Convergence of ray sequences of Padé approximants for 2 f 1(a, 1; c; z), (c > a > 0)

The Padé table of ₂ F ₁(a, 1; c; z) is normal for c > a > 0 (cf. [4]). For m ≥ n - 1 and c ? Z^-, the denominator polynomial Q _mn (z) in the [m/n] Padé approximant P _mn (z)/Q _mn (z) for ₂ F ₁(a, 1; c; z) and the remainder term Q _mn (z)₂ F ₁(a, 1; c; z)-P_mn (z) were explicitly evaluated by Padé (cf. [2], [6] or [9]). We show that for c > a > 0 and m ≥ n - 1, the poles of P_mn (z)/Q_mn (z) lie on the cut (1,∞). We deduce that the sequence of approximants P_mn (z)/Q_mn (z) converges to ₂ F ₁(a, 1; c; z) as m → ∞, n/m → ρ with 0 < ρ ≤ 1, uniformly on compact subsets of the unit disc |z| < 1 for c > a > 0.     

Asymptotic Representation of L<Subscript>p</Subscript>-Minimal Polynomials, 1 < p < ∞

While the theory of asymptotics for L₂-minimal polynomials is highly developed, so far not much is known about L_p-minimal polynomials, Indeed, Bernstein gave asymptotics for the minimum deviation, Fekete and Walsh gave nth root asymptotics and, recently, Lubinsky and Saff came up with asymptotics outside the support [-1,1]. But the main point of interest, the asymptotic representation on the support, still remains open. Here we settle it for weight functions of the form where w is positive and on [-1,1] with and 相似文献   

Stable Cohomology of Spaces of Non-Resultant Systems of Homogeneous Polynomials in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Stable rational cohomology groups of spaces of non-resultant homogeneous polynomial systems of growing degree in ?ⁿ are calculated.     

Approximation Properties of Julia Polynomials

Let G be a finite simply connected domain in the complex plane C, bounded by a rectifiable Jordan curve L, and let w = φ0 （z） be the Riemann conformal mapping of G onto D （0, r0） ：= {E-mail： ： ｜｜ 〈 r0}, normalized by the conditions φ0 （z0） = 0, φ＇0 （z0） = 1. In this work, the rate of approximation of φ0 by the polynomials, defined with the help of the solutions of some extremal problem, in a closed domain G is studied. This rate depends on the geometric properties of the boundary L.     

On local summability of Riesz potentials in the case Reα>0

The rangeI ^α (L _p ) of the Riesz potential operator, defined in the sense of distributions in the casep≥n/α, is shown to consist of regular distributions. Moreover, it is shown thatI ^α (L _p ) ?L _p ^loc (R ⁿ ) for all 1≤p<∞ and 0<α<∞. The distribution space used is that of Lizorkin, which is invariant with respect to the Riesz operator.     

On existence of a universal function for <Emphasis Type="Italic">L</Emphasis> <Superscript> <Emphasis Type="Italic">p</Emphasis> </Superscript>[0, 1] with <Emphasis Type="Italic">p</Emphasis>∈(0, 1)

We show that, for every number p ∈ (0, 1), there is g ∈ L¹[0, 1] (a universal function) that has monotone coefficients c_k(g) and the Fourier–Walsh series convergent to g (in the norm of L¹[0, 1]) such that, for every f ∈ L^p[0, 1], there are numbers δ_k = ±1, 0 and an increasing sequence of positive integers N_q such that the series ∑ _k=0^+∞δ_kc_k(g)W_k (with {W_k} theWalsh system) and the subsequence \(\sigma _{{N_q}}^{\left( \alpha \right)}\), α ∈ (?1, 0), of its Cesáro means converge to f in the metric of L^p[0, 1].     

On the maximal operator of (<Emphasis Type="Italic">C</Emphasis>, α)-means of Walsh–Kaczmarz–Fourier series

Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator σ^α,κ,* of the (C, α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space H _p to the space L _p for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved that this boundedness result does not hold if p ≤ 1 / (1 + α). However, in the endpoint case p = 1 / (1 + α ), the maximal operator σ^α,κ,* is bounded from the martingale Hardy space H _1/(1+α) to the space weak- L _1/(1+α). The main aim of this paper is to prove a stronger result, namely, that, for any 0 < p ≤ 1 / (1 + α), there exists a martingale f ∈ H _p such that the maximal operator σ^α,κ,* f does not belong to the space L _p .     

