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Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces K and L the product K\times L admits a sequence \langle \mu _n\colon n\in \mathbb {N}\rangle of normalized signed measures with finite supports which converges to 0 with respect to the weak* topology of the dual Banach space C(K\times L)^*. Our approach is completely constructive—the measures \mu _n are defined by an explicit simple formula. We also show that this result generalizes the classical theorem of Cembranos [Proc. Amer. Math. Soc. 91 (1984), pp. 556–558] and Freniche [Math. Ann. 267 (1984), pp. 479–486] which states that for every infinite compact spaces K and L the Banach space C(K\times L) contains a complemented copy of the space c_0. [ABSTRACT FROM AUTHOR]

We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if (X,d,\mu) is a locally complete and separable metric measure space, then continuous functions are dense in the Newtonian space N^{1,p}(X). Here the measure \mu is Borel and is finite and positive on all metric balls. In particular, we don't assume properness of X, doubling of \mu or any Poincaré inequalities. These resolve, partially or fully, questions posed by a number of authors, including J. Heinonen, A. Björn and J. Björn. In contrast to much of the past work, our results apply to locally complete spaces X and dispenses with the frequently used regularity assumptions: doubling, properness, Poincaré inequality, Loewner property or quasiconvexity. [ABSTRACT FROM AUTHOR]

Randomized quadratures for integrating functions in Sobolev spaces of order \alpha \ge 1, where the integrability condition is with respect to the Gaussian measure, are considered. In this function space, the optimal rate for the worst-case root-mean-squared error (RMSE) is established. Here, optimality is for a general class of quadratures, in which adaptive non-linear algorithms with a possibly varying number of function evaluations are also allowed. The optimal rate is given by showing matching bounds. First, a lower bound on the worst-case RMSE of O(n^{-\alpha -1/2}) is proven, where n denotes an upper bound on the expected number of function evaluations. It turns out that a suitably randomized trapezoidal rule attains this rate, up to a logarithmic factor. A practical error estimator for this trapezoidal rule is also presented. Numerical results support our theory. [ABSTRACT FROM AUTHOR]

We obtain a complete characterization of the bounded Hausdorff operators acting on a Fock space F^p_\alpha and taking its values into a larger one F^q_\alpha,\ 0 < p \leq q \leq \infty, as well as some necessary or sufficient conditions for a Hausdorff operator to transform a Fock space into a smaller one. Some results are written in the context of mixed norm Fock spaces. Also the compactness of Hausdorff operators on a Fock space is characterized. The compactness result for Hausdorff operators on the Fock space F^\infty _\alpha is extended to more general Banach spaces of entire functions with weighted sup norms defined in terms of a radial weight and conditions for the Hausdorff operators to become p-summing are also included. [ABSTRACT FROM AUTHOR]

In solid state physics, electronic properties of crystalline materials are often inferred from the spectrum of periodic Schrödinger operators. As a consequence of Bloch's theorem, the numerical computation of electronic quantities of interest involves computing derivatives or integrals over the Brillouin zone of so-called energy bands, which are piecewise smooth, Lipschitz continuous periodic functions obtained by solving a parametrized elliptic eigenvalue problem on a Hilbert space of periodic functions. Classical discretization strategies for resolving these eigenvalue problems produce approximate energy bands that are either non-periodic or discontinuous, both of which cause difficulty when computing numerical derivatives or employing numerical quadrature. In this article, we study an alternative discretization strategy based on an ad hoc operator modification approach. While specific instances of this approach have been proposed in the physics literature, we introduce here a systematic formulation of this operator modification approach. We derive a priori error estimates for the resulting energy bands and we show that these bands are periodic and can be made arbitrarily smooth (away from band crossings) by adjusting suitable parameters in the operator modification approach. Numerical experiments involving a toy model in 1D, graphene in 2D, and silicon in 3D validate our theoretical results and showcase the efficiency of the operator modification approach. [ABSTRACT FROM AUTHOR]

