ON STABLE LINE SEGMENTS IN ALL TRIANGULATIONS OF A PLANAR POINT SET

Abstract. Let S be a set of finite plauar points. A llne segment L(p, q) with p, q E Sis called a stable line segment of S, if there is no Line segment with two endpoints in S intersecting L(p, q). In this paper, some geometric properties of the set of all stable line segments     

The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimensions

In this paper we study the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere S ^r−1 ⊂ R^r. The hyperinterpolation approximation L _n ƒ, where ƒ ∈ C(S ^r −1), is derived from the exact L ² orthogonal projection Π ƒ onto the space P _n ^r (S ^r −1) of spherical polynomials of degree n or less, with the Fourier coefficients approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree ≤ 2n. We extend to arbitrary r the recent r = 3 result of Sloan and Womersley [9], by proving that under an additional “quadrature regularity” assumption on the quadrature rule, the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere is O(n ^r /2−1), which is the same as that of the orthogonal projection Π_n, and best possible among all linear projections onto P _n ^r (S ^r −1).     

On the commutant of certain multiplication operators on spaces of analytic functions

Let ℬ be a Banach space of analytic functions defined on the open unit disk. We characterize the commutant ofM _Z ² (the operator of multiplication by the square of independent variable defined on ℬ) and show that for an operatorS in the commutantM _Z ² ifSM _Z ^2k+1−M _Z ^2k+1 S is compact for some nonnegative integerk, thenS=M _ϕ whereϕ is a multiplier of ℬ. Letn be a positive integer andS be an operator in the commutant ofM _Z ⁿ defined on a functional Hilbert spaces of analytic functions. We show that under certain conditionsS has the formM _ϕ. Research supported by the Shiraz University Grant 78-SC-1188-657.     

Strong approximation ofGSBV functions by piecewise smooth functions

Let Ω be an open and bounded subset ofR ⁿ with locally Lipschitz boundary. We prove that the functionsv∈SBV(Ω,R ^m ) whose jump setS _vis essentially closed and polyhedral and which are of classW ^{k, ∞} (S _v,R ^m) for every integerk are strongly dense inGSBV ^p(Ω,R ^m ), in the sense that every functionu inGSBV ^p(Ω,R ^m ) is approximated inL ^p(Ω,R ^m ) by a sequence of functions {v _k{_j∈N with the described regularity such that the approximate gradients ∇v _jconverge inL ^p(Ω,R ^nm ) to the approximate gradient ∇u and the (n−1)-dimensional measure of the jump setsS _{v j} converges to the (n−1)-dimensional measure ofS _u. The structure ofS _v can be further improved in casep≤2.

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Sunto Sia Ω un aperto limitato diR ⁿ con frontiera localmente Lipschitziana. In questo lavoro si dimostra che le funzioniv∈SBV(Ω,R ^m ) con insieme di saltoS _v essenzialmente chiuso e poliedrale che sono di classeW ^{k, ∞} (S _v,R ^m ) per ogni interok sono fortemente dense inGSBV ^p(Ω,R ^m ), nel senso che ogni funzioneu∈GSBV ^p(Ω,R ^m ) è approssimata inL ^p(Ω,R ^m ) da una successione di funzioni {v _j}_j∈N con la regolaritá descritta tali che i gradienti approssimati ∇v _jconvergono inL ^p(Ω,R ^nm ) al gradiente approssimato ∇u e la misura (n−1)-dimensionale degli insiemi di saltoS _{v j}converge alla misura (n−1)-dimensionale diS _u. La struttura diS _vpuó essere migliorata nel caso in cuip≤2.

