We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group $\varGamma$ and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum $\varLambda\subseteq\varGamma$, the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1.

Let $\mu$ be a nonnegative Radon measure on $\mathbb{R}^d$ that satisfies the growth condition that there exist constants $C_0>0$ and $n\in(0,d]$ such that for all $x\in\mathbb{R}^d$ and $r>0$, ${\mu(B(x,\,r))\le C_0r^n}$, where $B(x,r)$ is the open ball centered at $x$ and having radius $r$. In this paper, the authors prove that if $f$ belongs to the $\textrm {BMO}$-type space $\textrm{RBMO}(\mu)$ of Tolsa, then the homogeneous maximal function $\dot{\mathcal{M}}_S(f)$ (when $\mathbb{R}^d$ is not an initial cube) and the inhomogeneous maximal function $\mathcal{M}_S(f)$ (when $\mathbb{R}^d$ is an initial cube) associated with a given approximation of the identity $S$ of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, $\dot{\mathcal{M}}_S$ and $\mathcal{M}_S$ are bounded from $\textrm{RBMO}(\mu)$ to the $\textrm {BLO}$-type space $\textrm{RBLO}(\mu)$. The authors also prove that the inhomogeneous maximal operator $\mathcal{M}_S$ is bounded from the local $\textrm {BMO}$-type space $\textrm{rbmo}(\mu)$ to the local $\textrm {BLO}$-type space $\textrm{rblo}(\mu)$.

For a positive finite measure $d\mu(\mathbf{u})$ on $\mathbb{R}^d$ normalized to satisfy $\int_{\mathbb{R}^d}d\mu(\mathbf{u})=1$, the dilated average of $f( \mathbf{x})$ is given by \[ A_tf(\mathbf{x})=\int_{\mathbb{R}^d}f(\mathbf{x}-t\mathbf{u})d\mu(\mathbf{u}). \] It will be shown that under some mild assumptions on $d\mu(\mathbf{u})$ one has the equivalence \[ \|A_tf-f\|_B\approx \inf \{ (\|f-g\|_B+t^2 \|P(D)g\|_B): P(D)g\in B\}\quad\text{for }t>0, \] where $\varphi(t)\approx \psi(t)$ means $c^{-1}\le\varphi(t)/\psi(t)\le c$, $B$ is a Banach space of functions for which translations are continuous isometries and $P(D)$ is an elliptic differential operator induced by $\mu$. Many applications are given, notable among which is the averaging operator with $d\mu(\mathbf{u})= \frac{1}{m(S)}\chi_S(\mathbf{u})d\mathbf{u}$, where $S$ is a bounded convex set in $\mathbb{R}^d$ with an interior point, $m(S)$ is the Lebesgue measure of $S$, and $\chi_S(\mathbf{u})$ is the characteristic function of $S$. The rate of approximation by averages on the boundary of a convex set under more restrictive conditions is also shown to be equivalent to an appropriate $K$-functional.

We study complex commutative Banach algebras (and, more generally, continuous inverse algebras) in which the holomorphic functions of a fixed $n$-tuple of elements are dense. In particular, we characterize the compact subsets of~$\C^n$ which appear as joint spectra of such $n$-tuples. The characterization is compared with several established notions of holomorphic convexity by means of approximation conditions.

We prove convergence of diagonal multipoint Pad\'e approximants of Stieltjes-type functions when a certain moment problem is determinate. This is used for the study of the convergence of Fourier--Pad\'e and nonlinear Fourier--Pad\'e approximants for such type of functions.

We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold ${\mathrm{Sym}}(p,{\mathbb R})^{++}\times {\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in ${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$

Continued fractions associated with $\GL_3 (\mathbf{Z})$ are introduced and applied to find fundamental units in a two-parameter family of complex cubic fields.

By means of the Pucci operator, we construct a function $u_0$, which plays an essential role in our considerations, and give the existence and regularity theorems for the bounded viscosity solutions of the generalized Dirichlet problems of second order fully nonlinear elliptic equations on the general bounded domains, which may be irregular. The approximation method, the accretive operator technique and the Caffarelli's perturbation theory are used.