We study Orlicz functions that do not satisfy the \Delta _2-condition at zero. We prove that for every Orlicz function M such that \limsup _{t\to 0}M(t)/t^p \!>0 for some p\ge 1, there exists a positive sequence T=(t_k)_{k=1}^\infty tending to zero and such that \begin{equation*} \sup _{k\in \mathbb {N}}\frac {M(ct_k)}{M(t_k)} <\infty,\text { for all }c>1, \end{equation*} that is, M satisfies the \Delta _2 condition with respect to T. Consequently, we show that for each Orlicz function with lower Boyd index \alpha _M < \infty there exists an Orlicz function N such that: (a) there exists a positive sequence T=(t_k)_{k=1}^\infty tending to zero such that N satisfies the \Delta _2 condition with respect to T, and (b) the space h_N is isomorphic to a subspace of h_M generated by one vector. We apply this result to find the maximal possible order of Gâteaux differentiability of a continuous bump function on the Orlicz space h_M(\Gamma) for \Gamma uncountable. [ABSTRACT FROM AUTHOR]

This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw to achieve an improved information complexity. We connect both approaches and propose a subsampling of structured points in an offline step. In particular, we start with quasi-Monte Carlo (QMC) points with inherent structure and stable L_2 reconstruction properties. The subsampling procedure consists of a computationally inexpensive random step followed by a deterministic procedure to further reduce the number of points while keeping its information. In these points functions (belonging to a reproducing kernel Hilbert space of bounded functions) will be sampled and reconstructed from whilst achieving state of the art error decay. Our method is dimension-independent and is applicable as soon as we know some initial quadrature points. We apply our general findings on the d-dimensional torus to subsample rank-1 lattices, where it is known that full rank-1 lattices lose half the optimal order of convergence (expressed in terms of the size of the lattice). In contrast to that, our subsampled version regains the optimal rate since many of the lattice points are not needed. Moreover, we utilize fast and memory efficient Fourier algorithms in order to compute the approximation. Numerical experiments in several dimensions support our findings. [ABSTRACT FROM AUTHOR]

We consider a family of regularized defocusing nonlinear Schrödinger (NLS) equations proposed in the context of the cubic NLS equation with a bounded dispersion relation. The time evolution is well-posed if the black soliton is perturbed by a small perturbation in the Sobolev space H^s(\mathbb {R}) with s > \frac {1}{2}. We prove that the black soliton is spectrally stable (unstable) if the regularization parameter is below (above) some explicitly specified threshold. We illustrate the stable and unstable dynamics of the perturbed black solitons by using the numerical finite-difference method. The question of orbital stability of the black soliton is left open due to the mismatch of the function spaces for the energy and momentum conservation. [ABSTRACT FROM AUTHOR]

It is shown that if the Fourier transform is a bounded map on a rearrangement-invariant space of functions on \mathbb R^n, modified by a weight, then the weight is bounded above and below and the space is equivalent to L^2. Also, if it is bounded from L^p to L^q, each modified by the same weight, then the weight is bounded above and below and 1\le p=q'\le 2. Applications prove the non-boundedness on these spaces of an operator related to the Schrödinger equation. [ABSTRACT FROM AUTHOR]

We consider quasi-isometries in real continuous functions spaces and show that such a quasi-isometry can be well approximated by an affine surjective isometry. On the other hand, we give an example of quasi-isometries of the unit ball B_H in a Hilbert space H that are far from any affine map of H and from any isometry of B_H. [ABSTRACT FROM AUTHOR]

Lei and Lin [Comm. Pure Appl. Math. 64 (2011), pp. 1297–1304] have recently given a proof of a global mild solution of the three-dimensional Navier-Stokes equations in function spaces based on the Wiener algebra. An alternative proof of existence of these solutions was then developed by Bae [Proc. Amer. Math. Soc. 143 (2015), pp. 2887–2892], and this new proof allowed for an estimate of the radius of analyticity of the solutions at positive times. We adapt the Bae proof to prove existence of the Lei-Lin solution in the spatially periodic setting, finding an improved bound for the radius of analyticity in this case. [ABSTRACT FROM AUTHOR]

