L p-approximation by Kantorovič operators

Оператор Канторович а дляf∈L _p(I), I=[0,1], определяе тся соотношением $$P_n (f,x) = (n + 1)\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} x^k (1 - x)^{n - 1} \int\limits_{I_k } {f(t)dt,} $$ гдеI _k=[k/(n}+1),(k+1)/(n+ 1)],n∈N. Доказывается, что есл ир>1 иf∈W _p ² (I), т.е.f абсол ютно непрерывна наI иf″∈L _p(I), то $$\left\| {P_n f - f} \right\|_p = O(n^{ - 1} ).$$ Далее, установлено, чт о еслиf∈L _p(I),p>1 и ∥P _n f-f∥_р=О(n ^?1), тоf∈S, гдеS={f∶f аб-солютно непрерывна наI, x(1?x)f′(x)=∝ ₀ ^x h(t)dt, гдеh∈L _p(I) и ∝ ₀ ¹ h(t)dt=0}. Если жеf∈L_p(I),p>1, то из условия ∥P _n(f)?f∥_pL=o(n^?1) вытекает, чтоf постоянна почти всюду.     

Order preserving nonexpansive operators inL 1

We study the limit behaviour ofT ^k f and of Cesaro averagesA _n f of this sequence, whenT is order preserving and nonexpansive inL ₁ ⁺ . IfT contracts also theL _∞-norm, the sequenceT ⁿ f converges in distribution, andA _n f converges weakly inL _p (1<p<∞), and also inL ₁ if the measure is finite. “Speed limit” operators are introduced to show that strong convergence ofA _n f need not hold. The concept of convergence in distribution is extended to infinite measure spaces. Much of this work was done during a visit of the first author at Ben Gurion University of the Negev in Beer Sheva, supported by the Deutsche Forschungsgemeinschaft.     

On density of interpolation points,a Kadec-type theorem,and Saff's principle of contamination inL p-approximation

Letf ∈L _p (I) and denote byB _n,p (f) the polynomial of bestL _p-approximation tof of degreen (1<p<∞,I=[?1,1], the norm is weightedL _p-norm with an arbitrary positive weight). Extending a result proved by Saff and Shekhtman forp=2 we show that for every 1<p<∞ andf ∈L _p (I) (not a polynomial) points of sign change of the error functionf-B _n,p (f) are dense inI asn→∞.     

Pointwise convergence of the iterates of a harris-recurrent Markov operator

LetP be a Markov operator recurrent in the sense of Harris, withσ-finite invariant measureμ. (1) Ifμ is finite andP aperiodic, then forf ∈L ₁(μ),P ⁿf →f fdμ a.e. (2) Ifμ is infinite,P ⁿf → 0 a.e. for everyf ∈L _p (μ), 1≦p <∞.     

On approximation of convex functions by rational ones in integral metrics

Пустьf - действительн означная конечная фу нкция на конечном отрезке Δ=[а, b] вещественной оси, |Δ|=b?a, M(f) = sup {|f(x)|: x∈Δ}, R_n(f,p Δ) = inf∥f?r∥_Lp(Δ) (0 < p < ∞), где нижняя грань бере тся по всем рациональ ным функциямr порядка не вышеп, K(М, Δ) класс всех выпуклых на отре зке Δ функцийf, для кот орыхM(f)≦M. Теорема.При любом вещ ественном р, 0<р<∞ и вс ехп=1, 2, ... sup {R_n(f, p, Δ):f∈K(M, Δ)} ≦ C(p)M|Δ|^1/pn^?2,где С(р) - величина, зави сящая лишь от р.     

