Zero location and asymptotic behavior for extremal polynomials with non-diagonal Sobolev norms

In this paper we are going to study the zero location and asymptotic behavior of extremal polynomials with respect to a non-diagonal Sobolev norm in the worst case, i.e., when the quadratic form is allowed to degenerate. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials. The multiplication operator by the independent variable is the main tool in order to obtain our results.     

Asymptotic of extremal polynomials in the complex plane

We study the zero location and asymptotic zero distribution of sequences of polynomials which satisfy an extremal condition with respect to a norm given on the space of all polynomials.     

Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality

We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e−φ(x), giving a unified treatment for the so-called Freud (i.e., when φ has polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.     

On the exact Markov inequality for k-monotone polynomials in uniform and L 1-norms

We consider the classical extremal problem of estimating norms of higher order derivatives of algebraic polynomials when their norms are given. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A. Markov, while Bernstein found the exact constant in the Markov inequality for monotone polynomials. In this note we give Markov-type inequalities for higher order derivatives in the general class of k-monotone polynomials. In particular, in case of first derivative we find the exact solution of this extremal problem in both uniform and L ₁-norms. This exact solution is given in terms of the largest zeros of certain Jacobi polynomials.     

Generation and evaluation of orthogonal polynomials in discrete Sobolev spaces I: algorithms

In this paper, we study theoretically the determination and evaluation of polynomials that are orthogonal with respect to a general discrete Sobolev inner product, that is, an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives. This Sobolev inner product has the property that the orthogonal polynomials with respect to it satisfy a linear recurrence relation of fixed order. We provide a complete set of formulas to compute the coefficients of this recurrence. Besides, we study the determination of the Fourier–Sobolev coefficients of a finite approximation of a function and the numerical evaluation of the resulting finite series at a general point.     

Sobolev orthogonal polynomials in the complex plane

Sobolev orthogonal polynomials with respect to measures supported on compact subsets of the complex plane are considered. For a wide class of such Sobolev orthogonal polynomials, it is proved that their zeros are contained in a compact subset of the complex plane and their asymptotic-zero distribution is studied. We also find the nth-root asymptotic behavior of the corresponding sequence of Sobolev orthogonal polynomials.     

On a Characterization of Integrability for the Reciprocal Weight of Orthogonal Polynomials on the Circle

We prove a necessary and sufficient condition for integrability of the reciprocal weight function of orthogonal polynomials. The condition is given in terms of the asymptotic behaviour of the norm of extremal polynomials with prescribed coefficients.     

Defect correction and Galerkin's method for second-order boundary value problems

The defect correction technique, based on the Galerkin finite element method, is analyzed as a procedure to obtain highly accurate numerical solutions to second-order elliptic boundary value problems. The basic solutions, defined over a rectangular region Ω, are computed using continuous piecewise bilinear polynomials on rectangles. These solutions are O(h²) accurate globally in the second-order discrete Sobolev norm. Corrections to these basic solutions are obtained using higher-order piecewise polynomials (Lagrange polynomials or splines) to form defects. An O(h²) improvement is gained on the first correction. The lack of regularity of the discrete problems (beyond the second-order Sobolev norm) makes it impossible to retain this order of improvement, but for problems satisfying certain periodicity conditions, straightforward arbitrary accuracy is obtained, since these problems possess high-order regularity. © 1992 John Wiley & Sons, Inc.     

Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities

We study the symmetry property of extremal functions to a family of weighted Sobolev inequalities due to Caffarelli-Kohn-Nirenberg. By using the moving plane method, we prove that all non-radial extremal functions are axially symmetric with respect to a line passing through the origin.

     

A generating function for nonstandard orthogonal polynomials involving differences: the Meixner case

In this paper we deal with a family of nonstandard polynomials orthogonal with respect to an inner product involving differences. This type of inner product is the so-called Δ-Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond to the Pascal distribution (the orthogonal polynomials associated to this distribution are known as Meixner polynomials). The aim of this work is to obtain a generating function for the Δ-Meixner–Sobolev orthogonal polynomials and, by using a limit process, recover a generating function for Laguerre–Sobolev orthogonal polynomials.     

On the solvability of a system of forward-backward linear equations with unbounded operator coefficients

An estimate of the norm of the Lagrange interpolation operator in the multidimensional weighted Sobolev space is obtained. It is shown that, under a certain choice of the sequence of multi-indices, the interpolating polynomials converge to the interpolated function and the rate of convergence is of the order of the best approximation of this function by algebraic polynomials in this space.     