We establish the existence of a scaling limit \mathcal {E}_p of discrete p-energies on the graphs approximating a generalized Sierpiński carpet for p > d_{\mathrm {ARC}}, where d_{\mathrm {ARC}} is the Ahlfors regular conformal dimension of the underlying generalized Sierpiński carpet. Furthermore, the function space \mathcal {F}_{p} defined as the collection of functions with finite p-energies is shown to be a reflexive and separable Banach space that is dense in the set of continuous functions with respect to the supremum norm. In particular, (\mathcal {E}_2, \mathcal {F}_2) recovers the canonical regular Dirichlet form constructed by Barlow and Bass [Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), pp. 225–257] or Kusuoka and Zhou [Probab. Theory Related Fields 93 (1992), pp. 169–196]. We also provide \mathcal {E}_{p}-energy measures associated with the constructed p-energy and investigate its basic properties like self-similarity and chain rule. [ABSTRACT FROM AUTHOR]

Let \mathcal {H} be a space of analytic functions on the unit ball {\mathbb {B}_d} in \mathbb {C}^d with multiplier algebra \mathrm {Mult}(\mathcal {H}). A function f\in \mathcal {H} is called cyclic if the set [f], the closure of \{\varphi f: \varphi \in \mathrm {Mult}(\mathcal {H})\}, equals \mathcal {H}. For multipliers we also consider a weakened form of the cyclicity concept. Namely for n\in \mathbb {N}_0 we consider the classes \begin{equation*} \mathcal {C}_n(\mathcal {H})=\{\varphi \in \mathrm {Mult}(\mathcal {H}):\varphi \ne 0, [\varphi ^n]=[\varphi ^{n+1}]\}. \end{equation*} Many of our results hold for N:th order radially weighted Besov spaces on {\mathbb {B}_d}, \mathcal {H}= B^N_\omega, but we describe our results only for the Drury-Arveson space H^2_d here. Letting \mathbb {C}_{stable}[z] denote the stable polynomials for {\mathbb {B}_d}, i.e. the d-variable complex polynomials without zeros in {\mathbb {B}_d}, we show that \begin{align*} &\text { if } d \text { is odd, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d-1}{2}}(H^2_d), \text { and }\\ &\text { if } d \text { is even, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d}{2}-1}(H^2_d). \end{align*} For d=2 and d=4 these inclusions are the best possible, but in general we can only show that if 0\le n\le \frac {d}{4}-1, then \mathbb {C}_{stable}[z]\nsubseteq \mathcal {C}_n(H^2_d). For functions other than polynomials we show that if f,g\in H^2_d such that f/g\in H^\infty and f is cyclic, then g is cyclic. We use this to prove that if f,g extend to be analytic in a neighborhood of \overline {{\mathbb {B}_d}}, have no zeros in {\mathbb {B}_d}, and the same zero sets on the boundary, then f is cyclic in \in H^2_d if and only if g is. Furthermore, if the boundary zero set of f\in H^2_d\cap C(\overline {{\mathbb {B}_d}}) embeds a cube of real dimension \ge 3, then f is not cyclic in the Drury-Arveson space. [ABSTRACT FROM AUTHOR]

Characterization of simultaneous similarity for commuting m- tuples of operators is an open problem even in finite-dimensional spaces; known as "A wild problem in linear algebra". In this paper we offer a criterion for simultaneous similarity of m-tuples of k-cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m-tuples of commuting square matrices. [ABSTRACT FROM AUTHOR]

It is proved that for every compact metric space K there exists a Banach space X whose Calkin algebra \mathcal {L}(X)/\mathcal {K}(X) is homomorphically isometric to C(K). This is achieved by appropriately modifying the Bourgain-Delbaen \mathscr {L}_\infty-space of Argyros and Haydon in such a manner that sufficiently many diagonal operators on this space are bounded. [ABSTRACT FROM AUTHOR]

For positive C_0-semigroups S and T on a Banach lattice such that S(t) \le T(t) for all times t, we prove that analyticity of T implies analyticity of S. This answers an open problem posed by Arendt in 2004. Our proof is based on a spectral theoretic argument: we apply spectral theory of positive operators to multiplication operators that are induced by S and T on a vector-valued function space. [ABSTRACT FROM AUTHOR]

