Approximation of positive operators by analytic vectors

The purpose of this paper is to present Bernstein and Hoeffding type inequalities for regenerative Markov chains. Furthermore, we generalize these results and establish exponential bounds for suprema of empirical processes over a class of functions F which size is controlled by its uniform entropy number. All constants involved in the bounds of the considered inequalities are given in an explicit form which can be advantageous for practical considerations. We present the theory for regenerative Markov chains, however the inequalities are also valid in the Harris recurrent case.

In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $K$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.

In this work, we discuss generalizations of the classical Bernstein and\nMarkov type inequalities for polynomials and we present some new inequalities\nfor the $k$th Fr\\'echet derivative of homogeneous polynomials on real and\ncomplex $L_{p}(\\mu)$ spaces. We also give applications to homogeneous\npolynomials and symmetric multilinear mappings in $L_{p}(\\mu)$ spaces. Finally,\nBernstein's inequality for homogeneous polynomials on both real and complex\nHilbert spaces has been discussed.\n

We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for complex algebraic polynomials of degree n having no zeros in the open unit disk. This is done by establishing the right Bernstein–Szegő–Videnskii type inequality for real trigonometric polynomials of degree at most n on intervals shorter than the period. The paper is closely related to recent work by B. Nagy and V. Totik. In fact, their asymptotically sharp Bernstein-type inequality for complex algebraic polynomials of degree at most n on subarcs of the unit circle is recaptured by using more elementary methods. Our discussion offers a somewhat new way to see V.S. Videnskii’s Bernstein and Markov type inequalities for trigonometric polynomials of degree at most n on intervals shorter than a period, two classical polynomial inequalities first published in 1960. A new Riesz–Schur type inequality for trigonometric polynomials is also established. Combining this with Videnskii’s Bernstein-type inequality gives Videnskii’s Markov-type inequality immediately.

We present a wavelet characterization of anisotropic Besov spaces $B_{p,q}^{\bm{\alpha}}(\mathbb{R}^n)$, valid for the whole range $0\ltp,q\lt\infty$, and in terms of multi-resolution analyses with dilation adapted to the anisotropy of the space. Our proofs combine classical techniques based on Bernstein and Jackson-type inequalities, and nonlinear methods for the cases $p\lt1$. Among the consequences of our results, we characterize $B_{p,q}^{\bm{\alpha}}$ as a linear approximation space, and derive embeddings and interpolation formulae for $B_{p,q}^{\bm{\alpha}}$, which appear to be new in the literature when $p\lt1$.AMS 2000 Mathematics subject classification: Primary 42B35; 42C40; 41A17

Real and complex interpolation methods for finite and infinite families of Banach spaces

Recently a number of authors have developed conditions of a generalized quasi-analytic nature which imply essential selfadjointness for semibounded symmetric operators in Hubert space.We give a unified derivation of these results by reducing them to the basic theorems of Nelson and Nussbaum.In addition we present an extension of Nussbaum's quasi-analytic vector theorem to the setting of semigroups in Banach spaces.1. Introduction.Let A be a linear operator in a Banach space with domain 3l(A).A vector x will be called a C vector for A provided x e ]n= j 3>(An) =S>'a(A).We shall distinguish the following subsets of 2 "(A):(1) analytic vectors: x e 3&(A) if 2"=o (in/!)M"x|| converges for some i>0;(2) quasi-analytic vectors: x e ^qaL4) if 2"= o M"*Il _1/" = oo;(3) semi-analytic vectors: xe@s!l(A) if 2= o (tnIQn)f)\Anx\ converges for some i>0;(4) Stieltjes vectors: xe@s(A) if 2"= o Mnx||~1,2n = co.The sets S>& and 3>%& are linear subspaces of St(A), but this is not necessarily true for ^qa and S>s.The following inclusion relations are immediate : ^a c ^aa n n ^sa c % Analytic vectors were invented by Nelson ; his paper [6] contains (among many things) the following fundamental fact.Theorem A. Let Abe a symmetric operator on Hubert space.If A has a dense set of analytic vectors then the closure of A is self adjoint.A significant generalization was made by Nussbaum [7], who introduced the notion of quasi-analytic vector and proved the analogue of Nelson's theorem : Theorem QA.Let Abe a symmetric operator on Hilbert space.If A has a total set of quasi-analytic vectors then the closure of A is selfadjoint.

We prove a new Bernstein type inequality in L p L^p spaces associated with the normal and the tangential derivatives on the boundary of a general compact C 2 C^2 -domain. We give two applications: Marcinkiewicz type inequality for discretization of L p L^p norms and positive cubature formulas. Both results are optimal in the sense that the number of function samples used has the order of the dimension of the corresponding space of algebraic polynomials.

