Invariant densities and mean ergodicity of markov operators

We prove that a Markov operatorT onL ₁ has an invariant density if and only if there exists a densityf that satisfies lim sup _n→∞‖T ⁿ f − f‖ < 2. Using this result, we show that a Frobenius-Perron operatorP is mean ergodic if and only if there exists a densityw such that lim sup _n→∞ ‖P ⁿ f − w‖<2 for every densityf. Corresponding results hold for strongly continuous semigroups.     

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.     

环域上一类拟线性椭圆型方程的正径向解的存在性

The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide.     

A question of Gol’dberg concerning entire functions with prescribed zeros

Let (z_j) be a sequence of complex numbers satisfying |z_j|→ ∞ asj → ∞ and denote by n(r) the number of z_j satisfying |z_j|≤ r. Suppose that lim inf_{r → ⇈} log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫^∞ (ϕ(t)t logt)⁻¹ dt < ∞. It is proved that there exists an entire functionf whose zeros are the z_j such that log log M(r,f) = o((log n(r))²ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here. These results answer a question by A. A. Gol’dberg.     

On large deviations and density functions

Suppose thatX ₁,X ₂, ... is a sequence of absolutely continuous or integer valued random variables with corresponding probability density functionsf _n (x). Let {φ _n } _n=1 ^∞ be a sequence of real numbers, then necessary and sufficient conditions are given forn ⁻¹ logf _n (φ _n )-n ⁻¹ log P (X _n >φ _n )=0(1) asn→∞.     

Minimax goodness-of-fit testing in multivariate nonparametric regression

We consider an unknown response function f defined on Δ = [0, 1] ^d , 1 ≤ d ≤ ∞, taken at n random uniform design points and observed with Gaussian noise of known variance. Given a positive sequence r _n → 0 as n → ∞ and a known function f ₀ ∈ L ₂(Δ), we propose, under general conditions, a unified framework for goodness-of-fit testing the null hypothesis H ₀: f = f ₀ against the alternative H ₁: f ∈ $ \mathcal{F} $ \mathcal{F} , ∥f − f ₀∥ ≥ r _n , where $ \mathcal{F} $ \mathcal{F} is an ellipsoid in the Hilbert space L ₂(Δ) with respect to the tensor product Fourier basis and ∥ · ∥ is the norm in L ₂(Δ). We obtain both rate and sharp asymptotics for the error probabilities in the minimax setup. The derived tests are inherently non-adaptive. Several illustrative examples are presented. In particular, we consider functions belonging to ellipsoids arising from the well-known multidimensional Sobolev and tensor product Sobolev norms as well as from the less-known Sloan-Woźniakowski norm and a norm constructed from multivariable analytic functions on the complex strip.     

On the solutions of the translation equation in rings of formal power series

We study in this paper solutions of the translation equation in rings of formal power series K[X] where K ∈R, C (so called one-parameter groups or flows), and even, more generally, homomorphisms Ф from an abelian group (G, +) into the group Г(K) of invertible power series in K[X]. This problem can equivalently be formulated as the question of constructing homomorphisms Ф from (G, +) into the differential group Г¹∞ describing the chain rules of higher order of C∞ functions with fixed point 0. In this paper we present the general form of these homomorphisms Ф : G → Г(K) (or L¹∞),Ф = (f_n ⁿ≤1,forwhich f₁ = l, f₂ = ... = f_p+l =0,f_p+2 ≠ 0 for fixed, but arbitrary p ≤ 0 (see Theorem 5, Corollary 6 and Theorem 6). This representation uses a sequence (w _n ^p )_n≥p+2 of universal polynomials in f_p+2 and a sequence of parameters, which determines the individual one-parameter group. Instead of (w _n ^p )_n≥p+2 we may also use another sequence (L _n ^p )_n≥p+2 of universal polynomials, and we describe the connection between these forms of the solutions.     

Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part II: Clenshaw-Lord type approximants

Laurent-Padé (Chebyshev) rational approximantsP _m (w, w ⁻¹)/Q _n (w, w ⁻¹) of Clenshaw-Lord type [2,1] are defined, such that the Laurent series ofP _m /Q _n matches that of a given functionf(w, w ⁻¹) up to terms of orderw ^±(m+n) , based only on knowledge of the Laurent series coefficients off up to terms inw ^±(m+n) . This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent series ofP _m matches that ofQ _n f up to terms of orderw ^±(m+n ), but based on knowledge of the series coefficients off up to terms inw ^±(m+2n). The Clenshaw-Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of the Padé-Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations are obtainable for allm≥0,n≥0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé-Chebyshev approximants. Remarkably similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw-Lord type methods, thus validating the use of either.     