This paper analyzes representations of continuous piecewise linear functions with infinite width, finite cost shallow neural networks using the rectified linear unit (ReLU) as an activation function. Through its integral representation, a shallow neural network can be identified by the corresponding signed, finite measure on an appropriate parameter space. We map these measures on the parameter space to measures on the projective n-sphere cross \mathbb {R}, allowing points in the parameter space to be bijectively mapped to hyperplanes in the domain of the function. We prove a conjecture of Ongie et al. [ A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case , arXiv, 2019] that every continuous piecewise linear function expressible with this kind of infinite width neural network is expressible as a finite width shallow ReLU neural network. [ABSTRACT FROM AUTHOR]

In this note, we prove that the following function space with absolutely convergent Fourier series \[ F_d≔\left \{ f\in L^2([0,1)^d)\middle | \|f\|≔\sum _{\boldsymbol {k}\in {\mathbb {Z}}^d}|\hat {f}({\boldsymbol {k}}) |\max \left (1,\min _{j\in \operatorname {supp}({\boldsymbol {k}})}\log |k_j|\right)<\infty \right \}\] with \hat {f}({\boldsymbol {k}}) being the {\boldsymbol {k}}-th Fourier coefficient of f and \operatorname {supp}({\boldsymbol {k}}) ≔\{j\!\in \!\{1,\ldots,d\}\mid k_j\neq 0\} is polynomially tractable for multivariate integration in the worst-case setting. Here polynomial tractability means that the minimum number of function evaluations required to make the worst-case error less than or equal to a tolerance \varepsilon grows only polynomially with respect to \varepsilon ^{-1} and d. It is important to remark that the function space F_d is unweighted , that is, all variables contribute equally to the norm of functions. Our tractability result is in contrast to those for most of the unweighted integration problems studied in the literature, in which polynomial tractability does not hold and the problem suffers from the curse of dimensionality. Our proof is constructive in the sense that we provide an explicit quasi-Monte Carlo rule that attains a desired worst-case error bound. [ABSTRACT FROM AUTHOR]

Recently, Kayumov [Lobachevskii J. Math. 38 (2017), pp. 466–468] obtained a sharp estimate for the n-th truncated area functional for normalized functions in the Bloch space for n\le 5 and then, together with Wirths [Lobachevskii J. Math. 40 (2019), pp. 1319–1323], extended the result for n=6. We prove that for the functions with non-negative Taylor coefficients, the same sharp estimate is valid for all n. For arbitrary functions, we obtain an estimate that is asymptotically of the same order but slightly larger (roughly by a factor of 4/e). We also consider related weighted estimates for functionals involving the powers n^t, t>0, and show that the exponent t=1 represents the critical case for the expected sharp estimate. [ABSTRACT FROM AUTHOR]

The heat semigroup on discrete hypercubes is well-known to be contractive over L_p-spaces for 1

In this paper we characterize the nonlinear superposition operators mapping the space \mathcal {Q}_s into the Dirichlet-type space \mathcal {D}^p_\alpha and answer basically the remaining open cases provided by Galanopoulos, Girela, and Auxiliadora Márquez [J. Math. Anal. Appl. 463 (2018), pp. 659–680]. Meanwhile, we give some sufficient-necessary conditions for the superposition operators between \mathcal {Q}_s spaces in all cases. [ABSTRACT FROM AUTHOR]

Let \Omega _i\subset \mathbb {R}^{n_i}, i=1,\ldots,m, be given domains. In this article, we study the low-rank approximation with respect to L^2(\Omega _1\times \dots \times \Omega _m) of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare Griebel and Harbrecht [IMA J. Numer. Anal. 34 (2014), pp. 28–54] and Griebel and Harbrecht [IMA J. Numer. Anal. 39 (2019), pp. 1652–1671], we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension. [ABSTRACT FROM AUTHOR]

We show that the Carathéodory space of place functions on the free product of two Boolean algebras is Riesz isomorphic with Fremlin's Archimedean Riesz space tensor product of their respective Carathéodory spaces of place functions. We provide a solution to Fremlin's problem 315Y(f) [ Measure Theory , Torres Fremlin, Colchester, 2004] concerning completeness in the free product of Boolean algebras by applying our results on the Archimedean Riesz space tensor product to Carathéodory spaces of place functions. [ABSTRACT FROM AUTHOR]

