设Sobolev型内积<f，g> = \int fg d mu_0+ int f' g' d mu_1，其中mu_0 = w + M delta_c，mu_1= N delta_c，其中w是Jacobi权，c是1或-1，M，N >= 0。我们得到了关于<...>的正交多项式在[-1，1]上的估计和渐近性质.和它们的内核。我们还比较了这些多项式与Jacobi正交多项式。
英文摘要：
Let the Sobolev-type inner product <f,g> = \int fg d mu_0+ int f' g' d mu_1 with mu_0 = w + M delta_c, mu_1= N delta_c where w is the Jacobi weight, c is either 1 or -1 and M, N >= 0. We obtain estimates and asymptotic properties on [-1,1] for the polynomials orthonormal with respect to <.,.> and their kernels. We also compare these polynomials with Jacobi orthonormal polynomials.
We define and study Sobolev spaces associated with Jacobi expansions. We prove that these Sobolev spaces are isomorphic to Jacobi potential spaces. As a technical tool, we also show some approximation properties of Poisson-Jacobi integrals.
We apply a symmetrization procedure to the setting of Jacobi expansions and study potential spaces in the resulting situation. We prove that the potential spaces of integer orders are isomorphic to suitably defined Sobolev spaces. Among further results, we obtain a fractional square function characterization, structural theorems and Sobolev type embedding theorems for these potential spaces.
The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Jacobi-Sobolev bilinear form with mass point at $-1$ and/or $+1$. In particular, we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that when the Jacobi parameters $α$ and $β$ are nonnegative integers the Jacobi-Sobolev orthogonal polynomials are eigenfunctions of a differential operator of finite order (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of…
In this contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are interested in Mehler-Heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these Sobolev orthogonal polynomials and the Jacobi ones. Moreover, this asymptotic behavior provides an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other well-known special functions.…
For every system $\{ p_n(z) \}_{n=0}^\infty$ of OPRL or OPUC, we construct Sobolev orthogonal polynomials $y_n(z)$, with explicit integral representations involving $p_n$. Two concrete families of Sobolev orthogonal polynomials (depending on an arbitrary number of complex parameters) which are generalized eigenvalues of a difference operator (in $n$) and generalized eigenvalues of a differential operator (in $n$) are given. Applications of a general connection between Sobolev orthogonal polynomials and orthogonal systems of functions in the direct sum of scalar $L^2_μ$ spaces are discussed.
We develop a theory of Sobolev orthogonal polynomials on the Sierpiński gasket ($SG$). These orthogonal polynomials arise through the Gram-Schmidt orthogonalisation process applied on the set of monomials on $SG$ using several notions of a Sobolev inner products. After establishing some recurrence relations for these orthogonal polynomials, we give estimates for their $L^2$, $L^\infty$ and Sobolev norms, and study their asymptotic behaviour. Finally, we study the properties of zero sets of polynomials and develop fast computational tools to explore applications to quadrature and interpolation.
In the paper we study the multipliers of Fourier series in polynomials orthogonal in continuous-discrete Sobolev's spaces. Multiplier Theorem for Fourier-Sobolev series is obtained. This result is based on the representation of the Fejér kernel, on the construction of the "humpbacked majorant" and weighted estimates of maximal functions.
The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Laguerre-Sobolev bilinear form with mass point at zero. In particular we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that they are eigenfunctions of a differential operator (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of the matrix which defines the discrete Laguerre-Sobolev bilinear form.
We present a new family $\left\{ S_{n}(x;q)\right\} _{n\geq 0}$ of monic polynomials in $x$, orthogonal with respect to a Sobolev-type inner product related to the $q$-Hermite I orthogonal polynomials, involving a first-order $q$-derivative on a mass-point $α\in \mathbb{R}$ located out of the corresponding orthogonality interval $[-1,1]$, for some fixed real number $q \in (0, 1)$. We present connection formulas, and the annihilation operator for this non-standard orthogonal polynomial family.
