Mohammad Masjed-Jamei, Wolfram Koepf
By using Fourier transforms of two symmetric sequences of finite orthogonal polynomials, we introduce two new classes of finite orthogonal functions and obtain their orthogonality relations via Parseval's identity.
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We obtain the exact-order estimates for approximations by Fourier sums, best approximations and best orthogonal trigonometric approximations in metrics of spaces L_s, 1\leq s<\infty, of classes of 2π-periodic functions, whose (ψ,β)-derivatives belong to unit ball of the space L_\infty.
We obtain order estimates in the spaces $L_p$ of the best $n$-term trigonometric orthogonal approximations of the classes ${\cal F}_{q}^ψ$ of periodic functions, whose Fourier coefficients decrease faster then any power function. We also obtain order estimates of the quantities of approximations by $n$-term Greedy approximants of such classes in the spaces $L_p$.
In this paper we establish exact order estimates for the best uniform orthogonal trigonometric approximations of the classes of $2π$-periodic functions, whose $(ψ,β)$-derivatives belong to unit balls of spaces $L_{p}$, $1\leq p<\infty$, in the case, when the sequence $ψ(k)$ tends to zero faster, than any power function, but slower than geometric progression. Similar estimates are also established in the $L_{s}$-metric, $1<s\leq\infty$ for the classes of differentiable functions, which $(ψ,β)$-derivatives belong to unit ball of space $L_{1}$.
We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-$D$ Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the 2-$D$ Hermite polynomials and a two variable extension of the Zernike or disc polynomials. We also give $q$-analogues of all these extensions. In each case in addition to generating functions and three term recursions we provide raising and lowering operators and show that the polynomials are eigenfunctions of second-order partial differential or $q$-differen…
We obtain order estimates for the best uniform orthogonal trigonometric approximations of $2π$-periodic functions, whose $(ψ,β)$-derivatives belong to unit balls of spaces $L_{p}, \ 1\leq p<\infty$, in case at consequences $ψ(k)$ are that product $ψ(n)n^{\frac{1}{p}}$ can tend to zero slower than any power function and $\sum\limits_{k=1}^{\infty}ψ^{p'}(k)k^{p'-2}<\infty$ when $1<p<\infty$, $\frac{1}{p}+\frac{1}{p'}=1$ and $\sum\limits_{k=1}^{\infty}ψ(k)<\infty$ when $p=1$. We also establish the analogical estimates in $L_{s}$-metric, $1< s\leq \infty$, for classes of the summable $(ψ,β)$-diffe…
Notions of the orthogonality and convolution orthogonality are explored with the use of the Kontorovich-Lebedev transform and its convolution. New classes of the corresponding orthogonal polynomials and functions are investigated. Integral representations, orthogonality relations and explicit expressions are established.
Finite orthogonal polynomials on a cone
本文研究锥面的有限正交多项式，定义了两类多项式并推导其微分方程和递推关系，其中一类在极限情况下退化为锥面上的拉格朗日多项式。
The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
我们列出了所谓的超几何正交多项式的Askey方案。第一章给出了该格式中各类正交多项式的定义、正交关系、三项递推关系和母函数。第二章给出了Askey方案中不同类正交多项式之间的全部极限关系。在第三章中，我们列出了Askey-方案中多项式的q-模拟。给出了它们的定义、正交关系、三项递推关系和生成函数。第四章给出了基本超几何正交多项式之间的极限关系。最后在第五章中我们指出Askey方案的“经典”超几何正交多项式是如何从它们的q-类似物得到的。
Skew-orthogonal polynomials for a quartic Freud weight: two classes of quasi-orthogonal polynomials
研究四次Freud权下斜正交多项式的评估方法，提出通过递推关系求解系数，并发现偶次和奇次多项式分别构成两种不同半经典拉盖尔权下的准正交多项式。
This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.
Considering a differential operator of third order that does not increase the degree of polynomials, we analyse some properties of elements of the dual space of 2-orthogonal polynomial eigenfunctions. In two classes of such generic operator, we prove that a 2-orthogonal polynomial solution fulfils Hahn's property.
We introduce a new family of orthogonal polynomials on the disk that has emerged in the context of wave propagation in layered media. Unlike known examples, the polynomials are orthogonal with respect to a measure all of whose even moments are infinite.
The polynomials $p_n$ orthogonal on the interval $[-1,1],$ normalized by $p_n(1)=1,$ satisfy Turán's inequality if $p_n^2(x)-p_{n-1}(x)p_{n+1}(x)\ge 0$ for $n\ge 1$ and for all $x$ in the interval of orthogonality. We give a general criterion for orthogonal polynomials to satisfy Turán's inequality. This extends essentially the results of \cite{szw}. In particular the results can be applied to many classes of orthogonal polynomials, by inspecting their recurrence relation.
In this paper new classes of $L_2$-orthogonal functions are constructed as iterated $L_2$-orthogonal systems. In order to do this we use the theory of the Riemann's zeta-function as well as our theory of Jacob's ladders. The main result is new one in the theory of the Riemann's zeta-function and simultaneously in the theory of $L_2$-orthogonal systems.
