A Note on the Quasi-Best Approximation Constant

Global feedback stabilizability results are derived for nonautonomous coupled systems arising from the linearization around a given time-dependent trajectory of FitzHugh-CNagumo type systems. The feedback is explicit and is based on suitable oblique (nonorthogonal) projections in Hilbert spaces. The actuators are, typically, a finite number of indicator functions and act only in the parabolic equation. Subsequently, local feedback stabilizability to time-dependent trajectories results are derived for nonlinear coupled parabolic-ODE systems of the FitzHugh-CNagumo type. Simulations are presented showing the stabilizing performance of the feedback control.

Numerical simulation of a mixing layer in an adaptive wavelet basis

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

Let H be a Hilbert space, W a closed subspace of H, and Q a (linear bounded) projection from H onto W with null space M⊥. We study decompositions like Qf = ∑n∊ℕ 〈 f, hn〉 fn, where {fn}n∊ℕ and {hn}n∊ℕ are frames for the subspaces W and M, respectively. This type of decompositions corresponds to sampling formulae. By considering the synthesis operator F (resp. H) of the sequence {fn}n∊ℕ (resp. {hn}n∊ℕ), the formula above can be expressed as the factorization Q = FH*. We study different properties of these factorizations and decompositions of oblique and orthogonal projections. Several characterizations of these decompositions are presented. By means of an operator inequality for positive operators, we get a result which minimizes the norm of F — H.

On solution sets of nonlinear equations with nonsmooth operators in Hilbert space and the quasi–solution method

We study the perturbation property of best approximation to a set defined by an abstract nonlinear constraint system. We show that, at a normal point, the perturbation property of best approximation is equivalent to an equality expressed in terms of normal cones. This equality is related to the strong conical hull intersection property. Our results generalize many known results in the literature on perturbation property of best approximation established for a set defined by a finite system of linear/nonlinear inequalities. The connection to minimization problem is considered.

Tensor Norms and Operator Ideals

Uniqueness of the maximal extension of a monotone operator

Quantum mechanics of Klein–Gordon fields I: Hilbert Space, localized states, and chiral symmetry

The center of the algebra of continuous functions on the quantum group \(\mathrm{SU}_q(2)\) is determined as well as centers of other related algebras. Several other results concerning this quantum group are given with direct proofs based on concrete realization of these algebras as algebras of operators on a Hilbert space.

A suboperator is a bounded linear transformation from a subspace of a Hilbert space into the whole space. The main purpose of this work is to raise some questions about the extension properties of suboperators (e.g., which are sub-positive?), and, in particular, to pose the problem of characterizing subnormal suboperators. The subdiscrete topology of operators on a Hilbert space is the topology of discrete convergence (pointwise ultimate equality). The problem of finding a simple proof of Bishop’s theorem (the set of subnormal operators is strongly closed) is reduced to the problem of finding a direct proof of the following assertion: if every finite restriction of an operator T is nearly subnormal, then every finite restriction of T is subnormal.

Let X be the complement of a a-Z-set in a locally compact separable ANR.It is proved that X satisfies the discrete n-cells property for each nonnegative integer n if and only if X satisfies the discrete approximation property.As a consequence, Hubert space manifolds that arise as complements of boundary sets in Hubert cube manifolds are characterized in terms of their homological structure coupled with a minimal amount of general positioning.1. Introduction.H. Torunczyk characterizes those infinite-dimensional manifolds modeled on the Hubert cube and Hilbert space entirely in terms of geometric general position properties.Hilbert cube manifolds are precisely those locally compact ANR's X for which arbitrary pairs of maps of the Hilbert cube into A" are approximable by maps whose images are disjoint [Toi].This general position property easily is seen to be equivalent to the disjoint -cells property (maps of the -cell into X are approximable by maps with disjoint images) holding simultaneously for all nonnegative integers n.Hilbert space manifolds are precisely those topologically complete separable ANR's X for which arbitrary sequences of maps of the Hilbert cube into X are strongly approximable by maps whose images form a discrete family [To2, BBMW].This latter general position property is known as the discrete approximation property, and the question naturally arises as to whether or not this property is equivalent to the analogous -dimensional properties holding simultaneously for all nonnegative integers .Recently, M. Bestvina, J. Mogilski, J. Walsh, and the author constructed examples of topologically complete separable ANR's that satisfy the discrete -cells property for each nonnegative integer , but fail to satisfy the discrete approximation property [BBMW].(Precise definitions of the aforementioned terms are found in 2.)None of the examples constructed in [BBMW] arise as complements of a-Z-sets in locally compact separable ANR's, a common source of Hilbert space manifolds.Our Main Result appears in 3 and shows that no such examples exist in this setting, henceforth referred to as the "boundary set setting".See [Cu].Main Result.Let F be a a-Z-set in the locally compact separable ANR Y. Then X = Y -F satisfies the discrete n-cells property for each nonnegative integer n if and only if X satisfies the discrete approximation property.

Greedy Algorithms and Approximation Properties for Frames in Hilbert Spaces

We analyze the complexity of the sparse-grid interpolation and sparse-grid quadrature of countably-parametric functions which take values in separable Banach spaces with unconditional bases. Assuming a suitably quantified holomorphic dependence on the parameters, we establish dimension-independent convergence rate bounds for sparse-grid approximation schemes. Analogous results are shown in the case that the parametric families are obtained as approximate solutions of corresponding parametric-holomorphic, nonlinear operator equations as considered in [A. Cohen and A. Chkifa and Ch. Schwab: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, J. Math. Pures Appl. 103 (2015) 400–428], for example by means of stable, finite-dimensional approximations. We discuss in detail nonlinear Petrov–Galerkin projections. Error and convergence rate bounds for constructive and explicit multilevel, sparse tensor approximation schemes combining sparse-grid interpolation in the parameter space and general, multilevel discretization schemes in the physical domain are proved. The present results unify and generalize earlier works in terms of the admissible multilevel approximations in the physical domain (comprising general stable Petrov–Galerkin and discrete Petrov–Galerkin schemes, collocation and stable domain approximations) and in terms of the admissible operator equations (comprising smooth, nonlinear locally well-posed operator equations). Additionally, a novel computational strategy to localize sequences of nested index sets for the anisotropic Smolyak interpolation in parameter space is developed which realizes best [Formula: see text]-term benchmark convergence rates. We also consider Smolyak-type quadratures in this general setting, for which we establish improved convergence rates based on cancellations in the integrands’ gpc expansions by symmetries of quadratures and the probability measure [J.Z̃ech and Ch.S̃chwab: Convergence rates of high dimensional Smolyak quadrature, Report 2017-27, SAM ETH Zürich (2017)]. Several examples illustrating the abstract theory include domain uncertainty quantification (UQ) for general, linear, second-order, elliptic advection–reaction–diffusion equations on polygonal domains, where optimal convergence rates of FEM are known to require local mesh refinement near corners. Further applications of the presently developed theory comprise evaluations of posterior expectations in Bayesian inverse problems.

Numerical experiments involving galerkin and collocation methods for linear integral equations of the first kind