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. These functions appear in various fields of science and technology, in particular they were discovered in quantum mechanics in the context of weak values introduced by Y. Aharonov and collaborators. The evolution problem of superoscillatory functions as initial conditions for the Schrödinger equation is intensively studied nowadays and the supershift property of the solution of Schrödinger equation encodes the persistence of superoscillatory phenomenon during the evolution. In this paper, we prove that the evolution of a superoscillatory initial datum for spinning particles in a magnetic field has the supershift property. Our techniques are based on the exact propagator of spinning particles, the associated infinite order differential operators and their continuity on suitable spaces of entire functions with growth conditions. [ABSTRACT FROM AUTHOR]

We prove that the Lipschitz-free space over a Banach space X of density \kappa, denoted by \mathcal {F}(X), is linearly isomorphic to its \ell _1-sum \left (\bigoplus _{\kappa }\mathcal {F}(X)\right)_{\ell _1}. This provides an extension of a previous result from Kaufmann in the context of non-separable Banach spaces. Further, we obtain a complete classification of the spaces of real-valued Lipschitz functions that vanish at 0 over a \mathcal {L}_p-space. More precisely, we establish that, for every 1\leq p\leq \infty, if X is a \mathcal {L}_p-space of density \kappa, then \mathrm {Lip}_0(X) is either isomorphic to \mathrm {Lip}_0(\ell _p(\kappa)) if p<\infty, or \mathrm {Lip}_0(c_0(\kappa)) if p=\infty. [ABSTRACT FROM AUTHOR]

Let \mathcal {M} be a semifinite von Nemann algebra equipped with an increasing filtration (\mathcal {M}_n)_{n\geq 1} of (semifinite) von Neumann subalgebras of \mathcal {M}. For 0

0, \[ \big (\mathsf {h}_\Phi ^c(\mathcal {M}), \mathsf {h}_\infty ^c(\mathcal {M})\big)_{\theta, r}=\mathsf {h}_{\Phi _0, r}^c(\mathcal {M}), \] where \mathsf {h}_{\Phi _0,r}^c(\mathcal {M}) is the noncommutative column Hardy space associated with the Orlicz-Lorentz space L_{\Phi _0,r}. [ABSTRACT FROM AUTHOR]

Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function f in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type f=\varphi g, where g is cyclic, \varphi is a contractive multiplier, and \|f\|=\|g\|. In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization. [ABSTRACT FROM AUTHOR]

Let K and S be locally compact Hausdorff spaces and let T be a real linear isomorphism of an extremely regular subspace A of C_{0}(K) into C_{0}(S) satisfying \|T \| \ \|T^{-1}\|<2. Put L=T \ \|T^{-1}\| and pick 0 \leq \varepsilon <1 such that \|T \| \ \|T^{-1}\|=1+\varepsilon. We prove that there exist a subset S_0 of S, a proper map \varphi of S_{0} onto K and a map \lambda :S_{0} \to \{-1, 1\} such that \begin{equation*} |Lf(s)-\lambda (s) f(\varphi (s))| \leq \varepsilon \|f\|, \ \forall s \in S_{0} \ \text {and} \ f \in A. \end{equation*} For \varepsilon < 1/2 the maps \varphi and \lambda are continuous, and this approximation of L by such a weighted composition operator improves the well-known Holsztyński theorem [Studia Math. 26 (1966), pp. 133–136], the case where \varepsilon =0, K and S are compact and A=C_{0}(K). The approximation of L in the general case where 0 \leq \varepsilon <1 is an extension of a Benyamini's result [Proc. Amer. Math. Soc. 83 (1981), pp. 479–485] which allowed us to strengthen a classical Jarosz's theorem [Proc. Amer. Math. Soc. 90 (1984), pp. 373–377] by proving that K is a continuous image not just of a subset of S but of a locally compact subset of S. Moreover, when K is compact, it is a continuous image not just a closed subset of S but of a compact subset of S, namely the set S_0 having the above properties. [ABSTRACT FROM AUTHOR]