     

Isoperimetric inequalities on minimal submanifolds of space forms

For a domainU on a certaink-dimensional minimal submanifold ofS ⁿ orH ⁿ, we introduce a “modified volume”M(U) ofU and obtain an optimal isoperimetric inequality forU k ^k ω _k M (D) ^k-1 ≤Vol(∂D) ^k , where ω _k is the volume of the unit ball ofR ^k . Also, we prove that ifD is any domain on a minimal surface inS ₊ ⁿ (orH ⁿ, respectively), thenD satisfies an isoperimetric inequality2π A≤L ²+A² (2π A≤L²−A² respectively). Moreover, we show that ifU is ak-dimensional minimal submanifold ofH ⁿ, then(k−1) Vol(U)≤Vol(∂U). Supported in part by KME and GARC     

Carathéodory balls and norm balls in a class of convex bounded Reinhardt domains

A family of the spherical fractional integrals on the unit sphere Σ _n in ℝ ⁿ⁺¹ is investigated. This family includes the spherical Radon transform (α = 0) and the Blaschke-Levy representation (α>1). Explicit inversion formulas and a characterization ofT ^αƒ are obtained for ƒ belonging to the spacesC ^∞,C, L^p and for the case when ƒ is replaced by a finite Borel measure. All admissiblen ≥ 2,α ε ℂ, andp are considered. As a tool we use spherical wavelet transforms associated withT ^α. Wavelet type representations are obtained forT ^α ƒ, ƒ εL ^p, in the case Reα ≤ 0, provided thatT ^α is a linear bounded operator inL ^p. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis, sponsored by the Minerva Foundation (Germany).     

A modulus of smoothness on the unit sphere

For functions onS ^d−1 (the unit sphere inR ^d) and, in particular, forf∈L _p(S ^d−1), we define new simple moduli of smoothness. We relate different orders of these moduli, and we also relate these moduli to best approximation by spherical harmonics of order smaller thann. Our new moduli lead to sharper results than those now available for the known moduli onL _p(S ^d−1). Supported by NSERC Grant A4816 of Canada.     

About the Blowup of Quasimodes on Riemannian Manifolds

On any compact Riemannian manifold (M,g) of dimension n, the L ²-normalized eigenfunctions φ _λ satisfy ||f_l||_￥ ￡ Cl^\fracn-12\|\phi_{\lambda}\|_{\infty}\leq C\lambda^{\frac{n-1}{2}} where −Δφ _λ =λ ² φ _λ . The bound is sharp in the class of all (M,g) since it is obtained by zonal spherical harmonics on the standard n-sphere S ⁿ . But of course, it is not sharp for many Riemannian manifolds, e.g., flat tori ℝ ⁿ /Γ. We say that S ⁿ , but not ℝ ⁿ /Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the (M,g) with maximal eigenfunction growth. In an earlier work, two of us showed that such an (M,g) must have a point x where the set ℒ _x of geodesic loops at x has positive measure in S^*_xMS^{*}_{x}M. We strengthen this result here by showing that such a manifold must have a point where the set ℛ _x of recurrent directions for the geodesic flow through x satisfies |{ℛ} _x |>0. We also show that if there are no such points, L ²-normalized quasimodes have sup-norms that are o(λ ^(n−1)/2), and, in the other extreme, we show that if there is a point blow-down x at which the first return map for the flow is the identity, then there is a sequence of quasimodes with L ^∞-norms that are Ω(λ ^(n−1)/2).     

On the maximal operator associated with certain rotational invariant measures

The aim of this work is to investigate the integrability properties of the maximal operator Mu,associated with a non-doubling measure μ defined on Rn. We start by establishing for a wide class of radial and increasing measures μ that Mu is bounded on all the spaces Lu^p（R^n）,P〉1.Also,we show that there is a radial and increasing measure p for which Mμ does not map Lμ^p（R^n） into weak Lμ^p（R^n）,1≤p〈∞.     

The Cauchy problem for nonhomogeneous Navier-Stokes equations inL q([0,T);L p)

In this paper the Cauchy problem for a class of nonhomogeneous Navier-Stokes equations in the infinite cylinderS _T =ℝⁿ x [0,T) is considered. We construct a unique local solution inL ^q([0,T);L ^p (ℝ ⁿ )) for a class of nonhomogeneous Navier-Stokes equations provided that initial data are inL ^r (ℝ ⁿ ), wherer>1 is an exponent determined by the structure of nonlinear terms andp,q are such that 2/q=n(1/r−1/p). Meanwhile under suitable conditions we also obtain thatu(t)≠L ^q([0,∞];L ^p (ℝ ⁿ )) provided that initial data are sufficiently small. This work is supported by the National Natural Sciences Foundation of China and the Foundation of LNM Laboratory of Institute of Mechanics of the Chinese Academy of Sciences.     