Uniform Spacing of Zeros of Orthogonal Polynomials

It is shown that for doubling weights, the zeros of the associated orthogonal polynomials are uniformly spaced in the sense that if cos θ _m,k , θ _m,k ∈[0,π] are the zeros of the m-th orthogonal polynomial associated with w, then θ _m,k −θ _m,k+1∼1/m. It is also shown that for doubling weights, neighboring Cotes numbers are of the same order. Finally, it is proved that these two properties are actually equivalent to the doubling property of the weight function.     

Asymptotic Form of Hermite-Pad? Polynomials

Permanent address: Department of Physics, Saint Louis University, St Louis, Missouri, U.S.A. A method is given which locates the area on which the zerosof Hermite-Pad? polynomials of large order lie. The method alsogives the density of these zeros along the ares, and so theasymptotic form of the polynomials for large order. Severalexamples which confirm and illustrate the method are given.These results are necessary for proving the convergence of schemesfor extracting information from power series based on Hermite-Pad?approximants (the so-called "quadratic" and "differential equation"or "integral" Pad? approximants).     

Example of Fitting Polynomials to Data

The table below shows the accumulated total number of reported casesof AIDS in the United States since 1983,based on Center for Disease Control statistics.At first thought,we might expect that the growth In the total number of reported cases of AIDSis exponential and so we would attempt to fit an exponential function to this set of data.The re-sulting best-fit exponential function,found with a calculator,is     

Sur les involutions de genresp α=P 4=1 existant sur une surface algébrique de genresp α=p g =p (1)=1,P 2>1

Sans résumé     

On Converse Theorem of Best Approximation by Polynomials in Bergman Spaces H_q~p(p>0,q>1)

A Bernstein type theorem and a converse theorem of best approximation by polynomials inBergman spaces H_q~p(p>0,q>1) are proved.Some proofs and results in [1] are in proved.     

Certain Solutions of Equation f（z,0,…,0,uk,0,…,0）=0

ＣｅｒｔａｉｎＳｏｌｕｔｉｏｎｓｏｆＥｑｕａｔｉｏｎｆ（ｚ，０，…，０，ｕ_ｋ，０，…，０）＝０ＬｉｕＧｕａｎｇｍｉｎ（刘光民）（ＫａｉｆｅｎｇＴｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ）ＬｉｕＧｕａｎｇｍｉｎ（ＫａｉｆｅｎｇＴｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ）?..     

Convergence in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>([0, <Emphasis Type="Italic">T</Emphasis>]) of Wavelet Expansions of <Emphasis Type="Italic">φ</Emphasis>-Sub-Gaussian Random Processes

The article presents new results on convergence in L _p ([0,T]) of wavelet expansions of φ-sub-Gaussian random processes. The convergence rate of the expansions is obtained. Specifications of the obtained results are discussed.     

Banach Disks and Marrelledness Properties of Metrizalbe <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript> (Ω, <Emphasis Type="Italic">X</Emphasis>)

Let be a non-empty set and X a metrizable locally convex space. We show that the metrizable locally convex space c₀ (, X) is p-barrelled (totally barrelled) if and only if X is p-barrelled (totally barrelled). Some applications for closed graph theorems are included.     

On the solvability of a singular boundary-value problem for the equation <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">t,x, x</Emphasis>′, <Emphasis Type="Italic">x</Emphasis>″) = 0

In this work, we consider boundary-value problems of the form

Approximation by (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Lorentz Polynomials on a Compact Disk

The (p, q)-factors were introduced in order to generalize or unify several forms of q-oscillator algebras well known in the physics literature related to the representation theory of single parameter quantum algebras. This notion has been recently used in approximation by positive linear operators via (p, q)-calculus which has emerged a very active area of research. In this paper, we introduce a new analogue of Lorentz polynomials based on (p, q)-integers. We obtain quantitative estimate in the Voronovskaja’s type theorem and exact orders in simultaneous approximation by the complex (p, q)-Lorentz polynomials of degree \(n\in \mathbb {N}\) (\(q>p>1)\), attached to analytic functions on compact disks of the complex plane. In this way, we put in evidence the overconvergence phenomenon for the (p, q)-Lorentz polynomial, namely the extensions of approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane.