this paper, we investigate biprojectivity and biflatness of \(\theta\)-Lau product of Banach algebras \(A\times_\theta B\). Indeed, we show that \(A\times_\theta B\) is biprojective if and only if \(A\) is contractible and \(B\) is biprojective. This generalizes some known results. Moreover, we characterize biflatness of \(\theta\)-Lau product of Banach algebras under some conditions. As an application, we give an example of biflat Banach algebras \(A\) and \(X\) such that the generalized module extension Banach algebra \(X\rtimes A\) is not biflat. Finally, we characterize pseudo-contractibility of \(\theta\)-Lau product of Banach algebras and give an affirmative answer to an open question.Multiplicative operator functions and abstract Cauchy problemshttps://zbmath.org/1496.470692022-11-17T18:59:28.764376Z"Früchtl, Felix"https://zbmath.org/authors/?q=ai:fruchtl.felixSummary: We use the duality between functional and differential equations to solve several classes of abstract Cauchy problems related to special functions. As a general framework, we investigate operator functions which are multiplicative with respect to convolution of a hypergroup. This setting contains all representations of (hyper)groups, and properties of continuity are shown; examples are provided by translation operator functions on homogeneous Banach spaces and weakly stationary processes indexed by hypergroups. Then we show that the concept of a multiplicative operator function can be used to solve a variety of abstract Cauchy problems, containing discrete, compact, and noncompact problems, including \(C_{0}\)-groups and cosine operator functions, and more generally, Sturm-Liouville operator functions.The derivatives of the heat kernel on complete manifoldshttps://zbmath.org/1496.580072022-11-17T18:59:28.764376Z"Fotiadis, Anestis"https://zbmath.org/authors/?q=ai:fotiadis.anestisThe author uses a new iteration argument to obtain the estimates for the time derivatives of the heat kernel on complete non-compact manifolds. Then the author applies these estimates to study the \(L^p\)-boundedness of the Littlewood-Paley-Stein operators on a class of locally symmetric spaces.
Reviewer: Shu-Yu Hsu (Chiayi)Riesz probability distributionshttps://zbmath.org/1496.600012022-11-17T18:59:28.764376Z"Hassairi, Abdelhamid"https://zbmath.org/authors/?q=ai:hassairi.abdelhamidFrom the cover of the book:\\
``Unique in the literature, this book provides an introductory, comprehensive and essentially self-contained exposition of the Riesz probability distribution on a symmetric cone and of its derivatives, with an emphasis on the case of the cone of positive definite symmetric matrices. \\
This distribution is an important generalization of the Wishart whose definition relies on the notion of generalized power. \\
Researchers in probability theory and harmonic analysis will find this book to be an important resource. \\
Given the connection between the Riesz probability distribution and the multivariate Gaussian samples with missing data, the book is also accessible and useful for statisticians.'' \\
\\
The preface finishes with:\\
``I hope that the book will be useful as a source of statements and applications of results in multivariate probability distributions and multivariate statistical analysis, as well as a reference to some material of harmonic analysis on symmetric cones adapted to the needs of researchers in these fields.'' \\
\\
The book is very large structured in Contents, Preface, Acknowledgment, 11 Chapters (with 45 subchapters), Bibliography (with 117 references), Index (with more than 70 items), Index of notations: \\
Chapter 1. Jordan algebras and symmetric cones -- Chapter 2. Generalized power -- Chapter 3. Riesz probability distributions -- Chapter 4. Riesz natural exponential families -- Chapter 5. Tweedie scale -- Chapter 6. Moments and constancy of regression -- Chapter 7. Beta Riesz probability distributions -- Chapter 8. Beta-Wishart distributions -- Chapter 9. Beta-hypergeometric distributions -- Chapter 10. Riesz-Dirichlet distributions -- Chapter 11. Riesz inverse Gaussian distribution \\
\\
The book can be very recommended all readers who are interested in this field.
Reviewer: Ludwig Paditz (Dresden)