A Lower Bound for the Norm of the Minimal Residual Polynomial

Let S be a compact infinite set in the complex plane with 0∉S, and let R _n be the minimal residual polynomial on S, i.e., the minimal polynomial of degree at most n on S with respect to the supremum norm provided that R _n (0)=1. For the norm L _n (S) of the minimal residual polynomial, the limit k(S):=lim_n?￥ⁿ?{L_n(S)}\kappa(S):=\lim_{n\to\infty}\sqrt[n]{L_{n}(S)} exists. In addition to the well-known and widely referenced inequality L _n (S)≥κ(S) ⁿ , we derive the sharper inequality L _n (S)≥2κ(S) ⁿ /(1+κ(S)²ⁿ ) in the case that S is the union of a finite number of real intervals. As a consequence, we obtain a slight refinement of the Bernstein–Walsh lemma.     

Summability of Fourier-Laplace Series with the Method of Lacunary Arithmetical Means at Lebesgue Points

Let ∑ _{n −1} be the unit sphere in the n-dimensional Euclidean space ℝ ⁿ . For a funcion ƒ∈L(∑ _{n −1}) denote by σ^δ _N (ƒ) the Cesàro means of order δ of the Fourier-Laplace series of ƒ. The special value of δ is known as the critical index. In the case when n is even, this paper proves the existence of the ‘rare’ sequence {n _k } such that the summability takes place at each Lebesgue point satisfying some antipole conditions. Received June 28, 1999, Revised August 11, 1999, Accepted February 16, 2000     

L P boundedness of area function associated with operators on product spaces

Let L be the infinitesimal generator of an analytic semigroup on L2 (Rn) with suitable upper bounds on its heat kernels. Assume that L has a bounded holomorphic functional calculus on L2(Rn). In this paper,we define the Littlewood- Paley g function associated with L on Rn × Rn, denoted by GL(f)(x1, x2), and decomposition, we prove that ‖SL(f)‖p ≈‖GL(f)‖p ≈‖f‖p for 1 ＜ p ＜∞.     

A class of Hamiltonian and edge symmetric Cayley graphs on symmetric groups

Abstract. Let Sn be the symmetric group     

On the pontryagin-steenrod-wu theorem

We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) LetM be a connected orientable closed smooth (n + 1)-manifold,n≥3. Define the degree map deg: π ⁿ (M) →H ⁿ (M; ℤ) by the formula degf =f*[S ⁿ ], where [S ⁿ ] εH ⁿ (M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ εH ₂(M, ℤ/2ℤ) such thatβ ·w ₂(M) ε 0. If suchβ does not exist, then deg is a 2-1 map; and (b) LetM be an orientable closed smooth (n+2)-manifold,n≥3. An elementα lies in the image of the degree map if and only ifρ ₂ α ·w ₂(M)=0, whereρ ₂: ℤ → ℤ/2ℤ is reduction modulo 2.     

On extremal sort sequences

A sort sequence S_n is a sequence of all unordered pairs of indices inI _n = {1, 2, ..., n}. With a sort sequenceS_n = (s₁ ,s₂ ,...,s_{( 2ⁿ )} )S_n = (s_1 ,s_2 ,...,s_{\left( {_2^n } \right)} ), one can associate a predictive sorting algorithm A(S_n). An execution of the algorithm performs pairwise comparisons of elements in the input setX in the order defined by the sort sequence S_n except that the comparisons whose outcomes can be inferred from the results of the preceding comparisons are not performed. A sort sequence is said to be extremal if it maximizes a given objective function. First we consider the extremal sort sequences with respect to the objective function ω(S_n) — the expected number of active predictions inS _n. We study ω-extremal sort sequences in terms of their prediction vectors. Then we consider the objective function Ω(S_n) — the minimum number of active predictions in S_n over all input orderings.     

A cardinal number connected to the solvability of systems of difference equations in a given function class

Let ℝ^ℝ denote the set of real valued functions defined on the real line. A map D: ℝ^ℝ → ℝ^ℝ is said to be a difference operator if there are real numbers a _i, b _i (i = 1, …, n) such that (Dƒ)(x) = ∑ _i=1 ⁿ a _i ƒ(x + b _i) for every ƒ ∈ ℝ^ℝand x ∈ ℝ. By a system of difference equations we mean a set of equations S = {D _i ƒ = g _i: i ∈ I}, where I is an arbitrary set of indices, D _i is a difference operator and g _i is a given function for every i ∈ I, and ƒ is the unknown function. One can prove that a system S is solvable if and only if every finite subsystem of S is solvable. However, if we look for solutions belonging to a given class of functions then the analogous statement is no longer true. For example, there exists a system S such that every finite subsystem of S has a solution which is a trigonometric polynomial, but S has no such solution; moreover, S has no measurable solutions. This phenomenon motivates the following definition. Let be a class of functions. The solvability cardinal sc( ) of is the smallest cardinal number κ such that whenever S is a system of difference equations and each subsystem of S of cardinality less than κ has a solution in , then S itself has a solution in . In this paper we determine the solvability cardinals of most function classes that occur in analysis. As it turns out, the behaviour of sc( ) is rather erratic. For example, sc(polynomials) = 3 but sc(trigonometric polynomials) = ω ₁, sc({ƒ: ƒ is continuous}) = ω ₁ but sc({f : f is Darboux}) = (2 ^ω )⁺, and sc(ℝ^ℝ) = ω. We consistently determine the solvability cardinals of the classes of Borel, Lebesgue and Baire measurable functions, and give some partial answers for the Baire class 1 and Baire class α functions. Partially supported by Hungarian Scientific Foundation grants no. 49786,37758,F 43620 and 61600. Partially supported by Hungarian Scientific Foundation grant no. 49786.     

Arens multiplication on Banach algebras related to locally compact semigroups

Let S be a locally compact semigroup, let ω be a weight function on S, and let M_a (S, ω) be the weighted semigroup algebra of S. Let L^∞₀ (S;M_a (S, ω)) be the C^*‐algebra of allM_a (S, ω)‐measurable functions g on S such that g /ω vanishes at infinity. We introduce and study an Arens multiplication on L^∞₀ (S;M_a (S, ω))^* under which M_a (S, ω) is a closed ideal. We show that the weighted measure algebra M (S, ω) plays an important role in the structure of L^∞₀ (S;M_a (S, ω))^*. We then study Arens regularity of L^∞₀ (S;M_a (S, ω))^* and ist relation with Arens regularity of M_a (S, ω), M (S, ω) and the discrete convolution algebra ℓ¹(S, ω). As the main result, we prove that L^∞₀ (S;M_a (S, ω))^* is Arens regular if and only if S is finite, or S is discrete and Ω is zero cluster. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiparameter inverse problem of approximation by functions with given supports

Let L _p (S), 0 < p < +∞, be a Lebesgue space of measurable functions on S with ordinary quasinorm ∥·∥ _p . For a system of sets {B _t |t ∈ [0, +∞) ⁿ } and a given function ψ: [0, +∞) ⁿ ↦ [ 0, +∞), we establish necessary and sufficient conditions for the existence of a function f ∈ L _p (S) such that inf {∥f − g∥ _p ^p g ∈ L _p (S), g = 0 almost everywhere on S\B _t } = ψ (t), t ∈ [0, +∞) ⁿ . As a consequence, we obtain a generalization and improvement of the Dzhrbashyan theorem on the inverse problem of approximation by functions of the exponential type in L ₂. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1116–1127, August, 2006.     

On a theorem of Marcinkiewicz and Zygmund

Let {ϕn(x)} be an orthonormal system on the closed interval [0,1], and let ∥ϕ _n ∥_∞ ≤M _n . In 1937 Marcinkiewicz and Zygmund obtained an estimate of the norm inL _q [0,1] of the sum of the series ∑ _n=1 ^∞ c _n ϕ _n (x) under the condition that {M _n } is monotone increasing. In this paper it is shown that this condition cannot be discarded. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 386–390, March, 1998. The author wishes to thank V. I. Kolyada for setting the problem and for his permanent attention to this work.