In this paper, we propose and investigate the optimal tuning PID controller of unstable fractional order system by desired transient characteristics using the real interpolation method (RIM). The research shows that the main advantages of this method are drawn as the followings: 1) Carrying out an investigation of the stable region of coefficients of a PID controller using D-decomposition method; 2) Applying the method to investigate an unstable fractional order system.

We introduce a decoupling method on the Wiener space to define a wide class of anisotropic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov spaces we introduce contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that characterize quantitative properties of directional derivatives in the Malliavin sense without computing or accessing these Malliavin derivatives explicitly. Regarding BSDEs, we deduce regularity properties of the solution processes from the Besov regularity of the initial data, in particular upper bounds for their L p L_p -variation, where the generator might be of quadratic type and where no structural assumptions, for example in terms of a forward diffusion, are assumed. As an example we treat sub-quadratic BSDEs with unbounded terminal conditions. Among other tools, we use methods from harmonic analysis. As a by-product, we improve the asymptotic behaviour of the multiplicative constant in a generalized Fefferman inequality and verify the optimality of the bound we established.

In this paper we prove a general Strichartz estimate for certain perturbed wave equations, and here we can drop the nontrapping hypothesis and handle trapping obstacles with some loss of derivatives for data in the local energy decay estimates. We then give the obstacle version of the sharp life span for semilinear wave equations when $n=3,p<p_c$, by using the real interpolation method, and by getting corresponding finite time Strichartz estimates (see Section 3). Finally, as another application, we get the Strauss conjecture for semilinear wave equations with several convex obstacles when $n=3,4$ (see Section 4).

New and sharp estimates are derived for the growth in the complex plane of polynomials known to have a curved majorant on a given ellipse. These so-called Bernstein type inequalities are closely connected with certain constrained Chebyshev approximation problems on ellipses. Also presented are some new results for approximation problems of this type.

Bernstein type inequalities for restrictions of polynomials to complex submanifolds of [formula omitted

Direct numerical simulation (DNS) of fluid flow in porous media with many scales is often not feasible, and an effective or homogenized description is more desirable. To construct the homogenized equations, effective properties must be computed. Computation of effective properties for nonperiodic microstructures can be prohibitively expensive, as many local cell problems must be solved for different macroscopic points. In addition, the local problems may also be computationally expensive. When the microstructure varies slowly, we develop an efficient numerical method for two scales that achieves essentially the same accuracy as that for the full resolution solve of every local cell problem. In this method, we build a dense hierarchy of macroscopic grid points and a corresponding nested sequence of approximation spaces. Essentially, solutions computed in high accuracy approximation spaces at select points in the the hierarchy are used as corrections for the error of the lower accuracy approximation spaces at nearby macroscopic points. We give a brief overview of slowly varying media and formal Stokes homogenization in such domains. We present a general outline of the algorithm and list reasonable and easily verifiable assumptions on the PDEs, geometry, and approximation spaces. With these assumptions, we achieve the same accuracy as the full solve. To demonstrate the elements of the proof of the error estimate, we use a hierarchy of macro--grid points in $[0,1]^2$ and finite element (FE) approximation spaces in $[0,1]^2$. We apply this algorithm to Stokes equations in a slowly porous medium where the microstructure is obtained from a reference periodic domain by a known smooth map. Using the arbitrary Lagrange--Eulerian (ALE) formulation of the Stokes equations (cf. [G. P. Galdi and R. Rannacher, Fundamental Trends in Fluid-Structure Interaction, Contemporary Challenges in Mathematical Fluid Dynamics and Its Applications 1, World Scientific, Singapore, 2010]), we obtain modified Stokes equations with varying coefficients in the periodic domain. We show that the algorithm can be utilized in this setting. Finally, we implement the algorithm on the modified Stokes equations, using a simple stretch deformation mapping, and compute the effective permeability. We show that our efficient computation is of the same order as the full solve.

The following dichotomy is proved. Every Banach space either contains a subspace isomorphic to ℓ 1 \ell ^1 , or it has an infinite-dimensional closed subspace which is a quotient of a Hereditarily Indecomposable (H.I.) separable Banach space. In the particular case of L p ( λ ) , 1 &gt; p &gt; ∞ L^p(\lambda ),\ 1&gt;p&gt;\infty , it is shown that the space itself is a quotient of a H.I. space. The factorization of certain classes of operators, acting between Banach spaces, through H.I. spaces is also investigated. Among others it is shown that the identity operator I : L ∞ ( λ ) → L 1 ( λ ) I: L^{\infty }(\lambda )\to L^1(\lambda ) admits a factorization through a H.I. space. The same result holds for every strictly singular operator T : ℓ p → ℓ q , 1 &gt; p , q &gt; ∞ T: \ell ^p\to \ell ^q,\ 1&gt;p,q&gt;\infty . Interpolation methods and the geometric concept of thin convex sets together with the techniques concerning the construction of Hereditarily Indecomposable spaces are used to obtain the above mentioned results.