Скорость рациональн ой аппроксимации и дифференциальные с войства функций

The paper deals with the order of best rational approximation of some classes of functions, depending on their differentiability properties. Improvements and generalizations of some results by P. P. Petrushev, V. A. Popov and the author are obtained. The proofs are based on the author's direct rational approximation theorems received recently. One of the results reads as follows. LetR _n (f,L _p ) denote the value of the best approximation of a functionf inL _p ,f∈L _p [0,1], by rational fractions of degree not exceedingn, n≧1. Suppose that 0<p≦∞,s∈NU{0}, andp≠∞ fors=0. Iff is thes-th primitive of some function of bounded variation on [0,1], then $$\sum\limits_{n = 1}^\infty {\frac{1}{n}(n^{s + 1} R_n (f,L_p ))^2< \infty } $$ . This statement is exact. Namely, for everys, s∈NU {0}, and every sequence {a _n } _n=1 ^∞ , $$a_n \geqq a_{n + 1} and \sum n^{ - 1} (n^{s + 1} a_n )^2< \infty ,$$ , there exists a functiong of the classC ^s+1 [0,1] satisfying the inequalities $$R_n (g, L_p ) \geqq c(p)a_{12} , n = 1, 2, \ldots ,$$ , for everyp, p∈(0, ∞).     

Methods of summation and best approximation

Let λ={λ _k ⁿ } be a triangular method of summation,f ε L_p (1 ≤ p ≤ ∞), $$U_n (f,x,\lambda ) = \frac{{a_0 }}{2} + \sum _{k = 1}^n \lambda _k^n (a_k \cos kx + b_k \sin kx).$$ Consideration is given to the problem of estimating the deviations ∥f ? U_n (f, λ) ∥ L_p in terms of a best approximation E_n (f) L_p in abstract form (for a sequence of projectors in a Banach space). Various generalizations of known inequalities are obtained.     

Kolmogorov-type inequalities for norms of Riesz derivatives of functions of several variables with Laplacian bounded in L ∞ and related problems

Let L _∞,∞ ^Δ (? ^m ) be the space of functions f ∈ L _∞(? ^m ) such that Δf ∈ L _∞(? ^m ). We obtain new sharp Kolmogorov-type inequalities for the L _∞-norms of the Riesz derivatives D ^α f of the functions f ∈ L _∞,∞ ^Δ (? ^m ) and solve the Stechkin problem of approximating an unbounded operator D ^α by bounded operators on the class f ∈ L _∞(? ^m ) such that ‖Δf‖_∞ ≤ 1, and also the problem of the best recovery of the operator D ^α from elements of this class given with error δ.     

Le théorème ergodique pour les opérateurs positifs sur les espaces Lp (1<p<∞) revisitéErgodic theorem for positive operators on Lp-spaces Lp (1<p<∞) revisited

We consider positive linear operators on L_p-spaces (1<p<∞), (A(L_p⁺)?L_p⁺), satisfying the inequality A^m+n<A^m+Aⁿ for all $\text{m,n∈}\text{N}\text{.}$ We describe the structure of these operators (Theorem 1). As a consequence we obtain for all f∈L^p,Aⁿf converges a.e. The last statement contains the theorem of a.e. convergence of Cesaro averages for positive mean bounded operators. To cite this article: A. Brunel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 205–207.     

Real analytic expansion of spectral projections and extension of the Hecke-Bochner identity

In this article, we review the Weyl correspondence of bigraded spherical harmonics and use it to extend the Hecke-Bochner identities for the spectral projections f × φ _k ^n?1 for function f ∈ L ^p (? ⁿ ) with 1 ≤ p ≤ ∞. We prove that spheres are sets of injectivity for the twisted spherical means with real analytic weight. Then, we derive a real analytic expansion for the spectral projections f × φ _k ^n?1 for function f ∈ L ²(? ⁿ ). Using this expansion we deduce that a complex cone can be a set of injectivity for the twisted spherical means.     

Non-L 1 functions with rotation sets of Hausdorff dimension one

Suppose that f: ? → ? is a given measurable function, periodic by 1. For an α ∈ ? put M _n ^α f(x) = 1/n+1 Σ _k=0 ⁿ f(x + kα). Let Γ _f denote the set of those α’s in (0;1) for which M _n ^α f(x) converges for almost every x ∈ ?. We call Γ _f the rotation set of f. We proved earlier that from |Γ _f | > 0 it follows that f is integrable on [0; 1], and hence, by Birkhoff’s Ergodic Theorem all α ∈ [0; 1] belongs to Γ _f . However, Γ _f \? can be dense (even c-dense) for non-L ¹ functions as well. In this paper we show that there are non-L ¹ functions for which Γ _f is of Hausdorff dimension one.     