We show that the norm in the Hardy space H^p satisfies \begin{equation} \|f\|_{H^p}^p\asymp \int _0^1M_q^p(r,f')(1-r)^{p\left (1-\frac 1q\right)}\,dr+|f(0)|^p\tag {\dagger } \end{equation} for all univalent functions provided that either q\ge 2 or \frac {2p}{2+p}

This paper deals with the long-term behavior of positive operator semigroups on spaces of bounded functions and of signed measures, which have applications to parabolic equations with unbounded coefficients and to stochastic analysis. The main results are a Tauberian type theorem characterizing the convergence to equilibrium of strongly Feller semigroups and a generalization of a classical convergence theorem of Doob. None of these results requires any kind of time regularity of the semigroup. [ABSTRACT FROM AUTHOR]

We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich [J. Approx. Theory 240 (2019), pp. 96–113] shows that rank-1 lattice rules with a randomly chosen number of points and good generating vector achieve almost the optimal order of the randomized error in weighted Korobov spaces, and moreover, that the error is bounded independently of the dimension if the weight parameters, \gamma _j, satisfy the summability condition \sum _{j=1}^{\infty }\gamma _j^{1/\alpha }<\infty, where \alpha is a smoothness parameter. The argument is based on the existence result that at least half of the possible generating vectors yield almost the optimal order of the worst-case error in the same function spaces. In this paper we provide a component-by-component construction algorithm of such randomized rank-1 lattice rules, without any need to check whether the constructed generating vectors satisfy a desired worst-case error bound. Similarly to the above-mentioned work, we prove that our algorithm achieves almost the optimal order of the randomized error and that the error bound is independent of the dimension if the same condition \sum _{j=1}^{\infty }\gamma _j^{1/\alpha }<\infty holds. We also provide analogous results for tent-transformed lattice rules for weighted half-period cosine spaces and for polynomial lattice rules in weighted Walsh spaces, respectively. [ABSTRACT FROM AUTHOR]

Let \varphi be a function in the Hardy space H^2(\mathbb {T}^d). The associated (small) Hankel operator \mathbf {H}_\varphi is said to have minimal norm if the general lower norm bound \|\mathbf {H}_\varphi \| \geq \|\varphi \|_{H^2(\mathbb {T}^d)} is attained. Minimal norm Hankel operators are natural extremal candidates for the Nehari problem. If d=1, then \mathbf {H}_\varphi has minimal norm if and only if \varphi is a constant multiple of an inner function. Constant multiples of inner functions generate minimal norm Hankel operators also when d\geq 2, but in this case there are other possibilities as well. We investigate two different classes of symbols generating minimal norm Hankel operators and obtain two different refinements of a counter-example due to Ortega-Cerdà and Seip. [ABSTRACT FROM AUTHOR]

We provide explicit sequence space representations for the test function and distribution spaces occurring in the Valdivia-Vogt structure tables by making use of Wilson bases generated by compactly supported smooth windows. Furthermore, we show that these kind of bases are common unconditional Schauder bases for all separable spaces occurring in these tables. Our work implies that the Valdivia-Vogt structure tables for test functions and distributions may be interpreted as one large commutative diagram. [ABSTRACT FROM AUTHOR]

Let C_b(X) denote the Banach space of all bounded continuous real-valued functions on a completely regular Hausdorff space X. We characterize nuclear tight operators T:C_b(X)\rightarrow E in terms of their representing Radon vector measures. As an application, it is shown that some natural kernel operators on C_b(X) are tight and nuclear. [ABSTRACT FROM AUTHOR]

We investigate (uniform) mean ergodicity of weighted composition operators on the space of smooth functions and the space of distributions, both over an open subset of the real line. Among other things, we prove that a composition operator with a real analytic diffeomorphic symbol is mean ergodic on the space of distributions if and only if it is periodic with period 2. Our results are based on a characterization of mean ergodicity in terms of Cesàro boundedness and a growth property of the orbits for operators on Montel spaces which is of independent interest. [ABSTRACT FROM AUTHOR]

