To the problem of a strong differentiability of integrals along different directions

It is proved that for any given sequence (σ _n ,n ∈ ℕ)=Γ₀ ⊂ Γ, where Γ is the set of all directions in ℝ² (i.e., pairs of orthogonal straight lines) there exists a locally integrable functionf on ℝ² such that: (1) for almost all directionsσ ∈ Γ\Γ₀ the integral ∫f is differentiable with respect to the familyB _2σ of open rectangles with sides parallel to the straight lines fromσ: (2) for every directionσ _n ∈ Γ₀ the upper derivative of ∫f with respect toB _{2σ n} equals +∞; (3) for every directionσ ∈ Γ the upper derivative of ∫ |f| with respect toB _2σ equals +∞.     

On the strong law of large numbers for dependent random variables

Let {X _n} _n ^=1/∞ be a sequence of random variables with partial sumsS _n, and let {ie241-1} be the σ-algebra generated byX ₁,…,X _n. Letf be a function fromR toR and suppose {ie241-2}. Under conditions off and moment conditions on theX' _ns, we show thatS _n/n converges a.e. (almost everywhere). We give several applications of this result. Research supported by N.S.F. Grant MCS 77-26809     

Joint ergodicity and mixing

The study of jointly ergodic measure preserving transformations of probability spaces, begun in [1], is continued, and notions of joint weak and strong mixing are introduced. Various properties of ergodic and mixing transformations are shown to admit analogues for several transformations. The case of endomorphisms of compact abelian groups is particularly emphasized. The main result is that, given such commuting endomorphisms σ₁σ₂,...,σ, ofG, the sequence ((1/N)Σ _n=0 ^N−1 σ ₁ ⁿ f ¹·σ ₂ ⁿ f ²· ··· · σ _s ⁿ f ^sconverges inL ²(G) for everyf ₁,f ₂,…,f _s∈L ^∞(G). If, moreover, the endomorphisms are jointly ergodic, i.e., if the limit of any sequence as above is Π _i=1 ^s ∫ _G f ₁ d μ, where μ is the Haar measure, then the convergence holds also μ-a.e.     

Almost everywhere convergence of Cesàro means of Fourier series on the group of 2-adic integers

We prove the almost everywhere convergence of the Cesàro (C, α)-means of integrable functions σ _n ^α f → f for f ∈ L ¹(I), where I is the group of 2-adic integers for every α > 0. This theorem for the case of α = 1 was proved by the author [1]. For the case of the (C, 1) Fejér means there are several generalizations known with respect to some orthonormal systems. One could mention the papers [2, 9]. Research supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. T 048780.     

On almost sure convergence of Cesaro averages of subsequences of vector-valued functions

A remarkable theorem proved by Komlòs [4] states that if {f_n} is a bounded sequence in L¹(R), then there exists a subsequence {f_nk} and f L¹(R) such that f_nk (as well as any further subsequence) converges Cesaro to f almost everywhere. A similar theorem due to Révész [6] states that if {f_n} is a bounded sequence in L²(R), then there is a subsequence {f_nk} and f L²(R) such that Σ_k=1^∞ a_k(f_nk − f) converges a.e. whenever Σ_k=1^∞ | a_k |² < ∞. In this paper, we generalize these two theorems to functions with values in a Hilbert space (Theorems 3.1 and 3.3).     

Fejér means of functions on noncommutative Vilenkin groups with respect to the character system

Summary It is known that the Fejér means - with respect to the character system of the Walsh, and bounded Vilenkin groups - of an integrable function converge to the function a.e. In this work we discuss analogous problems on the complete product of finite, not necessarily Abelian groups with respect to the character system for functions that are constant on the conjugacy classes. We find that the nonabelian case differs from the commutative case. We prove the a.e. convergence of the (C,1) means of the Fourier series of square integrable functions. We also prove the existence of a function f∊L^q for some q >1 such that sup |σ_n f | = + ∞ a.e. This is a sharp contrast between the Abelian and the nonabelian cases.     

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     

A BMO estimate for the multilinear singular integral operator

The behavior on the space L∞((R)n) for the multilinear singular integral operator defined by TAf(x)=∫Rn Ω(x-y)/|x-y|n 1(A(x)-A(y)-(△)A(y)(x-y))f(y)dy is considered, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, A has derivatives of order one in BMO((R)n). It is proved that if Ω satisfies some minimum size condition and an L1-Dini type regularity condition, then for f ∈ L∞((R)n), TAf is either infinite almost everywhere or finite almost everywhere, and in the latter case, TAf ∈ BMO((R)n).     

An ergodic theorem on Banach lattices

E is a Banach lattice that is weakly sequentially complete and has a weak unitu. TLf _n=ϕ means that the infimum of |f _n−ϕ| andu converges strongly to zero.T is a positive contraction operator onE andA _n=(1/n)(I+T+...+T ⁿ⁻¹). Without an additional assumption onE, the “truncated limit” TLA _nf need not exist forf inE. This limit exists for eachf ifE satisfies the following additional assumption (C): For everyf inE ₊ and for every numberα>0, there is a numberβ=β(f, α) such that ifg is inE ₊, ‖g‖≦1, 0≦f′≦f and ‖f′‖>α then ‖f′+g‖≧‖g‖+β. Research of this author is partially supported by NSERC Grant A3974. Research of this author is partially supported by NSF Grant 8301619.     

The riesz turndown collar theorem giving an asymptotic estimate of the powers of an operator under the ritt condition

For Banach space operatorsT satisfying the Tadmor-Ritt condition ‖(zI−T)⁻¹‖≤C|z−1|⁻¹, |z|>1, we show how to use the Riesz turndown collar theorem to estimate sup _n≥0‖T ⁿ‖. A similar estimate is shown for lim sup _n ‖T ⁿ‖ in terms of the Ritt constantM=lim sup _z→1‖(1−z)(zI−T)⁻¹‖. We also obtain an estimate of the functional calculus for these operators proving, in particular, that ‖f(T)‖≤C _q‖f‖ _Mult , where ‖·‖ _Mult stands for the multiplier norm of the Cauchy-Stieltjes integrals over a Lusin type cone domain depending onC and a parameterq, 0<q<1. Notation.D denotes the open unit disc of the complex plane,D={z∈ℂ:|z|<1}, andT={z∈ℂ:|z|=1} is the unit circle.H ^∞ is the Banach algebra of bounded analytic functions onD equipped with the supremum norm ‖.‖_∞.     

Fourier Series with the Continuous Primitive Integral

Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted A_c(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}) and is a Banach space under the Alexiewicz norm, ||f||_\mathbbT=sup_{|I| ￡ 2p}|ò_I f|\|f\|_{\mathbb{T}}=\sup_{|I|\leq2\pi}|\int_{I} f|, the supremum being taken over intervals of length not exceeding 2π. It contains the periodic functions integrable in the sense of Lebesgue and Henstock–Kurzweil. Many of the properties of L ¹ Fourier series continue to hold for this larger space, with the L ¹ norm replaced by the Alexiewicz norm. The Riemann–Lebesgue lemma takes the form [^(f)](n)=o(n)\hat{f}(n)=o(n) as |n|→∞. The convolution is defined for f ? A_c(\mathbbT)f\in{\mathcal{A}}_{c}(\mathbb{T}) and g a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate ||f*g||_￥ ￡ ||f||_\mathbbT ||g||_BV\|f\ast g\|_{\infty}\leq\|f\|_{\mathbb{T}} \|g\|_{\mathcal{BV}}. For g ? L¹(\mathbbT)g\in L^{1}(\mathbb{T}), ||f*g||_\mathbbT ￡ ||f||_{\mathbb T} ||g||₁\|f\ast g\|_{\mathbb{T}}\leq\|f\|_{\mathbb {T}} \|g\|_{1}. As well, [^(f*g)](n)=[^(f)](n) [^(g)](n)\widehat{f\ast g}(n)=\hat{f}(n) \hat{g}(n). There are versions of the Salem–Zygmund–Rudin–Cohen factorization theorem, Fejér’s lemma and the Parseval equality. The trigonometric polynomials are dense in A_c(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}). The convolution of f with a sequence of summability kernels converges to f in the Alexiewicz norm. Let D _n be the Dirichlet kernel and let f ? L¹(\mathbbT)f\in L^{1}(\mathbb{T}). Then ||D_n*f-f||_\mathbbT?0\|D_{n}\ast f-f\|_{\mathbb{T}}\to0 as n→∞. Fourier coefficients of functions of bounded variation are characterized. The Appendix contains a type of Fubini theorem.     