Unconditional structures of translates for L p (ℝ d )

We prove that a sequence (f _i ) _i=1 ^∞ of translates of a fixed f ∈ L _p (?) cannot be an unconditional basis of L _p (?) for any 1 ≤ p < ∞. In contrast to this, for every 2 < p < ∞, d ∈ ? and unbounded sequence (λ _n ) _n∈? ? ? ^d we establish the existence of a function f ∈ L _p (? ^d ) and sequence (g _n *) _n∈? ? L _p *(? ^d ) such that \({({T_{{\lambda _n}}}f,g_n^*)_{n \in {\Bbb N}}}\) forms an unconditional Schauder frame for L _p (? ^d ). In particular, there exists a Schauder frame of integer translates for L _p (?) if (and only if) 2 < p < ∞.     

Improved error bounds for near-minimax approximations

Givenf εC ⁽ⁿ⁺¹⁾[?1, 1], a polynomialp _n, of degree ≤n, is said to be near-minimax if (*) $$\left\| {f - p_n } \right\|_\infty = 2^{ - n} |f^{(n + 1)} (\xi )|/(n + 1)!,$$ for some ζ ε (?1,1). For three sets of near-minimax approximations, by considering the form of the error ∥f ?p _n∥_∞ in terms of divided differences, it is shown that better upper and lower bounds can be found than those given by (*).     

Uniform ergodic convergence and averaging along Markov chain trajectories

LetP(x, A) be a transition probability on a measurable space (S, Σ) and letX _n be the associated Markov chain.Theorem. Letf∈B(S, Σ). Then for anyx∈S we haveP _x a.s. $$\mathop {\underline {\lim } }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {f(X_k ) \geqslant } \mathop {\underline {\lim } }\limits_{n \to \infty } \mathop {\inf }\limits_{x \in S} \frac{1}{n}\sum\limits_{k = 1}^n {P^k f(x)} $$ and (implied by it) a corresponding inequality for the lim. If 1/n∑ _k=1 ⁿ P ^k f converges uniformly, then for everyx∈S, 1/n ∑ _k=1 ⁿ f(X _k ) convergesP _x a.s. Applications are made to ergodic random walks on amenable locally compact groups. We study the asymptotic behavior of 1/n∑ _k=1 ⁿ μ ^k *f and of 1/n∑ _k=1 ⁿ f(X _k ) via that ofΨ _n *f(x)=m(A _n )^?1∫ _An f(xt), where {A _n } is a Følner sequence, in the following cases: (i)f is left uniformly continuous (ii) μ is spread out (iii)G is Abelian. Non-Abelian Example: Let μ be adapted and spread-out on a nilpotent σ-compact locally compact groupG, and let {A _n } be a Følner sequence. If forf∈B(G, ∑) m(A _n )^?1∫ _An f(xt)dm(t) converges uniformly, then 1/n∑ _k=1 ⁿ f(X _k ) converges uniformly, andP _x convergesP _x a.s. for everyx∈G.     

Maximum modulus sets in pseudoconvex boundaries

LedD be a strictly pseudoconvex domain in ? ⁿ withC ^∞ boundary. We denote byA ^∞(D) the set of holomorphic functions inD that have aC ^∞ extension to \(\bar D\) . A closed subsetE of ?D is locally a maximum modulus set forA ^∞(D) if for everyp∈E there exists a neighborhoodU ofp andf∈A ^∞(D∩U) such that |f|=1 onE∩U and |f|<1 on \(\bar D \cap U\backslash E\) . A submanifoldM of ?D is an interpolation manifold ifT _p (M)?T _p ^c (?D) for everyp∈M, whereT _p ^c (?D) is the maximal complex subspace of the tangent spaceT _p (?D). We prove that a local maximum modulus set forA ^∞ (D) is locally contained in totally realn-dimensional submanifolds of ?D that admit a unique foliation by (n?1)-dimensional interpolation submanifolds. LetD =D ₁ x ... xD _r ? ? ⁿ whereD _i is a strictly pseudoconvex domain withC ^∞ boundary in ? ⁿ _i ,i=1,…,r. A submanifoldM of ?D ₁×…×?D _r verifies the cone condition if \(II_p (T_p (M)) \cap \bar C[Jn_1 (p),...,Jn_r (p)] = \{ 0\} \) for everyp∈M, wheren _i (p) is the outer normal toD _i atp, J is the complex structure of ? ⁿ , \(\bar C[Jn_1 (p),...,Jn_r (p)]\) is the closed positive cone of the real spaceV _p generated byJ _{n 1}(p),…,J _{n r}(p), and II _p is the orthogonal projection ofT _p (?D) onV _p . We prove that a closed subsetE of ?D ₁×…×?D _r which is locally a maximum modulus set forA ^∞(D) is locally contained inn-dimensional totally real submanifolds of ?D ₁×…×?D _r that admit a foliation by (n?1)-dimensional submanifolds such that each leaf verifies the cone condition at every point ofE. A characterization of the local peak subsets of ?D ₁×…×?D _r is also given.     