Sobolev orthogonal polynomials have been studied extensively in the past 20 years. The research in this field has sprawled into several directions and generates a plethora of publications. This paper contains a survey of the main developments up to now. The goal is to identify main ideas and developments in the field, which hopefully will lend a structure to the mountainous publications and help future research.
We study the regularity properties of the Hamilton-Jacobi flow equation and infimal convolution in the case where initial datum function is continuous and lies in given Sobolev-space $W^{1,p}(\rn)$. We prove that under suitable assumptions it holds for solutions $w(x,t)$ that $D_xw(\cdot,t)\to Du(\cdot)$ in $L^p(\rn)$. Moreover, we construct examples showing that our results are essentially optimal.
The q-Hermite I-Sobolev type polynomials of higher order are consider for their study. Their hypergeometric representation is provided together with further useful properties such as several structure relations which give rise to a three-term recurrence relation of their elements. Two different q-difference equations satisfied by the q-Hermite I-Sobolev type polynomials of higher order are also established.
In this paper we study the following family of hypergeometric polynomials: $y_n(x) = \frac{ (-1)^ρ}{ n! } x^n {}_2 F_0(-n,ρ;-;-\frac{1}{x})$, depending on a parameter $ρ\in\mathbb{N}$. Differential equations of orders $ρ+1$ and $2$ for these polynomials are given. A recurrence relation for $y_n$ is derived as well. Polynomials $y_n$ are Sobolev orthogonal polynomials on the unit circle with an explicit matrix measure.
We analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which involves the outward normal derivatives on the sphere. Using their representation in terms of spherical harmonics, algebraic and analytic properties will be deduced. First, we deduce explicit connection formulas relating classical multivariate ball polynomials and our family of Sobolev orthogonal polynomials. Then explicit representations for the norms and the kernels will be obtained. Finally, the asymptotic behaviour of the corresponding Christoffel functions is studied.
In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: ${}_2 F_2(-n,1;q,r;x)$ and ${}_3 F_2(-n,n-1+a+b,1;a,c;x)$ ($a,b,c,q,r>0$, $n=0,1,...$), which generalize Laguerre and Jacobi polynomials, respectively. These polynomials satisfy higher-order differential equations of the following form: $L y + λ_n D y = 0$, where $L,D$ are linear differential operators with polynomial coefficients not depending on $n$. For positive integer values of the parameters $r,c$ these polynomials are Sobolev orthogonal polyn…
The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function $W_\bg(x) = x_1^{\g_1} ... x_d^{\g_d} (1- |x|)^{\g_{d+1}}$ when all $\g_i > -1$ and they are eigenfunctions of a second order partial differential operator $L_\bg$. The singular cases that some, or all, $\g_1,...,\g_{d+1}$ are -1 are studied in this paper. Firstly a complete basis of polynomials that are eigenfunctions of $L_\bg$ in each singular case is found. Secondly, these polynomials are shown to be orthogonal with respect to an inner product which is explicitly determined. This inner prod…
We study approximation properties of weighted $\mathrm{L}^2$-orthogonal projectors onto spaces of polynomials of bounded degree in the Euclidean unit ball, where the weight is of the reflection-invariant form $(1-\lVert x \rVert^2)^α\prod_{i=1}^d \lvert x_i \rvert^{γ_i}$, $α, γ_1, \dots, γ_d > -1$. Said properties are measured in Dunkl-Sobolev-type norms in which the same weighted $\mathrm{L}^2$ norm is used to control all the involved differential-difference Dunkl operators, such as those appearing in the Sturm-Liouville characterization of similarly weighted $\mathrm{L}^2$-orthogonal polynom…
In this contribution we deal with sequences of polynomials orthogonal with respect to a Sobolev type inner product. A banded symmetric operator is associated with such a sequence of polynomials according to the higher order difference equation they satisfy. Taking into account the Darboux transformation of the corresponding matrix we deduce the connection with a sequence of orthogonal polynomials associated with a Christoffel perturbation of the measure involved in the standard part of the Sobolev inner product. A connection with matrix orthogonal polynomials is stated. The Laguerre-Sobolev ty…
For the weight function $W_μ(x) = (1-|x|^2)^μ$, $μ> -1$, $λ> 0$ and $b_μ$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product $$ \la f,g \ra = {b_μ[\int_{\BB^d} f(x) g(x) W_μ(x) dx + λ\int_{\BB^d} \nabla f(x) \cdot \nabla g(x) W_μ(x) dx]} $$ are constructed in terms of spherical harmonics and a sequence of Sobolev orthog onal polynomials of one variable. The latter ones, hence, the orthogonal polynomials with respect to $\la \cdot,\cdot\ra$, can be generated through a recursive formula.
