Space Lq Research Articles

No greedy bases for matrix spaces with mixed [formula omitted] and [formula omitted] norms

We show that none of the spaces (⨁n=1∞ℓp)ℓq, 1≤p≠q<∞ have a greedy basis. This solves a problem raised by Dilworth, Freeman, Odell and Schlumprecht. Similarly, the spaces (⨁n=1∞ℓp)c0, 1≤p<∞, and (⨁n=1∞co)ℓq, 1≤q<∞, do not have greedy bases. It follows from that and known results that a class of Besov spaces on Rn lack greedy bases as well.

On the converse theorem of approximation in various metrics for nonperiodic functions

The modulus of smoothness in the norm of space Lq of nonperiodic functions of several variables is estimated by best approximations by entire functions of exponential type in the metric of space Lp, 1 ? p ? q &lt; ?.

A generalization of Bihari’s lemma to the case of Volterra operators in Lebesgue spaces

For operators acting in the Lebesgue space Lq(Π), 1 < q < ∞, an abstract analog of Bihari’s lemma is stated and proved. We show that it can be used to derive a uniform pointwise estimate of the increment of the solution of a controlled functional-operator equation in the Lebesgue space. The procedure of reducing controlled initial boundary-value problems to this equation is illustrated by the Goursat-Darboux problem.

Best [formula omitted]-term trigonometric approximation of periodic functions of several variables from Nikol’skii–Besov classes for small smoothness

We find the asymptotic orders of best m-term trigonometric approximation for Nikol’skii–Besov classes Bp,θr of periodic functions of several variables in the space Lq, 1≤p≤2<q<∞, for the cases of “small” smoothness dp−dq<r<dp and for the “critical” value of the smoothness r=dp. These results are complementary to the estimates of the best m-term trigonometric approximation for classes Bp,θr obtained by R.A. DeVore and V.N. Temlyakov.

Best trigonometric and bilinear approximations of classes of functions of several variables

Order-sharp estimates of the best orthogonal trigonometric approximations of the Nikol’skii-Besov classes Bp,θr of periodic functions of several variables in the space Lq are obtained. Also the orders of the best approximations of functions of 2d variables of the form g(x, y) = f(x−y), x, y ∈ \(\mathbb{T}\)d = Πj=1d[−π, π], f(x) ∈ Bp,θr, by linear combinations of products of functions of d variables are established.

Conformal Weights and Sobolev Embeddings

We study embeddings of the Sobolev space $$ {\mathop{W}\limits_{~}^{\circ}}{{~}_2^1}\left( \Omega \right) $$ into weighted Lebesgue spaces L q (Ω, h) with the so-called universal conformal weight h defined as the Jacobian of a conformal homeomorphism φ from Ω onto the unit disk in the complex plane. Applications to elliptic boundary value problems are discussed. Bibliography: 17 titles.

Essential properties of [formula omitted] spaces (the amalgams) and the implicit function theorem for equilibrium analysis in continuous time

To extend the analysis of continuous-time general-equilibrium macro models we study 2 parameter variants Lp,q of the Lebesgue spaces, thus gaining separate control on the asymptotic behaviour (p) and the local behaviour (q): they behave w.r.t. p like the spaces ℓp and w.r.t. q like the spaces Lq on a probability space. Such spaces might naturally contain equilibrium variables (paths) as well as time-dependent policies of a macro model. Convolution behaves very well on those spaces, which can be used as a basis for the classical “comparative statics” (see e.g. Mertens and Rubinchik (2011)). Finally, we generalise the classical implicit function theorem (ift) for a family of Banach spaces, with the resulting implicit function having derivatives that are locally Lipschitz to very strong operator norms.

Embedding theorem for weighted Sobolev classes on a John domain with weights that are functions of the distance to some h-set

Let Ω be a John domain, let Γ ⊂ ∂Ω be an h-set, and let g and v be weights on Ω that are distance functions to the set Γ of special form. In the paper, sufficient conditions are obtained under which the Sobolev weighted class Wp,gr(Ω) is continuously embedded in the space Lq,v(Ω). Moreover, bounds for the approximation of functions in Wp,gr(Ω) by polynomials of degree not exceeding r − 1 in Lq,v(\(\tilde \Omega \)) are found, where \(\tilde \Omega \) is a subdomain generated by a subtree of the tree T defining the structure of Ω.

Brennan's Conjecture and universal Sobolev inequalities

Brennan's Conjecture states integrability of derivatives of plane conformal homeomorphisms φ:Ω→D that map a simply connected plane domain with non-empty boundary Ω⊂C to the unit disc D⊂R2. We prove that Brennan's Conjecture leads to existence of compact embeddings of Sobolev spaces W˚p1(Ω) into weighted Lebesgue spaces Lq(Ω,h) with universal conformal weights h(z):=J(z,φ)=|φ′(z)|2. For p=2 the number q is an arbitrary number between 1 and ∞ (Gol'dshtein and Ukhlov, in press [12]), for p≠2 the number q depends on p and the integrability exponent for Brennan's Conjecture.

