Best Uniform Rational Approximations of Functions by Orthoprojections

Let C[-1,1] be the Banach space of continuous complex functions $f$ on the interval [-1,1] equipped with the standard maximum norm $\left\| f \right\|$ ; let $\omega \left( \cdot \right) = \omega \left( { \cdot ,f} \right)$ be the modulus of continuity of $f$ ; and let $R_n = R_n \left( f \right)$ be the best uniform approximation of $f$ by rational functions (r.f.) whose degrees do not exceed $n = 1, 2, \ldots $ . The space C[-1,1] is also regarded as a pre-Hilbert space with respect to the inner product given by $\left( {f,g} \right) = \left( {1/\pi } \right)\int_{ - 1}^1 {f\left( x \right)g\left( x \right)} \left( {1 - x^2 } \right)^{ - 1/2} dx$ . Let $z_n = \{ z_1 , z_2 , \ldots z_n \} $ be a set of points located outside the interval [-1,1]. By $F\left( { \cdot ,f,z_n } \right)$ we denote an orthoprojection operator acting from the pre-Hilbert space C[-1,1] onto its ( ${n + 1}$ )-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set $z_n $ . In this paper, we show that if $f$ is not a rational function of degree $ \leqslant n$ , then we can find a set of points $z_n = z_n \left( f \right)$ such that $\left\| {f\left( \cdot \right) - F\left( { \cdot ,f,z_n } \right)} \right\| \leqslant 12R_n ln\frac{3}{{\omega ^{ - 1} \left( {R_n /3} \right)}}.$      

Orthogonality properties of linear combinations of orthogonal polynomials II

Let and be polynomials orthogonal on the unit circle with respect to the measures dσ and dμ, respectively. In this paper we consider the question how the orthogonality measures dσ and dμ are related to each other if the orthogonal polynomials are connected by a relation of the form , for , where . It turns out that the two measures are related by if , where and are known trigonometric polynomials of fixed degree and where the 's are the zeros of on . If the 's and 's are uniformly bounded then (under some additional conditions) much more can be said. Indeed, in this case the measures dσ and dμ have to be of the form and , respectively, where are nonnegative trigonometric polynomials. Finally, the question is considered to which weight functions polynomials of the form where denotes the reciprocal polynomial of , can be orthogonal. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Total Incomparability of Mixed Tsirelson Spaces

We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T\left[ {\left( {\mathcal{M}_k ,\theta _k } \right)_{k = 1}^l } \right]$ with index $i\left( {\mathcal{M}_k } \right)$ finite are either c ₀ or $\ell _p $ saturated for some p and we characterize when any two spaces of such a form are totally incomparable in terms of the index $i\left( {\mathcal{M}_k } \right)$ and the parameter θ _k . Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T\left[ {\left( {\mathcal{A}_k ,\theta _k } \right)_{k = 1}^\infty } \right]$ in terms of the asymptotic behaviour of the sequence $\left\| {\sum\limits_{j = 1}^n {e_i } } \right\|$ where (e _i is the canonical basis.     

Exact and asymptotic estimates forn-widths of some classes of periodic functions

Let \(\tilde W_p^r : = \left\{ {f\left| {f \in C^{r - 1} } \right.} \right.\left[ {0,2\pi } \right],f^{(i)} (0) = f^{(i)} (2\pi ),i = 0, \ldots ,r - 1,f^{(r - 1)}\) , abs. cont. on [0, 2π] andf ^(r)∈L ^p[0, 2π]}, and set \(\tilde B_p^r : = \left\{ {f\left| {f \in \tilde W_p^r ,} \right.\left\| {f^{(r)} } \right\|_p \leqslant 1} \right\}\) . We find the exact Kolmogrov, Gel'fand, and linearn-widths of \(\tilde B_p^r\) inL ^p forn even and allp∈(1, ∞). The strong asymptotic estimates forn-widths of \(\tilde B_p^r\) inL ^p are also obtained.     