Almost Everywhere Convergence of Fejér Means of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> Functions on Rarely Unbounded Vilenkin Groups

A highly celebrated problem in dyadic harmonic analysis is the pointwise convergence of the Fejér (or (C, 1)) means of functions on unbounded Vilenkin groups. There are several papers of the author of this paper concerning this. That is, we know the a.e. convergence σ _n f → f (n → ∞) for functions f ∈ L ^p , where p > 1 (Journal of Approximation Theory, 101(1), 1–36, (1999)) and also the a.e. convergence σM _n f → f (n → ∞) for functions f ∈ L ¹ (Journal of Approximation Theory, 124(1), 25–43, (2003)). The aim of this paper is to prove the a.e. relation lim _n → σ _n f = f for each integrable function f on any rarely unbounded Vilenkin group. The concept of the rarely unbounded Vilenkin group is discussed in the paper. Basically, it means that the generating sequence m may be an unbounded one, but its "big elements" are not "too dense". Research supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. M 36511/2001 and T 048780     

On the iteration of holomorphic self-maps of ℂ*

Letf be a holomorphic self-map of the punctured plane ℂ^*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI ₀(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞} andI _∞(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞}. We try to find the relation betweenI ₀(f),I _∞(t) andJ(f). It is proved that both the boundary ofI ₀(f) and the boundary ofI _∞)f) equal toJ(f),I ₀(f) ∩J(f) ≠ θ andI _∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI ₀(f) andI _∞(f) are not doubly-bounded. Supported by the National Natural Science Foundation of China     

Growth Behavior and Zero Distribution of Rational Approximants

We investigate the growth and the distribution of zeros of rational uniform approximations with numerator degree ≤n and denominator degree ≤m _n for meromorphic functions f on a compact set E of ℂ where m _n =o(n/log n) as n→∞. We obtain a Jentzsch–Szegő type result, i.e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain E _ρ(f) of meromorphy of f if f has a singularity of multivalued character on the boundary of E _ρ(f). The paper extends results for polynomial approximation and rational approximation with fixed degree of the denominator. As applications, Padé approximation and real rational best approximants are considered.     

Norms of powers of absolutely convergent fourier series

Letf(t) = ∑a _k e ^ikt be infinitely differentiable on R, |f(t)|<1. It is known that under these assumptions ‖ⁿ‖ converges to a finite limitl asn → ∞ (l ² = sec(arga),a = (f′(0))² -f″(0)). We obtain here more precise results: (i) an asymptotic series (in powers ofn ^-1/2) for the Fourier coefficientsa _nk off ⁿ , which holds uniformly ink asn → ∞; (ii) an asymptotic series (this time only powers ofn ^-1 are present!) for ‖f ⁿ ‖; (iii) the fact that ifi ^j f ^(j)(0) is real forj = 1,2,..., 2h + 2 then ‖f ⁿ ‖ = l + o(n ^-h ),n → ∞. More generally, we obtain analogous finite asymptotic expansions whenf is assumed to be differentiable only finitely many times.     

On the bestL 1-approximations of functional classes by splines under restrictions imposed on their derivatives

We find the exact asymptotics (asn→∞) of the bestL ₁-approximations of classesW ₁ ^r of periodic functions by spliness∈S _{2n, r∼-1} (S _{2n, r∼-1} is a set of 2π-periodic polynomial splines of orderr−1, defect one, and with nodes at the pointskπ/n,k∈ℤ) such that V ₀ ^2π s( ^r-1)≤1+ɛ _n , where {ɛ _n } _n=1 ^∞ is a decreasing sequence of positive numbers such that ɛ _n n ²→∞ and ɛ _n →0 asn→∞. Dnepropetrovsk University, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 435–444, April, 1999.     

On the order of growth of solutions of algebraic differential equations

Assume thatf is an integer transcendental solution of the differential equationP _n (z, f, f′)=P _n−1(z, f, f′, ... f ^(p)), whereP _n andP _n−1 are polynomials in all variables, the degree ofP _n with respect tof andf′ is equal ton, and the degree ofP _n−1 with respect tof, f′, ... f ^(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E _*, whereE _* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z ^1/2 (β+o(1)), β∈C, holds forz=re ^iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re ^iν)‖=o(r ^1/2),r→+∞,r>0, , where Δ is a certain set on the semiaxisr>0 with mes Δ<∞. “L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77, January, 1999.     

On the Proximal Point Algorithm

Let A be a maximal monotone operator in a real Hilbert space H and let {u _n } be the sequence in H given by the proximal point algorithm, defined by u _n =(I+c _n A)⁻¹(u _n−1−f _n ), ∀ n≥1, with u ₀=z, where c _n >0 and f _n ∈H. We show, among other things, that under suitable conditions, u _n converges weakly or strongly to a zero of A if and only if lim inf  _n→+∞|w _n |<+∞, where w _n =(∑ _k=1 ⁿ c _k )⁻¹∑ _k=1 ⁿ c _k u _k . Our results extend previous results by several authors who obtained similar results by assuming A ⁻¹(0)≠φ.     

Immagini inverse dei gruppi di omotopia

Riassunto SeM edN sono varietà poliedriche chiuse connesse ed orientate di dimensioni rispettivem edn, conm≥n>2, edf∶M→N è una trasformazione continua, allora per ognir, minore din e non inferiore a 2, si definisce un omomorfismo indotto ϕ_rπ:_r (N)→H _m-n+r (M) dal quale si ricavano certi invarianti topologici.