Weak generalized localization for multiple Fourier series whose rectangular partial sums are considered with respect to some subsequence

In this paper, we obtain the structural and geometric characteristics of some subsets of $ \mathbb{T} $ ^N = [?π, π] ^N (of positive measure), on which, for the classes L _p ( $ \mathbb{T} $ ^N ), p > 1, where N ≥ 3, weak generalized localization for multiple trigonometric Fourier series is valid almost everywhere, provided that the rectangular partial sums S _n (x; f) (x ∈ $ \mathbb{T} $ ^N , f ∈ L _p ) of these series have a “number” n = (n ₁,…, n _N ) ∈; ? ₊ ^N such that some components n _j are elements of lacunary sequences. For N = 3, similar studies are carried out for generalized localization almost everywhere.     

Nearly Umbilical Ovaloids in the n-Space are close to Spheres

An ovaloid F in the euclidean n-space ?ⁿ (n ≧ 3) is called δ - umbilical (δ ≧ 0) if δ equals the L_∞-norm, i.e. the maximum of the difference of the greatest and smallest radius of curvature at each point of F. We prove that every δ-umbilical ovaloid F is ∈-close to a sphere (i.e. with minimal Hausdorff distance ∈ ≧ 0 to any sphere in ?ⁿ) with $\epsilon {<\over -} (2(1+2\root +\of{2}))^{n-3}\ \delta$ . Hereby the factor $1+2+\root +\of {2}$ may be removed if F is centrally symmetric. In the case n = 3 the L_∞-norm for δ may be replaced by the (weaker) L₁-norm.     

Tail field representations and the zero-two law

The zero-two law was proved for a positiveL ₁-contractionT by Ornstein and Sucheston, and gives a condition which impliesT ⁿ f−T ⁿ⁺¹ f → 0 for allf. Extensions of this result to the case of a positiveL _p -contraction, 1≤p<∞, have been obtained by several authors. In the present paper we prove a theorem which is related to work of Wittmann. We will say that a positive contractionT contains a circle of lengthm if there is a nonzero functionf such that the iterated valuesf, T f,…,T ^m-1 f have disjoint support, whileT ^m f=f. Similarly, a contractionT contains a line if for everym there is a nonzero functionf (which may depend onm) such thatf,…,T ^m-1 f have disjoint support. Approximate forms of these conditions are defined, which are referred to as asymptotic circles and lines, respectively. We show (Theorem 3) that if the conclusionT ⁿ f−T ⁿ⁺¹ f→0 of the zero-two law does not hold for allf inL _p , then eitherT contains an asymptotic circle orT contains an asymptotic line. The point of this result is that any condition onT which excludes circles and lines must then imply the conclusion of the zero-two law. Theorem 3 is proved by means of the representation of a positiveL _p -contraction in terms of anL _p -isometry. Asymptotic circles and lines forT correspond to exact circles and lines for the isometry on tail-measurable functions, and exact circles and lines for the isometry are obtained using the Rohlin tower construction for point transformations. Research supported in part by NSERC.     

Analogues of the “zero-two” law for positive linear contractions inL p andC(X)

LetT be a positive linear contraction inL ^p (1≦p<∞), then we show that lim ‖T ^pf −T ⁿ⁺¹ f‖ _p ≦(1 − ε)2^1/p (f∈L _p ⁺ , ε>0 independent off) implies already lim_n n→∞ ‖T ⁿf −T ⁿ⁺¹ n+1f ‖_p p=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown.