Embedding lp, {infty} into Lp, {infty}[0, 1] Complementably

We show that l^p, embeds into L^p, [0, 1] as a complemented sublattice.     

The Adamjan-Arov-Krein Theorem on Hp-Spaces (2 <= p <= {infty}) and on the Disc Algebra

This paper is devoted to the following extension of the AAKtheorem. Let (p,q)[2, + ]². Let u: H^p x H^q C bea hankelianbilinear form and n N*. There is a hankelian bilinear formv: H^p x H^q C with rk(v) < n and ||u–v|| Ca_n(U) forsome constant C > 0 depending only on (p,q). Moreover, H^por H^q may be replaced by A in this statement.     

On the Reflexivity of the Space {pi}p(E, F) of p-Absolutely Summing Operators, 1 <= p < + {infty}

We give a representation of the dual of the space _p(E,F) ofp-absolutely summing operators (1 p < + ) under certainconditions on E and F. One deduces that the space _p(E, F), 1 p < + , is reflexive if and only if E and F are reflexive.We improve results of Gordon, Lewis, Retherford and Saphar.     

Lipschitzian Properties of Integral Functionals on Lebesgue Spaces $\bf{L_p, 1 \leqslant {p} < \infty}$

We give complete characterizations of integral functionals which are Lipschitzian on a Lebesgue space L _p with p ≠ ∞. When the measure is atomless, we characterize the integral functionals which are locally Lipschitzian on such Lebesgue spaces. In every cases, the Lipchitzian properties of the integral functional can be described by growth conditions on the subdifferentials of the integrand which are equivalent to Lipschitzian properties of the integrand.     

Conditions for Superlinear Convergence in l1 and l{infty} Solutions of Overdetermined Non-linear Equations

Let n be the number of unknowns in an overdetermined systemof non-linear equations. If fewer than n equations are satisfiedin an l₁ solution or if fewer than (n+1) maximum residuals occurin an lsolution, then iterative methods of calculation convergesuperlinearly only if some second derivative information isused. This paper establishes some conditions on second derivativeestimates that are necessary and sufficient for superlinearconvergence.     

Navigating in the Cayley Graphs of {\rm SL}_{N}(\mathbb{Z}) and {\rm SL}_{N}(\mathbb{F}_{p})

We give a nondeterministic algorithm that expresses elements of , for N ≥ 3, as words in a finite set of generators, with the length of these words at most a constant times the word metric. We show that the nondeterministic time-complexity of the subtractive version of Euclid’s algorithm for finding the greatest common divisor of N ≥ 3 integers a₁, ..., a_N is at most a constant times . This leads to an elementary proof that for N ≥ 3 the word metric in is biLipschitz equivalent to the logarithm of the matrix norm – an instance of a theorem of Mozes, Lubotzky and Raghunathan. And we show constructively that there exists K>0 such that for all N ≥ 3 and primes p, the diameter of the Cayley graph of with respect to the generating set is at most .Mathematics Subject Classification: 20F05     

Local-global double algebras for slow H{infty} adaptation; the case of l2 disturbances

In this paper, a common algebraic framework is described forthe frozen-time analysis of stability and H optimization inslowly time-varying systems, based on the notion of a normedalgebra on which local and global products are defined. Relationsbetween local stability, local (near) optimality. local coprimefactorization, and global versions of these properties are sought.The framework is valid for time-domain disturbances in l² (Z)^R. The first part of the paper elaborates the double-algebra conceptfor Volterra operators which approximately commute with theshift. The main algebraic properties and norm inequalities aresummarized. Local conditions for global inveribility are obtained.Classical frozen-time stability conditions are incorporatedin relations between local and global spectra. The paper continuesby establishing an explicit formula linking global and localsensitivity for systems with stable plants, where local sensitivityis a Lipschitz continuous function of data. Frequency-domainestimates of time-domain sensitivity norms. which become accurateas rates of time-variation approach zero, are obtained. Notionsof adaptive versus nonadaptive (robust) control are introduced.It is shown that adaptive control can achieve better sensitivitythan optimal nonadaptive control. It is demonstrated by an examplethat, in general, H-optimal interpolants do not depend Lipschitz-continuouslyon data. However, -suboptimal interpolants of the AAK central(maximal entropy) type are shown to satisfy a tractable Lipschitzcondition.     