On the finiteness of a field-theoretic invariant for commutative rings

If T is a (commutative unital) ring extension of a ring R, then Λ(T /R) is defined to be the supremum of the lengths of chains of intermediate fields between R _P /P R _P and T _Q /QT _Q , where Q varies over Spec(T) and P:= Q ∩ R. The invariant σ(R):= sup Λ(T/R), where T varies over all the overrings of R. It is proved that if Λ(S/R)< ∞ for all rings S between R and T, then (R, T) is an INC-pair; and that if (R, T) is an INC-pair such that T is a finite-type R-algebra, then Λ(T/R)< ∞. Consequently, if R is a domain with σ(R) < ∞, then the integral closure of R is a Prüfer domain; and if R is a Noetherian G-domain, then σ(R) < ∞, with examples showing that σ(R) can be any given non-negative integer. Other examples include that of a onedimensional Noetherian locally pseudo-valuation domain R with σ(R)=∞.     

Convergence theorems for the Birkhoff integral

We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence (f _n ) of functions from a measure space to a Banach space. In one result the equi-integrability of f _n ’s is involved and we assume f _n → f almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of (f _n ) to f is assumed.     

On the almost monotone convergence of sequences of continuous functions

A sequence (f _n ) _n of functions f _n : X → ℝ almost decreases (increases) to a function f: X → ℝ if it pointwise converges to f and for each point x ∈ X there is a positive integer n(x) such that f _n+1(x) ≤ f _n (x) (f _n+1(x) ≥ f _n (x)) for n ≥ n(x). In this article I investigate this convergence in some families of continuous functions.     

T-Colorings and T-Edge Spans of Graphs

 Suppose G is a graph and T is a set of non-negative integers that contains 0. A T-coloring of G is an assignment of a non-negative integer f(x) to each vertex x of G such that |f(x)−f(y)|∉T whenever xy∈E(G). The edge span of a T-coloring−f is the maximum value of |f(x) f(y)| over all edges xy, and the T-edge span of a graph G is the minimum value of the edge span of a T-coloring of G. This paper studies the T-edge span of the dth power C ^d _n of the n-cycle C _n for T={0, 1, 2, …, k−1}. In particular, we find the exact value of the T-edge span of C _n ^d for n≡0 or (mod d+1), and lower and upper bounds for other cases. Received: May 13, 1996 Revised: December 8, 1997     

Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem

We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l ¹(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||_A(\mathbbT) = || [^(f)] ||_{l¹ (\mathbbZ)}\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || e^inf ||_A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || e^inf ||_A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.     

A minimax theorem for irreducible compact operators inL p-spaces

Let (X, Σ, μ) be a σ-finite measure space,T a compact irreducible (positive, linear) operator onL ^p (μ) (1≦p<+∞). It is shown that the spectral radiusr ofT is characterized by the minimax property {fx196-1} where ∑₀ denotes the ring of sets of finite measure and whereQ denotes the set of all, almost everywhere positive functions inL ^p. Moreover, ifr>0 then equality on either side is assumed ifff is the (essentially unique) positive eigenfunction ofT. Various refinements are given in terms of corresponding relations for irreducible finite rank operators approximatingT. Dedicated to H. G. Tillmann on his 60th birthday     

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 the existence of invariant probability measures

Let (X,A) be a measureable space andT:X →X a measurable mapping. Consider a family ℳ of probability measures onA which satisfies certain closure conditions. IfA ₀⊂A is a convergence class for ℳ such that, for everyA ∈A ₀, the sequence ((1/n) Σ _i ^=0/n−1 1 _A ∘T ⁱ) converges in distribution (with respect to some probability measurev ∈ ℳ), then there exists aT-invariant element in ℳ. In particular, for the special case of a topological spaceX and a continuous mappingT, sufficient conditions for the existence ofT-invariant Borel probability measures with additional regularity properties are obtained.     

A generalization of peripherally-multiplicative surjections between standard operator algebras

Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σ_π (T) of T is defined by σ_π (T) = z ∈ σ(T): |z| = max_w∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σ_π (φ(S)ϕ(T)) = σ_π (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A ₁ TA ₂ ⁻¹ and ϕ(T) = A ₂ TA ₁ ⁻¹ for some bijective bounded linear operators A ₁; A ₂ of X onto Y, or of the form φ(T) = B ₁ T*B ₂ ⁻¹ and ϕ(T) = B ₂ T*B ⁻¹ for some bijective bounded linear operators B ₁;B ₂ of X* onto Y.