We provide an abstract approach to approximation with a wide range of regularity classes X in spaces of pseudocontinuable functions K^p_\vartheta, where \vartheta is an inner function and p>0. More precisely, we demonstrate a general principle, attributed to A. Aleksandrov, which asserts that if a certain linear manifold X is dense in K^{q}_\vartheta for some q>0, then X is in fact dense in K^p_{\vartheta } for all p>0. Moreover, for a rich class of Banach spaces of analytic functions X, we describe the precise mechanism that determines when X is dense in a certain space of pseudocontinuable functions. As a consequence, we obtain an extension of Aleksandrov's density theorem to the class of analytic functions with uniformly convergent Taylor series. [ABSTRACT FROM AUTHOR]

We prove two sharp estimates for the subspace of a standard weighted Bergman space that consists of functions vanishing at a given point (with prescribed multiplicity). [ABSTRACT FROM AUTHOR]

Given the germ of an analytic function on affine space with a smooth critical locus, we prove that the constancy of the reduced cohomology of the Milnor fiber in lowest possible non-trivial degree off a codimension two subset of the critical locus implies that the vanishing cycles are concentrated in lowest degree and are constant. [ABSTRACT FROM AUTHOR]

Let X be a rearrangement-invariant Banach function space on the unit circle \mathbb {T} and let H[X] be the abstract Hardy space built upon X. We prove that if the Cauchy singular integral operator (Hf)(t)=\frac {1}{\pi i}\int _{\mathbb {T}}\frac {f(\tau)}{\tau -t}\,d\tau is bounded on the space X, then the norm, the essential norm, and the Hausdorff measure of non-compactness of the operator aI+bH with a,b\in \mathbb {C}, acting on the space X, coincide. We also show that similar equalities hold for the backward shift operator (Sf)(t)=(f(t)-\widehat {f}(0))/t on the abstract Hardy space H[X]. Our results extend those by Krupnik and Polonskiĭ [Funkcional. Anal. i Priloz̆en. 9 (1975), pp. 73-74] for the operator aI+bH and by the second author [J. Funct. Anal. 280 (2021), p. 11] for the operator S. [ABSTRACT FROM AUTHOR]

A topological space X is defined to have a neighborhood P-base at any x\in X from some partially ordered set (poset) P if there exists a neighborhood base (U_p[x])_{p\in P} at x such that U_p[x]\subseteq U_{p'}[x] for all p\geq p' in P. We prove that a compact space is countable, hence metrizable, if it has countable scattered height and a \mathcal {K}(M)-base for some separable metric space M. Banakh [Dissertationes Math. 538 (2019), p. 141] gives a positive answer to Problem 8.6.8. Let A(X) be the free Abelian topological group on X. It is shown that if Y is a retract of X such that the free Abelian topological group A(Y) has a P-base and A(X/Y) has a Q-base, then A(X) has a P\times Q-base. Also if Y is a closed subspace of X and A(X) has a P-base, then A(X/Y) has a P-base. It is shown that any Fréchet-Urysohn topological group with a \mathcal {K}(M)-base for some separable metric space M is first-countable, hence metrizable. And if P is a poset with calibre (\omega _1, \omega) and G is a topological group with a P-base, then any precompact subset in G is metrizable, hence G is strictly angelic. Applications in function spaces C_p(X) and C_k(X) are discussed. We also give an example of a topological Boolean group of character \leq \mathfrak {d} such that the precompact subsets are metrizable but G doesn't have an \omega ^\omega-base if \omega _1<\mathfrak {d}. Gabriyelyan, Kakol, and Liederman [Fund. Math. 229 (2015), pp. 129–158] give a consistent negative answer to Problem 6.5. [ABSTRACT FROM AUTHOR]

We obtain closed expressions for weighted orthogonal polynomials and optimal approximants associated with the function ƒ(z) = 1− 1/√2 (z1 + z2) and a scale of Hilbert function spaces in the unit 2-ball having reproducing kernel (1−⟨z,w⟩)−γ, γ > 0. Our arguments are elementary but do not rely on reduction to the one-dimensional case. [ABSTRACT FROM AUTHOR]