Given a homogeneous elliptic partial differential operator $L$ with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain $\Omega$ in $\mathbf{R}^n$ and which belong locally to a Banach space $V$, we consider the problem of approximating in the norm of $V$ the functions in this class by ``analytic'' and ``meromorphic'' solutions of the equation $Lu=0$. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces $V$ and operators $L$. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

We construct meromorphic functions with asymptotic power series expansion in $z^{-1}$ at $\infty$ on an Arakelyan set $A$ having prescribed zeros and poles outside $A$. We use our results to prove approximation theorems where the approximating function fulfills interpolation restrictions outside the set of approximation.

The convergence behavior of best uniform rational approximations $r^\ast_{mn}$ with numerator degree~$m$ and denominator degree~$n$ to the function $|x|^\alpha$, $\alpha>0$, on $[-1,1]$ is investigated. It is assumed that the indices $(m,n)$ progress along a ray sequence in the lower triangle of the Walsh table, {\it i.e.} the sequence of indices $\{ (m,n)\}$ satisfies $$ {m\over n}\rightarrow c\in [1, \infty)\quad\hbox{as } m+ n\rightarrow\infty. $$ In addition to the convergence behavior, the asymptotic distribution of poles and zeros of the approximants and the distribution of the extreme points of the error function $|x|^\alpha - r^\ast_{mn} (x)$ on $[-1,1]$ will be studied. The results will be compared with those for paradiagonal sequences $(m=n+2[\alpha/2])$ and for sequences of best polynomial approximants.

Let $A$ be a simple $C^*$-algebra with real rank zero, stable rank one and weakly unperforated $K_0(A)$ of countable rank. We show that a monomorphism $\phi\colon C(S^2) \to A$ can be approximated pointwise by homomorphisms from $C(S^2)$ into $A$ with finite dimensional range if and only if certain index vanishes. In particular, we show that every homomorphism $\phi$ from $C(S^2)$ into a UHF-algebra can be approximated pointwise by homomorphisms from $C(S^2)$ into the UHF-algebra with finite dimensional range. As an application, we show that if $A$ is a simple $C^*$-algebra of real rank zero and is an inductive limit of matrices over $C(S^2)$ then $A$ is an AF-algebra. Similar results for tori are also obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$ for lower dimensional spaces is also studied.

We consider the shift-invariant space, $\bbbs(\Phi)$, generated by a set $\Phi=\{\phi_1,\ldots,\phi_r\}$ of compactly supported distributions on $\RR$ when the vector of distributions $\phi:=(\phi_1,\ldots,\phi_r)^T$ satisfies a system of refinement equations expressed in matrix form as $$ \phi=\sum_{\alpha\in\ZZ}a(\alpha)\phi(2\,\cdot - \,\alpha) $$ where $a$ is a finitely supported sequence of $r\times r$ matrices of complex numbers. Such {\it multiple refinable functions} occur naturally in the study of multiple wavelets. The purpose of the present paper is to characterize the {\it accuracy} of $\Phi$, the order of the polynomial space contained in $\bbbs(\Phi)$, strictly in terms of the refinement mask $a$. The accuracy determines the $L_p$-approximation order of $\bbbs(\Phi)$ when the functions in $\Phi$ belong to $L_p(\RR)$ (see Jia~[10]). The characterization is achieved in terms of the eigenvalues and eigenvectors of the subdivision operator associated with the mask $a$. In particular, they extend and improve the results of Heil, Strang and Strela~[7], and of Plonka~[16]. In addition, a counterexample is given to the statement of Strang and Strela~[20] that the eigenvalues of the subdivision operator determine the accuracy. The results do not require the linear independence of the shifts of $\phi$.

Positive, copositive, onesided and intertwining (co-onesided) polynomial and spline approximations of functions $f\in\Wp^k\mll$ are considered. Both uniform and pointwise estimates, which are exact in some sense, are obtained.