Approximation of 1/x by exponential sums in [1, {infty})

^** Email: braess{at}num.rub.de^*** Email: wh{at}mis.mpg.de Approximations of 1/x by sums of exponentials are well studiedfor finite intervals. Here the error decreases like (exp(–ck))with the order k of the exponential sum. In this paper we investigateapproximations of 1/x in the interval [1, ). We prove estimatesof the error by and confirm this asymptotic estimate by numerical results. Numericalresults lead to the conjecture that the constant in the exponentequals .     

Approximation in $L_{p} (\mathbb{R}^{d}), 0 < p < 1$, by linear combinations of characteristic functions of balls

We prove that translates of the characteristic function of a ball span the space L _p ( \mathbbR^d\mathbb{R}^{d}) provided that 0 <p<1 and d ≥ 2. Similar approximation problems are considered for some other functions. Bibliography: 5 titles.     

Embedding l{infty} into the Space of Bounded Operators on Certain Banach Spaces

Sufficient conditions are given on a Banach space X which ensurethat embeds in L(X), the space of all bounded linear operatorson X. A basic sequence e_n is said to be quasisubsymmetric iffor any two increasing sequences (k_n) and (l_n) of positive integerswith (k_n) (l_n) for all n, () dominates (). If a Banach space X has a seminormalized quasisubsymmetric basis then l embedsin l(X). 2000 Mathematics Subject Classification 46B28 (primary),46B03 (secondary).     

The Mesa Problem: Diffusion Patterns for ut = {nabla} {middle dot} (um{nabla}u) as m -> +{infty}

In this paper we consider the limit m+ of solutions of the porous-mediumequation u_t = · (u^mu) (xR^N), with N > 1. We conjecturethat, for initial data with a unique maximum, the evolutionis characterized by the onset of a ‘mesa’ region,in which the solution is nearly spatially independent, surroundedby a region in which u is nearly equal to its initial value.The transition between these regions occurs near a surface whichis identified with the free boundary in a certain Stefan problemwhich can be studied using variational inequalities. Moreover,singular-perturbation theory can be used to describe the structureof the transition region.     

The Hopf algebra Rep U_{q} \widehat{\frak g \frak l}_\infty

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of in the limitN→∞. The resulting Hopf algebra Rep is a tensor product of its Hopf subalgebras Rep_a ,a ∈ ℂ^×/q^2ℤ. Whenq is generic (resp.,q ² is a primitive root of unity of orderl), we construct an isomorphism between the Hopf algebra Rep _a and the algebra of regular functions on the prounipotent proalgebraic group (resp., ). Whenq is a root of unity, this isomorphism identifies the Hopf subalgebra of Rep _a spanned by the modules obtained by pullback with respect to the Frobenius homomorphism with the algebra generated by the coefficients of the determinant of an element of considered as anl×l matrix over the Taylor series. This gives us an explicit formula for the Frobenius pullbacks of the fundamental representations. In addition, we construct a natural action of the Hall algebra associated to the infinite linear quiver (resp., the cyclic quiver withl vertices) on Rep _a and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra.     

Iterative algorithm for solving a class of general Sylvester-conjugate matrix equation \sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C

This paper is concerned with iterative solution to general Sylvester-conjugate matrix equation of the form $\sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C$ . An iterative algorithm is established to solve this matrix equation. When this matrix equation is consistent, for any initial matrices, the solutions can be obtained within finite iterative steps in the absence of round off errors. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm.     

Russian text ignored Lp, 0 < p < 1

Certain weak forms of normality are discussed and are shown to coincide with normality in an appropriate setting by devising a strong form of semilocal connectedness. It is shown that every compact Hausdorff locally connected space is strongly semilocally connected. This improves a result of Mrówka and Pervin. Properties of s-continuous mappings are studied and in the process improvements of certain results of Pervin and Levine, Friedler, and Kohli are obtained.     

Dilation operators and integral operators on amalgam space $$(L_{p},l_{q})$$(Lp,lq)

This paper establishes the Hardy–Littlewood–Pólya inequalities, the Hardy inequalities and the Hilbert inequalities on amalgam spaces. Moreover, it also gives the mapping properties of the Mellin convolutions, the Hadamard fractional integrals and the Hausdorff operators on amalgam spaces. We establish these properties by some estimates for the operator norms of the dilation operators on amalgam spaces.

     

The $H^{infty}-$calculus and sums of closed operators

We develop a very general operator-valued functional calculus for operators with an calculus. We then apply this to the joint functional calculus of two sectorial operators when one has an calculus. Using this we prove theorem of Dore-Venni type on sums of sectorial operators and apply our results to the problem of maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structure of spaces and spaces, to obtain some more detailed results in this setting. Received: 3 July 2000 / Revised version: 31 January 2001 / Published online: 23 July 2001