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Résumé Soientm≥n>r≥2 des entiers etM, N des variétés polyédrales closes connexes orientées satisfaisant dimM=m et dimN=n, de plusH _i(M) le groupe de Betti à i dimensions deM,M,π _i (N) le groupe de Hurewicz ài dimensions deN, etf∶M→N une application continue. Alorsf définit, pour,r=2, 3, …n−1, un homomorphisme réciproque ϕ_rπ:_r (N)→H _m-n+r (M) comme il suit. Etant donné un élément α du groupe π_r (N) et uner-sphère continue orientéeS de α, on peut supposer quef ⁻¹(S) soit un polyèdre finiA àm−n+r dimensions. Parf est induit dansA un (m−n+r)-cyclez à coefficients entiers, et la classe d'homologie dez est justement l'image ϕr(α) de α par ϕr. Pourr=1, on obtient un homomorphisme réciproque ϕ_rπ:_r (N)→H _m-n+r (M) du groupe fondamentalF(N) deN dans le groupe d'homologie àm−n+1 dimensions deM. A l'aide des homomorphismes ϕ,,ϕ2,ϕ,3...,ϕn-i, on parvient à certaines expressions caractéristiques dépendantes seulement de la classe d'homotopie def, en particulier on obtient des constantes pour les images des bases de Betti deM, pour Fimage du groupe de torsion deM, et pour l'image réciproque du groupe fondamental deN.

     

Summation rational operators of the Jackson type

Summation rational positive operatorsD _4n−n(x; f) of the Jackson type are constructed on the real axis. The corresponding approximations of continuous functionsf onℝwith coinciding finite limits lim_x→−∞ f(x) and lim_x→+∞ f(x) are estimated. Translated fromMatematischeskie Zametki, Vol. 61, No. 2, pp. 270–277, February, 1997. Translated by N. K. Kulman     

On semi-fredholm properties of a boundary value problem inR + n

The paper considers a boundary value problem with the help of the smallest closed extensionL ^∼ :H _k →H _{k 0}×B _{h 1}×...×B _{h N} of a linear operatorL :C ₍₀₎ ^∞ (R ₊ ⁿ ) →L(R ₊ ⁿ )×L(R ⁿ⁻¹)×...×L(R ⁿ⁻¹). Here the spacesH _k (the spaces ℬ _h ) are appropriate subspaces ofD′(R ₊ ⁿ ) (ofD′(R ⁿ⁻¹), resp.),L(R ₊ ⁿ ) andC ₍₀₎ ^∞ (R ₊ ⁿ )) denotes the linear space of smooth functionsR ⁿ →C, which are restrictions onR ₊ ⁿ of a function from the Schwartz classL (fromC ₀ ^∞ , resp.),L(R ⁿ⁻¹) is the Schwartz class of functionsR ⁿ⁻¹ →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ^∼) and for the uniqueness of solutionsL ^∼ U=F are expressed. In addition, ana priori estimate for the corresponding boundary value problem is established.     

On a Delay Reaction-Diffusion Difference Equation of Neutral Type

In this paper, we first consider a delay difference equation of neutral type of the form: Δ(y _n + py _n−k + q _n y _n−l = 0 for n∈ℤ⁺(0) (1*) and give a different condition from that of Yu and Wang (Funkcial Ekvac, 1994, 37(2): 241–248) to guarantee that every non-oscillatory solution of (1*) with p = 1 tends to zero as n→∞. Moreover, we consider a delay reaction-diffusion difference equation of neutral type of the form: Δ₁(u _n,m + pu _n−k,m ) + q _n,m u _n−l,m = a ²Δ² ₂ u _{n +1, m−1} for (n,m) ∈ℤ⁺ (0) ×Ω, (2*) study various cases of p in the neutral term and obtain that if p≥−1 then every non-oscillatory solution of (2*) tends uniformly in m∈Ω to zero as n→∞; if p = −1 then every solution of (2*) oscillates and if p < −1 then every non-oscillatory solution of (2*) goes uniformly in m∈Ω to infinity or minus infinity as n→∞ under some hypotheses. Received July 14, 1999, Revised August 10, 2000, Accepted September 30, 2000     

Potential spaces and traces of Lévy processes on <Emphasis Type="Italic">h</Emphasis>-sets

Potential spaces and Dirichlet forms associated with Lévy processes subordinate to Brownian motion in ℝ ⁿ with generator f(−Δ) are investigated. Estimates for the related Rieszand Bessel-type kernels of order s are derived which include the classical case f(r) = r ^α/2 with 0 < α < 2 corresponding to α-stable Lévy processes. For general (tame) Bernstein functions f potential representations of the trace spaces, the trace Dirichlet forms, and the trace processes on fractal h-sets are derived. Here we suppose the trace condition ∫₀¹ r ⁻⁽ⁿ⁺¹⁾ f(r ⁻²)⁻¹ h(r) dr < ∞ on f and the gauge function h. Dedicated to the 80th birthday of Klaus Krickeberg