In this paper we deal with polynomials orthogonal with respect to an inner product involving derivatives, that is, a Sobolev inner product. Indeed, we consider Sobolev type polynomials which are orthogonal with respect to $$(f,g)=\int fg dμ+\sum_{i=0}^r M_i f^{(i)}(0) g^{(i)}(0), \quad M_i \ge 0,$$ where $μ$ is a certain probability measure with unbounded support. For these polynomials, we obtain the relative asymptotics with respect to orthogonal polynomials related to $μ$, Mehler--Heine type asymptotics and their consequences about the asymptotic behaviour of the zeros. To establish these re…
Is is shown in this paper that there is a connection between the Riemann zeta-function $\zf$ and the classical Jacobi's polynomials, i.e. the Legendre polynomials, Chebyshev polynomials of the first and the second kind,...
The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appell's function $F_{3}(:)$ due to Marichev-Saigo-Maeda (MSM), to the Jacobi type orthogonal polynomials. The results are expressed in terms of generalized hypergeometric function. Some of the interesting special cases of the main results also established.
We study approximation properties of weighted $L^2$-orthogonal projectors onto spaces of polynomials of bounded degree in the Euclidean unit ball, where the weight is of the generalized Gegenbauer form $x \mapsto (1-\|x\|^2)^α$, $α> -1$. Said properties are measured in Sobolev-type norms in which the same weighted $L^2$ norm is used to control all the involved weak derivatives. The method of proof does not rely on any particular basis of orthogonal polynomials, which allows for a short, streamlined and dimension-independent exposition.
Persymmetric Jacobi matrices are invariant under reflection with respect to the anti-diagonal. The associated orthogonal polynomials have distinctive properties that are discussed. They are found in particular to be also orthogonal on the restrictions either to the odd or to the even points of the complete orthogonality lattice. This is exploited to design very efficient inverse problem algorithms for the reconstruction of persymmetric Jacobi matrices from spectral points. Isospectral deformations of such matrices are also considered. Expressions for the associated polynomials and their weight…
We consider the discrete Sobolev inner product $$(f,g)_S=\int f(x)g(x)dμ+Mf^{(j)}(c)g^{(j)}(c), \quad j\in \mathbb{N}\cup\{0\}, \quad c\in\mathbb{R}, \quad M>0, $$ where $μ$ is a classical continuous measure with support on the real line (Jacobi, Laguerre or Hermite). The orthonormal polynomials with respect to this Sobolev inner product are eigenfunctions of a differential operator and obtaining the asymptotic behavior of the corresponding eigenvalues is the principal goal of this paper.
We say that the polynomial sequence $(Q^{(λ)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ <p, r>_S=<{\bf u} ,{p\, r}> +λ<{\bf u}, {{\mathscr D}p \,{\mathscr D}r}>, $$ where ${\bf u}$ is a semiclassical linear functional, ${\mathscr D}$ is the differential, the difference or the $q$--difference operator, and $λ$ is a positive constant. In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the sem…
In this paper, we consider the second-order differential expression \ell [y](x)=(1-x^2)(-(y'(x))'+k(1-x^2)^(-1)y(x))(x\in(-1,1)). This is the Jacobi differential expression with non-classical parameters α = β= -1 in contrast to the classical case when α, β > -1. For fixed k \geq 0 and appropriate values of the spectral parameter λ, the equation \ell[y]=λy has, as in the classical case, a sequence of (Jacobi) polynomial solutions {P_{n}^{(-1,-1)}}_{n=0}^{\infty}. These Jacobi polynomial solutions of degree \geq 2 form a complete orthogonal set in the Hilbert space L^2((-1,1);(1-x^2)^(-1). Unlik…
In this contribution we deal with sequences of monic polynomials orthogonal with respect to the Freud Sobolev-type inner product \begin{equation*} \left\langle p,q\right\rangle _{s}=\int_{\mathbb{R}}p(x)q(x)e^{-x^{4}}dx+M_{0}p(0)q(0)+M_{1}p^{\prime }(0)q^{\prime }(0), \end{equation*}% where $p,q$ are polynomials, $M_{0}$ and $M_{1}$ are nonnegative real numbers. Connection formulas between these polynomials and Freud polynomials are deduced and, as a consequence, a five term recurrence relation for such polynomials is obtained. The location of their zeros as well as their asymptotic behavior i…
Orthogonal polynomials on the product domain $[a_1,b_1] \times [a_2,b_2]$ with respect to the inner product $$ \langle f,g \rangle_S = \int_{a_1}^{b_1} \int_{a_2}^{b_2} \nabla f(x,y)\cdot \nabla g(x,y)\, w_1(x)w_2(y) \,dx\, dy + λf(c_1,c_2)g(c_1,c_2) $$ are constructed, where $w_i$ is a weight function on $[a_i,b_i]$ for $i = 1, 2$, $λ> 0$, and $(c_1, c_2)$ is a fixed point. The main result shows how an orthogonal basis for such an inner product can be constructed for certain weight functions, in particular, for product Laguerre and product Gegenbauer weight functions, which serve as primary e…
We consider a general discrete Sobolev inner product involving the Hahn difference operator, so this includes the well--known difference operators $\mathscr{D}_{q}$ and $Δ$ and, as a limit case, the derivative operator. The objective is twofold. On the one hand, we construct the ladder operators for the corresponding nonstandard orthogonal polynomials and we obtain the second--order difference equation satisfied by these polynomials. On the other hand, we generalise some related results appeared in the literature as we are working in a more general framework. Moreover, we will show that all th…
We investigate type I multiple orthogonal polynomials on $r$ intervals which have a common point at the origin and endpoints at the $r$ roots of unity $ω^j$, $j=0,1,\ldots,r-1$, with $ω= \exp(2πi/r)$. We use the weight function $|x|^β(1-x^r)^α$, with $α,β>-1$ for the multiple orthogonality relations. We give explicit formulas for the type I multiple orthogonal polynomials, the coefficients in the recurrence relation, the differential equation, and we obtain the asymptotic distribution of the zeros.
We investigate the uniform asymptotic of some Sobolev orthogonal polynomials. Three term recurrence relation is given, moreover we give a recurrence relation between the so-called Sobolev orthogonal polynomials and Freud orthogonal polynomials.