Of concern are some simple criteria about the convergence of sequences of positive linear operators and functionals in the framework of spaces of bounded functions which are continuous on a given subset of their domain. Among other things some applications concerning the behaviour of the iterates of Bernstein operators defined both on [0,1] and on the d-dimensional simplex and hypercube (d ≥ 1) are discussed. A final section treats the behaviour of integrated arithmetic means with respect to a probability Borel measure on a convex compact subset K of a locally convex space. As a consequence a general weak law of large numbers for sequences of K-valued random variables is derived. [ABSTRACT FROM AUTHOR]

In this note two results are established for energy functionals that are given by the integral of W(x,∇u(x)) over Ω ⊂ Rn with ∇ u ∈ BMO(Ω RN×n), the space of functions of Bounded Mean Oscillation of John and Nirenberg. A version of Taylor's theorem is first shown to be valid provided the integrand W has polynomial growth. This result is then used to demonstrate that every Lipschitz-continuous solution of the corresponding Euler-Lagrange equations at which the second variation of the energy is uniformly positive is a strict local minimizer of the energy in W1,BMO(Ω RN), the subspace of the Sobolev space W1,1(Ω RN) for which the weak derivative ∇ u ∈ BMO(Ω RN×n). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the strict inclusion relation associated with intersections and unions of a general family of function spaces. We answer partially a question left open in Korhonen and Rättyä [Comput. Methods Funct. Theory 5 (2005), pp. 459–469]. [ABSTRACT FROM AUTHOR]

We investigate the problem of classifying the Banach spaces Lip0(C(K)) for Hausdorff compacta K. In particular, sufficient conditions are established for a space Lip0(C(K)) to be isomorphic to Lip0(c0(Γ)) for some uncountable set Γ. [ABSTRACT FROM AUTHOR]

We analyze rates of convergence for quasi-Monte Carlo (QMC) integration for Bayesian inversion of linear, elliptic partial differential equations with uncertain input from function spaces. Adopting a Riesz or Schauder basis representation of the uncertain inputs, function space priors are constructed as product measures on spaces of (sequences of) coefficients in the basis representations. The numerical approximation of the posterior expectation, given data, then amounts to a high- or infinite-dimensional numerical integration problem. We consider in particular so-called Besov priors on the admissible uncertain inputs. We extend the QMC convergence theory from the Gaussian case, and establish sufficient conditions on the uncertain inputs for achieving dimension-independent convergence rates greater than 1/2 of QMC integration with randomly shifted lattice rules. We apply the theory to a concrete class of linear, second order elliptic boundary value problems with log-Besov uncertain diffusion coefficient. [ABSTRACT FROM AUTHOR]

One of the fundamental results of ergodic optimisation asserts that for any dynamical system on a compact metric space X and for any Banach space of continuous real-valued functions on X which embeds densely in C(X) there exists a residual set of functions in that Banach space for which the maximising measure is unique. We extend this result by showing that this residual set is additionally prevalent, answering a question of J. Bochi and Y. Zhang. [ABSTRACT FROM AUTHOR]

First we prove that if a separable Banach space X contains an isometric copy of an infinite-dimensional space A(S) of affine continuous functions on a Choquet simplex S, then its dual X* lacks the weak* fixed point property for nonexpansive mappings. Then, we show that the dual of a separable L1-predual X fails the weak * fixed point property for nonexpansive mappings if and only if X has a quotient isometric to some infinite-dimensional space A(S). Moreover, we provide an example showing that ''quotient'' cannot be replaced by ''subspace''. Finally, it is worth mentioning that in our characterization the space A(S) cannot be substituted by any space C(K) of continuous functions on a compact Hausdorff K. [ABSTRACT FROM AUTHOR]

We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space X, the function space Cp(X) is not countable dense homogeneous. This answers a question posed recently by R. Hernández-Gutiérrez. We also conclude that, for any infinite-dimensional Banach space E (dual Banach space E*), the space E equipped with the weak topology (E* with the weak * topology) is not countable dense homogeneous. We generalize some results of Hrušák, Zamora Avilés, and Hernández-Gutiérrez concerning countable dense homogeneous products. [ABSTRACT FROM AUTHOR]

We prove that a Banach space of continuous functions C(K) has a renorming that is uniformly rotund in every direction (URED) if and only if the compact space K supports a strictly positive measure. [ABSTRACT FROM AUTHOR]