The classical subspaces of the projective tensor products of lpand C(α) spaces, α< ω1

We completely determine the lq and C(K) spaces which are isomorphic to a subspace of lp⊗ˆπC(α), the projective tensor product of the classical lp space, 1≤p<∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α<ω1. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from lp to l1, 1≤p<∞. The first main theorem is an extension of a result of E. Oja and states that the only lq space which is isomorphic to a subspace of lp⊗ˆπC(α) with 1≤p≤q<∞ and ω≤α<ω1 is lp. The second main theorem concerning C(K) spaces improves a result of Bessaga and Pelczynski which allows us to classify, up to isomorphism, the separable spaces (X,Y) of nuclear operators, where X and Y are direct sums of lp and C(K) spaces. More precisely, we prove the following cancellation law for separable Banach spaces. Suppose that K1 and K3 are finite or countable compact metric spaces of the same cardinality and $1 (a) (lp⊕C(K1),lq⊕C(K2)) and (lp⊕C(K3),lq⊕C(K4)) are isomorphic.

Asymptotic properties of extremal polynomials corresponding to measures supported on analytic regions

Let G be a bounded region with simply connected closure G¯ and analytic boundary and let μ be a positive measure carried by G¯ together with finitely many pure points outside G. We provide estimates on the norms of the monic polynomials of minimal norm in the space Lq(μ) for q>0. In case the norms converge to 0, we provide estimates on the rate of convergence, generalizing several previous results. Our most powerful result concerns measures μ that are perturbations of measures that are absolutely continuous with respect to the push-forward of a product measure near the boundary of the unit disk. Our results and methods also yield information about the strong asymptotics of the extremal polynomials and some information concerning Christoffel functions.

Best approximation of periodic functions of several variables from the classes $ MB_{{p,\theta }}^{\omega } $

We obtain exact-order estimates for the best approximation of periodic functions of several variables from the classes $ MB_{{p,\theta }}^{\omega } $ by trigonometric polynomials with the “numbers” of harmonics from graded hyperbolic crosses in the metric of the space L q :

Edge Singular Behavior for the Heat Equation on Polyhedral Cylinders in $\boldsymbol{\mathbb{R}}^{\bf 3}$

We study the Heat equation in the polyhedral cylinder with a non-convex edge. We construct the singularity functions depending on the time and edge axis, and the coefficient of the singularity, called the stress intensity distributions, and show regularity results for the solution and the coefficient. The regularity is achieved in the (not weighted) Sobolev space in the L2 and Lq spaces, respectively. An application to the finite polyhedral cylinder is described.

An application of atomic decomposition in Bergman spaces to the study of differences of composition operators

In this paper we characterize bounded differences of composition operators from standard weighted Bergman spaces Aαp into Lebesgue spaces Lq(μ) when α>−1 and p>q. In the process we also find an interesting connection between the difference operator and certain weighted composition operators answering a question raised in Saukko (2011) [18]. By combining the results of this paper with those of Saukko (2011) [18], we obtain a complete characterization of bounded and compact differences between standard weighted Bergman spaces.

Vanishing viscosity limits for the degenerate lake equations with Navier boundary conditions

This paper concerns the vanishing viscosity limit of the two-dimensional degenerate viscous lake equations when the Navier slip conditions are prescribed on the impermeable boundary of a simply connected bounded regular domain. When the initial vorticity is in the Lebesgue space Lq with 2 < q ⩽ ∞, we show that the degenerate viscous lake equations possess a unique global solution and the solution converges to a corresponding weak solution of the inviscid lake equations. In a special case when the vorticity is in L∞, an explicit convergence rate is obtained.

Bernstein Widths of Some Classes of Functions Defined by a Self‐Adjoint Operator

We consider the classes of periodic functions with formal self‐adjoint linear differential operators Wp(ℒr), which include the classical Sobolev class as its special case. Using the iterative method of Buslaev, with the help of the spectrum of linear differential equations, we determine the exact values of Bernstein width of the classes Wp(ℒr) in the space Lq for 1 &lt; p ≤ q &lt; ∞.

Best m-term approximation of the classes $$ B_{\infty, \theta }^r $$ of functions of many variables by polynomials in the haar system

We obtain an exact-order estimate for the best m-term approximation of the classes $$ B_{\infty, \theta }^r $$ of periodic functions of many variables by polynomials in the Haar system in the metric of the space L q , 1 < q < ∞.

Strong solutions to the Stokes equations of a flow around a rotating body in weighted Lq spaces

We consider the motion of a fluid in the exterior of a rotating obstacle. This leads to a modified version of the Stokes system which we consider in the whole space \documentclass{article}\usepackage{amssymb,dsfont}\begin{document}\pagestyle{empty}${\mathds R}^n$\end{document}, n = 2 or n = 3 and in an exterior domain \documentclass{article}\usepackage{amssymb,dsfont}\begin{document}\pagestyle{empty}$D\subset {\mathds R}^3$\end{document}. For every q ∈ (1, ∞) we prove existence of solutions and estimates in function spaces with weights taken from a subclass of the Muckenhoupt class Aq. Moreover, uniqueness is shown modulo a vector space of dimension 3. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Best approximation by ridge functions in L p -spaces

We study the approximation of the classes of functions by the manifold R n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W r p from the set R n of ridge functions in the space L q (B d ) satisfies the sharp order n -r/(d-1).

Asymptotic estimates for the best trigonometric and bilinear approximations of classes of functions of several variables

We obtain exact order estimates for the best M-term trigonometric approximations of the Besov classes B ∞,θ r in the space L q . We also determine the exact orders of the best bilinear approximations of the classes of functions of 2d variables generated by functions of d variables from the classes B p,θ r with the use of translation of arguments.