This contribution deals with the sequence $\{\mathbb{U}_{n}^{(a)}(x;q,j)\}_{n\geq 0}$ of monic polynomials, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam--Carlitz I orthogonal polynomials, and involving an arbitrary number of $q$-derivatives on the two boundaries of the corresponding orthogonality interval. We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order $q$-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certai…
It turned out that the partial sums $g_n(z) = \sum_{k=0}^n \frac{(a_1)_k ... (a_p)_k}{(b_1)_k ... (b_q)_k} \frac{z^k}{k!}$, of the generalized hypergeometric series ${}_p F_q(a_1,...,a_p; b_1,...,b_q;z)$, with parameters $a_j,b_l\in\mathbb{C}\backslash\{ 0,-1,-2,... \}$, are Sobolev orthogonal polynomials. The corresponding monic polynomials $G_n(z)$ are polynomials of $R_I$ type, and therefore they are related to biorthogonal rational functions. Polynomials $g_n$ possess a differential equation (in $z$), and a recurrence relation (in $n$). We study integral representations for $g_n$, and some…
In this paper we study modified kernel polynomials: $u_n(x) = \sum_{k=0}^n c_k g_k(x)$, depending on parameters $c_k>0$, where $\{ g_k \}_0^\infty$ are orthonormal polynomials on the real line. Besides kernel polynomials with $c_k = g_k(t_0)>0$, for example, $c_k$ may be chosen to be some other solutions of the corresponding second-order difference equation of $g_k$. It is shown that all these polynomials satisfy a $4$-th order recurrence relation. The cases with $g_k$ being Jacobi or Laguerre polynomials are of a special interest. Suitable choices of parameters $c_k$ imply $u_n$ to be Sobolev…
We study a family of orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue equation involving a third order differential operator of Dunkl-type. The orthogonality measure of these polynomials consists in the continuous measure of the little -1 Jacobi polynomials to which is added an arbitrary mass located at the point $x=0$, the middle of the orthogonality interval. This provides the first nontrivial example of Krall-type polynomials with a point mass inside the orthogonality interval. These polynomials can be obtained by a Geronimus transform of the litt…
In this contribution we consider sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product \[ \langle f,g \rangle _{S}:= \langle {\bf u}, f g\rangle +N (\mathscr D_q f)(α) (\mathscr D _{q}g)(α),\qquad α\in \mathbb R, \quad N\ge 0, \] where $\bf u$ is a $q$-classical linear functional and $\mathscr D _{q}$ is the $q$-derivative operator. We obtain some algebraic properties of these polynomials such as an explicit representation, a five-term recurrence relation as well as a second order linear $q$-difference holonomic equation fulfilled by such polynomials. We presen…
Let $p\geq 1$, $\ell\in \NN$, $α,β>-1$ and $\varpi=(ω_0,ω_1, \dots, ω_{\ell-1})\in \RR^{\ell}$. Given a suitable function $f$, we define the discrete-continuous Jacobi-Sobolev norm of $f$ as: $$ \normSp{f}:= \left(\sum_{k=0}^{\ell-1} \left|f^{(k)}(ω_{k})\right|^{p} + \int_{-1}^{1} \left|f^{(\ell)}(x)\right|^{p} d\Jm(x)\right)^{\frac{1}{p}}, $$ where $ d\Jm(x)=(1-x)^α (1+x)^βdx$. Obviously, $\normSp[2]{\cdot}= \sqrt{\IpS{\cdot}{\cdot}}$, where $\IpS{\cdot}{\cdot}$ is the inner product. $$ \IpS{f}{g}:= \sum_{k=0}^{\ell-1} f^{(k)}(ω_{k}) \, g^{(k)}(ω_{k}) + \int_{-1}^{1} f^{(\ell)}(x) \,g^{(\ell)…
We consider the following discrete Sobolev inner product involving the Gegenbauer weight $$(f,g)_S:=\int_{-1}^1f(x)g(x)(1-x^2)^αdx+M\big[f^{(j)}(-1)g^{(j)}(-1)+f^{(j)}(1)g^{(j)}(1)\big],$$ where $α>-1,$ $j\in \mathbb{N}\cup \{0\},$ and $M>0.$ Let $\{Q_n^{(α,M,j)}\}_{n\geq0}$ be the sequence of orthogonal polynomials with respect to the above inner product. These polynomials are eigenfunctions of a differential operator $\mathbf{T}. $ We establish the asymptotic behavior of the corresponding eigenvalues. Furthermore, we calculate the exact value $$r_0 = \lim_{n\rightarrow \infty}\frac{\log \lef…
We introduce and develop a theory of orthogonality with respect to Sobolev inner products on the real line for sequences of functions with a tridiagonal, skew-Hermitian differentiation matrix. While a theory of such L2-orthogonal systems is well established, Sobolev orthogonality requires new concepts and their analysis. We characterise such systems completely as appropriately weighed Fourier transforms of orthogonal polynomials and present a number of illustrative examples, inclusive of a Sobolev-orthogonal system whose leading N coefficients can be computed in $\mathcal{O}(N \log N)$ operati…