Converse Quadrature sum Inequalities for Freud Weights. ii

Let $W: = \exp \left( { - Q} \right)$ , where $Q$ is of smooth polynomial growth at $\infty$ , for example $Q\left( x \right) = \left| x \right|^\beta ,\beta >1$ . We call $W^2 $ a Freud weight. Let $\left\{ {x_{j{\kern 1pt} n} } \right\}_{j = 1}^n $ and $\left\{ {\lambda _{j{\kern 1pt} n} } \right\}_{j = 1}^n $ denote respectively the zeros of the $n$ th orthonormal polynomial $p_n$ for $W^2 $ and the Christoffel numbers of order $n$ . We establish converse quadrature sum inequalities associated with W, such as $$\left\| {\left( {PW} \right)\left( x \right)\left( {1 + \left| x \right|} \right)^r } \right\|_{L_p \left( R \right)} $$ with $C$ independent of $n$ and polynomials P of degree $ < n$ , and suitable restrictions on $r$ , $R$ . We concentrate on the case ${ \geqq 4}$ , as the case ${p < 4}$ was handled earlier. We are able to treat a general class of Freud weights, whereas our earlier treatment dealt essentially with $\left( { - \left| x \right|^\beta } \right),\beta = 2,4,6,....$ Some applications to Lagrange interpolation are presented.     

Annihilators in BCK-Algebras

We introduce the concepts of an annihilator and a relative annihilator of a given subset of a BCK-algebra $\mathcal{A}$ . We prove that annihilators of deductive systems of BCK-algebras are again deductive systems and moreover pseudocomplements in the lattice ${\mathcal{D}\left( \mathcal{A} \right)}$ of all deductive systems on $\mathcal{A}$ . Moreover, relative annihilators of ${C \in \mathcal{D}\left( \mathcal{A} \right)}$ with respect to ${B\;{\text{in}}\;\mathcal{D}\left( \mathcal{A} \right)}$ are introduced and serve as relative pseudocomplements of C w.r.t. B in ${\mathcal{D}\left( \mathcal{A} \right)}$ .     

ОБ ИНтЕРпОлИРОВАНИИ В кОМплЕксНОИ ОБлАст И

LetD be a simply connected domain, the boundary of which is a closed Jordan curveγ; \(\mathfrak{M} = \left\{ {z_{k, n} } \right\}\) , 0≦k≦n; n=1, 2, 3, ..., a matrix of interpolation knots, \(\mathfrak{M} \subset \Gamma ; A_c \left( {\bar D} \right)\) the space of the functions that are analytic inD and continuous on \(\bar D; \left\{ {L_n \left( {\mathfrak{M}; f, z} \right)} \right\}\) the sequence of the Lagrange interpolation polynomials. We say that a matrix \(\mathfrak{M}\) satisfies condition (B _m ), \(\mathfrak{M}\) ∈(B _m ), if for some positive integerm there exist a setB _m containingm points and a sequencen _{p p=1} ^∞ of integers such that the series \(\mathop \Sigma \limits_{p = 1}^\infty \frac{1}{{n_p }}\) diverges and for all pairsn _i ,n _j ∈{n _p } _p=1 ^∞ the set \(\left( {\bigcap\limits_{k = 0}^{n_i } {z_{k, n_i } } } \right)\bigcap {\left( {\bigcup\limits_{k = 0}^{n_j } {z_{k, n_j } } } \right)} \) is contained inB _m . The main result reads as follows. {Let D=z: ¦z¦ \(\Gamma = \partial \bar D\) and let the matrix \(\mathfrak{M} \subset \Gamma \) satisfy condition (B_m). Then there exists a function \(f \in A_c \left( {\bar D} \right)\) such that the relation $$\mathop {\lim \sup }\limits_{n \to \infty } \left| {L_n \left( {\mathfrak{M}, f, z} \right)} \right| = \infty $$ holds almost everywhere on γ.     

Sensitivity of best recovery in the Sobolev spaces W_\infty ^{r,d} , \widetilde W_\infty ^{r,d}for perturbed sampling

Let $I^d $ be the d‐dimensional cube, $I^d = [0,1]^d $ , and let $F \ni f \mapsto Sf \in L_\infty (I^d ) $ be a linear operator acting on the Sobolev space F, where Fis either $$$$ or $$$$ where $$\left\| f \right\|_F = \sum\limits_{\left| m \right| = r} {\mathop {{\text{esssup}}}\limits_{x \in I^d } \left| {\frac{{\partial f^{\left| m \right|} }} {{\partial x_1^{m_1 } \partial x_2^{m_2 } \cdot \cdot \cdot \partial x_d^{m_d } }}(x)} \right|.} $$ We assume that the problem elements fsatisfy the condition $\sum\nolimits_{\left| m \right| = r} {{\text{esssup}}} _{x \in I^d } \left| {f^{(m)} (x)} \right| \leqslant 1 $ and that Sis continuous with respect to the supremum norm. We study sensitivity of optimal recovery of Sfrom inexact samples of ftaken at npoints forming a uniform grid on $I^d $ . We assume that the inaccuracy in reading the sample vector is measured in the pth norm and bounded by a nonnegative number δ. The sensitivity is defined by the difference between the optimal errors corresponding to the exact and perturbed readings, respectively. Our main result is that this difference is bounded by $\mathcal{A}\delta $ , where $\mathcal{A} $ is a positive constant independent of the number of samples. This indicates that the curse of dimension, which badly affects the optimal errors, does not extend to sensitivity.     