Orthogonality of Jacobi and Laguerre polynomials for general parameters via the Hadamard finite part
对复形a，b（a，b不是负整数，a+b不是-2，-3，.）使用积分的Hadamard有限部分，在经典情况下给出它们的正交性。找到了这些多项式所满足的黎曼-希尔伯特问题，其结果与经典情形（当a，b的实部大于-1时）的结果在形式上类似
A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line
Szego将单位圆上的正交多项式与[-1，1]上的正交多项式连接的方法推广到非对称测度上.它在洛朗多项式L的线性空间上生成所谓的半正交函数，并在模LxL中导出一种新的正交结构。这种结构可以用[-1，1]上的2 × 2矩阵测度来解释，并且半正交函数提供了相应的正交矩阵多项式序列。这给出了单位圆上的正交多项式与[-1，1]上的某类矩阵正交多项式之间的联系。作为应用，导出了这些矩阵正交多项式的强渐近性，得到了相应的Szego矩阵函数的显式表达式。
We find the exact order estimates of the approximations of the classes ${\cal F}_{q,r}^ψ$ of functions of several variables by greedy approximants in the integral metric. We also obtain the exact order estimates of the best $n$-term orthogonal trigonometric approximations of the classes ${\cal F}_{q,r}^ψ$ in the integral metric.
Orthogonal polynomials and the Fourier orthogonal series on a cone of revolution in $\mathbb{R}^{d+1}$ are studied. It is shown that orthogonal polynomials with respect to the weight function $(1-t)^γ(t^2-\|x\|^2)^{μ-\frac12}$ on the cone $\mathbb{V}^{d+1} = \{(x,t): \|x\| \le t \le 1\}$ are eigenfunctions of a second order differential operator, with eigenvalues depending only on the degree of the polynomials, and the reproducing kernels of these polynomials satisfy a closed formula that has a one-dimensional chacteristic. The latter leads to a convolution structure on the cone, which is then…
Finite biorthogonal polynomials suggested by the finite orthogonal polynomials $M_{n}^{(p,q)}(x)$
本文由有限正交多项式 $M_{n}^{(p,q)}(x)$ 启发推导出一对有限双正交多项式，给出其双正交关系、微分方程、生成函数，并利用 Fourier 变换与 Parseval 恒等式得到新函数族，同时计算了 Laplace 变换与分数阶微积分算子。
The article proves an assertion analogous to the Littlewood-Paley theorem for the orthoprojectors onto mutually orthogonal subspaces of piecewise polynomial functions on the cube $ I^d. $ This assertion provides an upper estimate for the norms of the functions in $ L_p(I^d) $ via corresponding norms of projections onto subspaces of piecewise polynomial multivariable functions. These relationships are used to obtain upper estimates of the Kolmogorov widths of Besov classes of non-periodic functions meeting the mixed Hoelder conditions.
We establish the Plancherel-Rotach-type asymptotics around the largest zero (the soft edge asymptotics) for some classes of polynomials satisfying three-term recurrence relations with exponentially increasing coefficients. As special cases, our results include this type of asymptotics for q^{-1}-Hermite polynomials of Askey, Ismail and Masson, q-Laguerre polynomials, and the Stieltjes-Wigert polynomials. We also introduce a one parameter family of solutions to the q-difference equation of the Ramanujan function.
Beta-integrals and finite orthogonal systems of Wilson polynomials
我们构造了3个有限的$4-F-3$超几何正交多项式系统。权重是1）由$5-H-5$ Dougall求和公式定义的权重; 2）Askey β积分中的被积函数; 3）权重$w(s)=|p(s)/q(s)|^2$，其中$p(s)=Γ(a+is)Γ(b+is)Γ(c+is)$和$q(s)=Γ(d+is)Γ(2is)$。我们还评估积分的功能$w(s)$使用雅可比变换（Olevsky变换），这个积分也可以减少到Nassrallah-拉赫曼积分。
Contiguity relations for finite families of orthogonal polynomials in the Askey scheme
本文分类了($q$-)Askey方案中有限多项式族的相邻关系，给出了存在的充要条件并求解出完整列表，同时证明所有相邻关系均对应于谱变换。
Duality of orthogonal polynomials on a finite set
证明了定义在有限集上的正交多项式的一个对偶关系。其结果是在一个直接证明的等价性的两种不同的方式计算的相关函数的离散正交多项式合奏。
We study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the generalized eigenfunctions written in terms of the orthogonal polynomials. The spectrum and its spectral type are studied for large classes of orthogonality measures.
The exponential orthogonal polynomials encode via the theory of hyponormal operators a shade function $g$ supported by a bounded planar shape. We prove under natural regularity assumptions that these complex polynomials satisfy a three term relation if and only if the underlying shape is an ellipse carrying uniform black on white. More generally, we show that a finite term relation among these orthogonal polynomials holds if and only if the first row in the associated Hessenberg matrix has finite support. This rigidity phenomenon is in sharp contrast with the theory of classical complex orthog…
Quadrature rules from finite orthogonality relations for Bernstein-Szego polynomials
本文通过将两种伯恩斯坦-齐格勒多项式家族结合，构建了关联有限维雅可比矩阵的特征基，从而得到该复合特征基的有限正交关系，进而推导出若干类似高斯的求积规则，用于精确积分有指定极点的有理函数。
Finite Markov chains and multiple orthogonal polynomials
本文利用正交多项式与多重正交多项式的递推矩阵构建有限马尔可夫链，推导其谱理论与多项性质，揭示生灭过程因式分解特性，并结合超几何多项式实例验证，构建对应常返链与反转链。
We introduce a nine-parameter Heun-type differential equation and obtain three classes of its solutions as series of square integrable functions written in terms of the Jacobi polynomial. The expansion coefficients of the series satisfy three-term recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Some of these are well-known polynomials while others are either new or modified versions of known ones.
We introduce sequences of functions orthogonal on a finite interval: proper orthogonal rational functions, orthogonal exponential functions, orthogonal logarithmic functions, and transmuted orthogonal polynomials
New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations
该研究通过增态和删态两类Darboux变换，基于12种经典正交多项式构造出新型有限型多指标正交多项式，给出其显式形式与满足的差分方程，并明确了其与离散量子力学形变哈密顿量本征态的对应关系。
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