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.     

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The Application of Kernel Smoothing to Time Series Data

There are already a lot of models to fit a set of stationary time series, such as AR, MA, and ARMA models. For the non-stationary data, an ARIMA or seasonal ARIMA models can be used to fit the given data. Moreover, there are also many statistical softwares that can be used to build a stationary or non-stationary time series model for a given set of time series data, such as SAS, SPLUS, etc. However, some statistical softwares wouldn＇t work well for small samples with or without missing data, especially for small time series data with seasonal trend. A nonparametric smoothing technique to build a forecasting model for a given small seasonal time series data is carried out in this paper. And then, both the method provided in this paper and that in SAS package are applied to the modeling of international airline passengers data respectively, the comparisons between the two methods are done afterwards. The results of the comparison show us the method provided in this paper has superiority over SAS＇s method.     

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).     

Solution of The Last Example

Solution Let E be the elevation of the platear(in meters).From the Slide Model for Subtraction.E-20=53.Solve this equation.     

On the Overconvergence of Hermite Interpolating Polynomials

In this paper we study the overconvergence of Hermite interpolating polynomials. A new extension of a theorem of J.L.Walsh is obtained and a conjecture is posed.     

Bispectrality of Multivariable Racah–Wilson Polynomials

We construct a commutative algebra A_x{\mathcal{A}}_{x} of difference operators in ℝ ^p , depending on p+3 parameters, which is diagonalized by the multivariable Racah polynomials R _p (n;x) considered by Tratnik (J. Math. Phys. 32(9):2337–2342, 1991). It is shown that for specific values of the variables x=(x ₁,x ₂,…,x _p ) there is a hidden duality between n and x. Analytic continuation allows us to construct another commutative algebra A_n{\mathcal{A}}_{n} in the variables n=(n ₁,n ₂,…,n _p ) which is also diagonalized by R _p (n;x). Thus, R _p (n;x) solve a multivariable discrete bispectral problem in the sense of Duistermaat and Grünbaum (Commun. Math. Phys. 103(2):177–240, 1986). Since a change of the variables and the parameters in the Racah polynomials gives the multivariable Wilson polynomials (Tratnik in J. Math. Phys. 32(8):2065–2073, 1991), this change of variables and parameters in A_x{\mathcal{A}}_{x} and A_n{\mathcal{A}}_{n} leads to bispectral commutative algebras for the multivariable Wilson polynomials.     

Moments of the Rudin–Shapiro Polynomials

We develop a new approach of the Rudin–Shapiro polynomials. This enables us to compute their moments of even order q for q 32, and to check a conjecture on the asymptotic behavior of these moments for q even and q 52.     

Asymptotic Properties of Orthogonal Polynomials with Respect to a Non-discrete Jacobi-Sobolev Inner Product

Let {Q _n ^(α,β) (x)} _n=0 ^∞ denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product

$\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)d\mu_{\alpha,\beta}(x)+\lambda\int_{-1}^{1}f'(x)g'(x)d\nu_{\alpha,\beta}(x)$

where λ>0 and d μ _α,β(x)=(x?a)(1?x)^α?1(1+x)^β?1 dx, d ν _α,β(x)=(1?x) ^α (1+x) ^β dx with aα,β>0. Their inner strong asymptotics on (?1,1), a Mehler-Heine type formula as well as some estimates of the Sobolev norms of Q _n ^(α,β) are obtained.

     

The Total volume of social and network retail sales——Analysis of Data Published on the Website of CNBS for Example

<正>Online shopping occurred in China just around the year 2000,but it has grown immensely over the past dozen years.I am a post-90sand often use the way of online shopping.However,my mom likes the way of shopping in the mall.After I helped her succeeded in shopping on the Internet once again,     

Do Orthogonal Polynomials Dream of Symmetric Curves?

The complex or non-Hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the freedom in the choice of the integration contour for such polynomials, the location of their zeros is a priori not clear. Nevertheless, numerical experiments, such as those presented in this paper, show that the zeros not simply cluster somewhere on the plane, but persistently choose to align on certain curves, and in a very regular fashion. The problem of the limit zero distribution for the non-Hermitian orthogonal polynomials is one of the central aspects of their theory. Several important results in this direction have been obtained, especially in the last 30 years, and describing them is one of the goals of the first parts of this paper. However, the general theory is far from being complete, and many natural questions remain unanswered or have only a partial explanation. Thus, the second motivation of this paper is to discuss some “mysterious” configurations of zeros of polynomials, defined by an orthogonality condition with respect to a sum of exponential functions on the plane, that appeared as a results of our numerical experiments. In this apparently simple situation the zeros of these orthogonal polynomials may exhibit different behaviors: for some of them we state the rigorous results, while others are presented as conjectures (apparently, within a reach of modern techniques). Finally, there are cases for which it is not yet clear how to explain our numerical results, and where we cannot go beyond an empirical discussion.     

Asymptotic Behavior of Müntz Orthogonal Polynomials

We provide a representation for the Müntz orthogonal polynomials as a real integral. This allows us to establish a general result on their asymptotic behavior within the interval of orthogonality. This is the first time that such asymptotics have been obtained for general Müntz exponents {λ _n }. We consider some special cases, particularly when the exponents satisfy the asymptotic relation $\lim_{n\rightarrow\infty}\frac{n}{\lambda_n}=\rho $ for some constant ρ>0.     

Expansions in Products of Heine—Stieltjes Polynomials

It is shown that products of polynomials introduced by Heine and Stieltjes form orthogonal bases in suitable function spaces. A theorem on the expansion of analytic function in these bases is proved. June 5, 1997. Date revised: November 18, 1997. Date accepted: January 14, 1998.     

Chebyshev’s Problem of the Moments of Nonnegative Polynomials

Mathematical Notes - We study the problem of P. L. Chebyshev (proposed in 1883) concerning the extreme values of moments of nonnegative polynomials with weight on the interval $$[-1,1]$$ at a fixed...     

Möbius Polynomials of Face Posets of Convex Polytopes

The Möbius polynomial is an invariant of ranked posets, closely related to the Möbius function. In this paper, we study the Möbius polynomial of face posets of convex polytopes. We present formulas for computing the Möbius polynomial of the face poset of a pyramid or a prism over an existing polytope, or of the gluing of two or more polytopes in terms of the Möbius polynomials of the original polytopes. We also present general formulas for calculating Möbius polynomials of face posets of simplicial polytopes and of Eulerian posets in terms of their f-vectors and some additional constraints.