Lower Bounds for the Perron Root of a Sum of Nonnegative Matrices

Let $A^{(l)} (l = 1, \ldots ,k)$ be $n \times n$ nonnegative matrices with right and left Perron vectors $u^{(l)} $ and $v^{(l)} $ , respectively, and let $D^{(l)} $ and $E^{(l)} (l = 1, \ldots ,k)$ be positive-definite diagonal matrices of the same order. Extending known results, under the assumption that $$u^{(1)} \circ v^{(1)} = \ldots = u^{(k)} \circ v^{(k)} \ne 0$$ (where `` $ \circ $ '' denotes the componentwise, i.e., the Hadamard product of vectors) but without requiring that the matrices $A^{(l)} $ be irreducible, for the Perron root of the sum $\sum\nolimits_{l = 1}^k {D^{(l)} A^{(l)} E^{(l)} } $ we derive a lower bound of the form $$\rho \left( {\sum\limits_{l = 1}^k {D^{(l)} A^{(l)} E^{(l)} } } \right) \geqslant \sum\limits_{l = 1}^k {\beta _{l\rho } (A^{(l)} ),{\text{ }}\beta _l >0.} $$ Also we prove that, for arbitrary irreducible nonnegative matrices $A^{{\text{ (}}l{\text{)}}} (l = 1, \ldots ,k),$ , $$\rho \left( {\sum\limits_{l = 1}^k {A^{(l)} } } \right) \geqslant \sum\limits_{l = 1}^k {\alpha _{l\rho } (A^{(l)} ),} $$ where the coefficients ∝₁>0 are specified using an arbitrarily chosen normalized positive vector. The cases of equality in both estimates are analyzed, and some other related results are established. Bibliography: 8 titles.     

On simultaneous approximation by lagrange interpolating polynomials

This paper considers to replace △_m(x)=(1-x~2)~2(1/2)/n +1/n~2 in the following result for simultaneousLagrange interpolating approximation with (1-x~2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then|f~(k)(x)-P_~(k)(f,x)|=O(1)△_(n)~(a-k)(x)ω(f~(a),△(x))(‖L_n-‖+‖L_n‖),0≤k≤q,where P_n( f ,x)is the Lagrange interpolating polynomial of degree n+ 2r-1 of f on the nodes X_nU Y_n(see the definition of the text), and thus give a problem raised in [XiZh] a complete answer.     

Orlicz norm estimates for eigenvalues of matrices

Let φ be a supermultiplicative Orlicz function such that the function $t \mapsto \varphi \left( {\sqrt t } \right)$ is equivalent to a convex function. Then each complexn×n matrixT=(τ _ij ) _{i, j} satisfies the following eigenvalue estimate: $\left\| {\left( {\lambda _i \left( T \right)} \right)_{i = 1}^n } \right\|_{\ell _\varphi } \leqslant C\left\| ( \right\|\left( {\tau _{ij} } \right)_{i = 1}^n \left\| {_{_{\ell _{\varphi *} } } )_{j = 1}^n } \right\|\ell _{\bar \varphi } $ . Here, ?_* stands for Young’s conjugate function of φ, ?, $\bar \varphi $ is the minimal submultiplicative function dominating φ andC>0 a constant depending only on φ. For the power function φ(t)=t ^p ,p≥2 this is a celebrated result of Johnson, König, Maurey and Retherford from 1979. In this paper we prove the above result within a more general theory of related estimates.     

On a theorem of Jackson

We prove that $$\mathop {L_n \in Z_n }\limits^{\inf } \mathop \omega \limits^{sup^* } \mathop {f \in H_\omega }\limits^{\sup } \frac{{\left\| {f - L_n \left( f \right)} \right\|}}{{\omega \left( {\frac{\pi }{{n + 1}}} \right)}} = 1\left( {n = 0,1,2,...} \right)$$ (n=0,1,2,...), where \(\mathop {L_n \in Z_n }\limits^{\inf } \) is taken over all linear polynomial approximation methods of degree not higher than n and \(\mathop \omega \limits^{sup^* } \) over all convex moduli of continuity ω(δ).     

On the Symmetry of Extremals in the Weight Embedding Theorem

Varying conditions on the weight function f, the author is interested in whether the extremal of the ratio $\inf \frac{{\left\| {\nabla u} \right\|_{Lp} \left( \Omega \right)}}{{\left\| {fu} \right\|_{Lp} \left( \Omega \right)}}{\lambda }_{pq} \left( {\Omega ,f} \right)$ remains symmetric. Here, f is positive almost everywhere in $u \in \mathop {W_p^1 }\limits^{\text{O}} \left( \Omega \right)$ and the infimum is taken over all functions $u \in \mathop {W_p^1 \left( \Omega \right)}\limits^{\text{O}}$ . Bibliography: 23 titles.     

Two inequalities for pseudo-differential operators

Изучается ограничен ность псевдодиффере нциальных операторов на \(L^2 (R^n )\) и на пр остранствах Харди в \(R^n \) . Пусть \(D_k = \{ \xi \in R^n :2^{k - 1} \leqq \left| \xi \right|< 2^k \} , k = 1,2,3, \ldots ,\) и \(D_0 = \{ \xi \in R^n :\left| \xi \right|< 1\} \) . Псевдодиффер енциальный операторP с символом p определяется соотно шением $$Pf(x) = \int\limits_{R^n } {e^{ix \cdot \xi } p(x,\xi )\hat f(\xi )d\xi ,x \in R^n .} $$ Будем говорить, что p пр инадлежит классу \(\bar S_{\varrho ,} {}_\delta (M,N), 0 \leqq \delta ,\varrho \leqq 1\) , ес ли $$\left| {D_x^a p(x,\xi )} \right| \leqq C_a (1 + \left| \xi \right|)^{\delta \left| a \right|} , x,\xi \in R^n ,\left| a \right| \leqq M,$$ и $$\int\limits_{D_k } {\left| {D_x^a D_\xi ^\beta p(x,\xi )} \right|d\xi \leqq C_{a\beta } 2^{kn} 2^{k(\delta |a| - \varrho |\beta |)} , x} \in R^n , k = 0,1,2, \ldots ;|a| \leqq M, |\beta | \leqq N.$$ Изучаются условия, ко торым должны удовлет ворять ?. δ,M иN, чтобы для каждого символа \(p \in \bar S_\varrho , {}_\delta (M,N)\) соответствующий оп ераторP был ограниче н на \(L^2 (R^n )\) . Далее, пусть \(p \in S_\varrho , {}_\delta \) , если дл я всех мультииндексо в а и β выполнено условие $$|D_x^a D_\xi ^\beta p(x,\xi )| \leqq C_{a\beta } (1 + |\xi |)^{\delta |\alpha | - \varrho |\beta |} , x,\xi \in R^n .$$ Доказывается, что при 0≦δ<1 операторP отображ ает пространство Харди \(H^p (R^n )\) в локальное пространство Харди ? ^p , если символp принадл ежит классуS ₁, δ.     

Strong approximation of Riesz means at critical index on Hp(T) (0<p≤1)

This paper is a continuation of [3]. Suppose f∈H^p(T), 0σ _r ^σ f,σ=1/p?1. When p=1, it is just the partial Fourier sums S_kf. In this paper we establish the sharp estimations on the degree of approximation: $$\left\{ { - \frac{1}{{logR}}\int\limits_1^R {\left\| {\sigma _r^\delta f - f} \right\|_{H^p (T)}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqq C{\mathbf{ }}{}_p\omega \left( {f,{\mathbf{ }}( - \frac{1}{{logR}})^{1/p} } \right)_{H^p (T)} ,0< p< 1,$$ and \(\frac{1}{{\log L}}\sum\limits_{k - 1}^L {\frac{{\left\| {S_k f - f} \right\|_H 1_{(T)} }}{k} \leqq Cp\omega (f; - \frac{1}{{\log L}})_H 1_{(T)} } \) Where $$\omega (f,{\mathbf{ }}h)_{H^p (T)} \begin{array}{*{20}c} { = Sup} \\ {0 \leqq \left| u \right| \leqq h} \\ \end{array} \left\| {f( \cdot + u) - f( \cdot )} \right\|_{H^p (T).} $$ .