The main result is as follows. Let X be a Banach space and let Y be a closed subspace of X. Assume that the pair $(X*,Y^{\perp })$ has the λ-bounded approximation property. Then there exists a net $\left(S_{\alpha }\right)$ of finite-rank operators on X such that $S_{\alpha }\left(Y\right)\subset Y$ and $\left|\right|S_{\alpha }\left|\right|\le \lambda$ for all α, and $\left(S_{\alpha }\right)$ and $\left(S{*}_{\alpha }\right)$ converge pointwise to the identity operators on X and X*, respectively. This means that the pair (X,Y) has the λ-bounded duality approximation property.

The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E\subset {ℂ}^N$. We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function ${\Phi }_E$. Moreover, we show that one of these extremal-like functions is equal to ${\Phi }_E$ if E is a nonpluripolar set with $li{m}_{n\to \infty }Mₙ{\left(E\right)}^{1/n}=1$ where $Mₙ\left(E\right):=sup{\left|\right|\left|gradP\right|\left|\right|}_E/{\left|\right|P\left|\right|}_E$, the supremum is taken over all polynomials P of N variables...

For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if $K\subset {\sum }_{n=1}^{\infty }\alpha ₙxₙ:\alpha ₙ\in Ball\left(l_{p^{\text{'}}}\right)$, where p’ = p/(p-1) and $xₙ\in l_p^s\left(X\right)$. An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering $xₙ\in l_p^w\left(X\right)$. It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of $l_p$ in a particular manner. The normed operator ideals $(K_p,{\kappa }_p)$ of p-compact operators and $(W_p,{\omega }_p)$ of weakly p-compact operators, arising from these factorizations,...

We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form $$\mathrm{div}\left(a\left(\right|\nabla u\left(x\right)\left|\right)\nabla u\left(x\right)\right)+f\left(u\left(x\right)\right)=0.$$ Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in ${ℝ}^2$ and ${ℝ}^3$ and of the Bernstein problem on the flatness of minimal area graphs in ${ℝ}^3$. A one-dimensional symmetry result in the half-space is also obtained as a byproduct...

We prove the existence of an effectively computable integer polynomial P(x,t₀,...,t₅) having the following property. Every continuous function $f:{ℝ}^s\to ℝ$ can be approximated with arbitrary accuracy by an infinite sum ${\sum }_{r=1}^{\infty }{H}_r(x₁,...,x_s)\in C^{\infty }\left({ℝ}^s\right)$ of analytic functions $H_r$, each solving the same system of universal partial differential equations, namely $P(x_{\sigma };H_r,\partial H_r/\partial x_{\sigma },...,\partial ⁵H_r/\partial x{⁵}_{\sigma }⁵)=0$ (σ = 1,..., s).

CONTENTSChapter 0...............................................................................................................................................................................5   0.1. Introduction..................................................................................................................................................................5   0.2. Preliminary results.......................................................................................................................................................9Chapter...

Let d be a positive integer and α a real algebraic number of degree d + 1. Set $\alpha ̲:=(\alpha ,\alpha ²,...,{\alpha }^d)$. It is well-known that $c\left(\alpha ̲\right):=limin{f}_{q\to \infty }{q}^{1/d}·\left|\right|q\alpha ̲\left|\right|>0$, where ||·|| denotes the distance to the nearest integer. Furthermore, $c\left(\alpha ̲\right)n^{-1/d}\le c\left(n\alpha ̲\right)\le nc\left(\alpha ̲\right)$ for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that $c\left(n\alpha ̲\right)\le C{n}^{-1/d}$ for any integer n ≥ 1.

A point $({\xi }_1,{\xi }_2)$ with coordinates in a subfield of $ℝ$ of transcendence degree one over $\mathbb{Q}$, with $1,{\xi }_1,{\xi }_2$ linearly independent over $\mathbb{Q}$, may have a uniform exponent of approximation by elements of ${\mathbb{Q}}^2$ that is strictly larger than the lower bound $1/2$ given by Dirichlet’s box principle. This appeared as a surprise, in connection to work of Davenport and Schmidt, for points of the parabola $\{(\xi ,{\xi }^2);\xi \in ℝ\}$. The goal of this paper is to show that this phenomenon extends to all real conics defined over $\mathbb{Q}$, and that the largest...

It is shown that there is a subspace $Z_q$ of ${\ell }_q$ for $1<q<2$ which is isomorphic to ${\ell }_q$ such that ${\ell }_q/Z_q$ does not have the approximation property. On the other hand, for $2<p<\infty$ there is a subspace $Y_p$ of ${\ell }_p$ such that $Y_p$ does not have the approximation property (AP) but the quotient space ${\ell }_p/Y_p$ is isomorphic to ${\ell }_p$. The result is obtained by defining random “Enflo-Davie spaces” $Y_p$ which with full probability fail AP for all $2<p\le \infty$ and have AP for all $1\le p\le 2$. For $1<p\le 2$, $Y_p$ are isomorphic to ${\ell }_p$.

We introduce the definition of $p$-limited completely continuous operators, $1\le p<\infty$. The question of whether a space of operators has the property that every $p$-limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$-limited completely continuous evaluation operators.

We study the following nonlinear method of approximation by trigonometric polynomials. For a periodic function f we take as an approximant a trigonometric polynomial of the form $Gₘ\left(f\right):={\sum }_{k\in \Lambda }f̂\left(k\right)e^{i(k,x)}$, where $\Lambda \subset {\mathbb{Z}}^d$ is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in...

In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators $T$ for which there exists a fundamental decomposition $𝕂={𝕂}^{+}\oplus {𝕂}^{-}$ of the Krein space $𝕂$ with ${𝕂}^{+}$ and ${𝕂}^{-}$ invariant under $T$.

We consider the Kantorovich and the Durrmeyer type modifications of the generalized Favard operators and we prove inverse approximation theorems for functions $f$ such that $w_{2m}f\in L^p(R)$, where $1\le p\le \mathrm{\infty }$ and $w_{2m}(x)={(1+x^{2m})}^{-1}$, $m\in N_0$.

Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define $X_M$ to be X ∩ M with topology generated by $U\cap M:U\in \cap M$. Suppose $X_M$ is homeomorphic to the irrationals; must $X=X_M$? We have partial results. We also answer a question of Gruenhage by showing that if $X_M$ is homeomorphic to the “Long Cantor Set”, then $X=X_M$.

We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as $T_{\epsilon }\left(x\right):={\sum }_{j\in D_{\epsilon }\left(x\right)}e{*}_j\left(x\right)e_j$, where $D_{\epsilon }\left(x\right):=j:|e{*}_j\left(x\right)|\ge \epsilon$. We study a generalized version of $T_{\epsilon }$ that we call the weak thresholding approximation. We modify the $T_{\epsilon }\left(x\right)$ in the following way. For ε > 0, t ∈ (0,1) we set $D_{t,\epsilon }\left(x\right):=j:t\epsilon \le |e{*}_j\left(x\right)|<\epsilon$ and consider...

Let ${𝕋}_n$ be the space of all trigonometric polynomials of degree not greater than $n$ with complex coefficients. Arestov extended the result of Bernstein and others and proved that $\parallel (1/n)T_n^{\text{'}}{\parallel }_p\le {\parallel T_n\parallel }_p$ for $0\le p\le \infty$ and $T_n\in {𝕋}_n$. We derive the multivariate version of the result of Golitschek and Lorentz $${∥{\left|T_{n}cos\alpha +\frac{1}{n}\nabla T_{n}sin\alpha \right|}_{l_{\infty }^{\left(m\right)}}∥}_p\le {\parallel T_n\parallel }_p,0\le p\le \infty$$ for all trigonometric polynomials (with complex coeffcients) in $m$ variables of degree at most $n$.

Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence ${\left({ϵ}_i\right)}_{i=1}^{\infty }\in {0,1}^{\mathbb{N}}$ a β-expansion for x if $x={\sum }_{i=1}^{\infty }{ϵ}_i{\beta }^{-i}$. We call a finite sequence ${\left({ϵ}_i\right)}_{i=1}^n\in {0,1}^n$ an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes. Given $\Psi :\mathbb{N}\to {ℝ}_{\ge 0}$, we introduce the following subset of ℝ: $W_{\beta }\left(\Psi \right):={\bigcap }_{m=1}^{\infty }{\bigcup }_{n=m}^{\infty }{\bigcup }_{{\left({ϵ}_i\right)}_{i=1}^n\in {0,1}^n}[{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right),{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right)+\Psi \left(n\right)]$In other words, $W_{\beta }\left(\Psi \right)$ is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities $0\le x-{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right)\le \Psi \left(n\right)$. When ${\sum }_{n=1}^{\infty }{2}^n\Psi \left(n\right)<\infty$, the Borel-Cantelli lemma tells us that the Lebesgue measure...

The Littlewood conjecture in Diophantine approximation claims that $$\underset{q\ge 1}{inf}q·\parallel q\alpha \parallel ·\parallel q\beta \parallel =0$$ holds for all real numbers $\alpha$ and $\beta$, where $\parallel ·\parallel$ denotes the distance to the nearest integer. Its $p$-adic analogue, formulated by de Mathan and Teulié in 2004, asserts that $$\underset{q\ge 1}{inf}q·\parallel q\alpha \parallel ·{\left|q\right|}_p=0$$ holds for every real number $\alpha$ and every prime number $p$, where ${|·|}_p$ denotes the $p$-adic absolute value normalized by ${\left|p\right|}_p=p^{-1}$. We survey the known results on these conjectures and highlight recent developments. ...

We consider simultaneous solutions of operator Sylvester equations $A_{i}X-X{B}_i=C_i$ (1 ≤ i ≤ k), where $(A₁,...,A_k)$ and $(B₁,...,B_k)$ are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and $(C₁,...,C_k)$ is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of $(A₁,...,A_k)$ and $(B₁,...,B_k)$ do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w*}(X*,Y)$, as well as the complementation of the space $K_{w*}(X*,Y)$ of w*-w compact operators in the space $L_{w*}(X*,Y)$ of w*-w operators from X* to Y.

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

Let $\alpha \&gt;1$ be irrational. Several authors studied the numbers $${\ell }^m\left(\alpha \right)=inf\left\{\right|y|:y\in {\Lambda }_m,y\ne 0\},$$ where $m$ is a positive integer and ${\Lambda }_m$ denotes the set of all real numbers of the form $y={ϵ}_0{\alpha }^n+{ϵ}_1{\alpha }^{n-1}+\cdots +{ϵ}_{n-1}\alpha +{ϵ}_n$ with restricted integer coefficients $|{ϵ}_i|\le m$. The value of ${\ell }^1\left(\alpha \right)$ was determined for many particular Pisot numbers and ${\ell }^m\left(\alpha \right)$ for the golden number. In this paper the value of ${\ell }^m\left(\alpha \right)$ is determined for irrational numbers $\alpha$, satisfying ${\alpha }^2=a\alpha \pm 1$ with a positive integer $a$.

The following results are proved: (i) if $\alpha$, $\beta \in {ℝ}^{+}$ and $A$ is a sectorial operator, then the set $\{t^{\alpha }{A}^{\beta }(t+A);t>0\}$ is bounded; (ii) the same set of operators is R-bounded if $A$ is R-sectorial.