Products of polynomials in uniform norms

We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. This subject includes the well known Gel'fond-Mahler inequalities for the unit disk and Kneser inequality for the segment . Using tools of complex analysis and potential theory, we prove a sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set of positive logarithmic capacity in the complex plane. The above classical results are contained in our theorem as special cases.

It is shown that the asymptotically extremal sequences of polynomials, for which this inequality becomes an asymptotic equality, are characterized by their asymptotically uniform zero distributions. We also relate asymptotically extremal polynomials to the classical polynomials with asymptotically minimal norms.

     

Random Linear Combinations of Functions from L 1

In this paper, properties of random polynomials with respect to a general system of functions are studied. Some lower bounds for the mathematical expectation of the uniform norm and the recently introduced integral-uniform norm of random polynomials are established.     

Computing orthogonal polynomials in Sobolev spaces

Summary. Numerical methods are considered for generating polynomials orthogonal with respect to an inner product of Sobolev type, i.e., one that involves derivatives up to some given order, each having its own (positive) measure associated with it. The principal objective is to compute the coefficients in the increasing-order recurrence relation that these polynomials satisfy by virtue of them forming a sequence of monic polynomials with degrees increasing by 1 from one member to the next. As a by-product of this computation, one gains access to the zeros of these polynomials via eigenvalues of an upper Hessenberg matrix formed by the coefficients generated. Two methods are developed: One is based on the modified moments of the constitutive measures and generalizes what for ordinary orthogonal polynomials is known as "modified Chebyshev algorithm". The other - a generalization of "Stieltjes's procedure" - expresses the desired coefficients in terms of a Sobolev inner product involving the orthogonal polynomials in question, whereby the inner product is evaluated by numerical quadrature and the polynomials involved are computed by means of the recurrence relation already generated up to that point. The numerical characteristics of these methods are illustrated in the case of Sobolev orthogonal polynomials of old as well as new types. Based on extensive numerical experimentation, a number of conjectures are formulated with regard to the location and interlacing properties of the respective zeros. Received July 13, 1994 / Revised version received September 26, 1994     

Generation and evaluation of orthogonal polynomials in discrete Sobolev spaces II: numerical stability

In this paper, we concern ourselves with the determination and evaluation of polynomials that are orthogonal with respect to a general discrete Sobolev inner product, that is, an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives. In a previous paper we provided a complete set of formulas to compute the coefficients of this recurrence. Here, we study the numerical stability of these algorithms for the generation and evaluation of a finite series of Sobolev orthogonal polynomials. Besides, we propose several techniques for reducing and controlling the rounding errors via theoretical running error bounds and a carefully chosen recurrence.     

Sobolev Spaces with Respect to Measures in Curves and Zeros of Sobolev Orthogonal Polynomials

In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial results about these Sobolev spaces with respect to measures; in particular, we prove a main result in the theory: they are Banach spaces. J.M. Rodriguez supported in part by three grants from M.E.C. (MTM 2006-13000-C03-02, MTM 2006-11976 and MTM 2007-30904-E), Spain, and by a grant from U.C.III M./C.A.M. (CCG07-UC3M/ESP-3339), Spain. J.M. Sigarreta supported in part by a grant from M.E.C. (MTM 2006-13000-C03-02), Spain, and by a grant from U.C.III M./C.A.M. (CCG07-UC3M/ESP-3339), Spain.     

Pseudo-uniform Convexity in Hp and Some Extremal Problems on Sobolev Spaces

We extend Newman and Keldysh theorems to the behavior of sequences of functions in H^p (μ) which explain geometric properties of discs in these spaces. Through Keldysh's theorem we obtain asymptotic results for extremal polynomials in Sobolev spaces.     

Estimate of the norm of the Lagrange interpolation operator in a multidimensional Sobolev space

We obtain an estimate of the norm of the Lagrange interpolation operator in a multidimensional Sobolev space. It is shown that, under a suitable choice of the sequence of multi-indices, interpolation polynomials converge to the interpolated function and their rate of convergence is of the order of the best approximation of this function.     

GENERALIZED WEIGHTED SOBOLEV SPACES AND APPLICATIONS TO SOBOLEV ORTHOGONAL POLYNOMIALS Ⅱ

We present a definition of general Sobolev spaces with respect to arbitrary measures ,W^k,p(Ω,μ) for 1≤p≤∞,In[RARP] we proved that these spaces are complete under very light conditions.Now we prove that if we consider certain general types of measures,then Cc^∞(R) is dense in these spaces,As an application to Sobolev orthogonal polynomials,we study the boundedness of the multiplication poerator,THis gives an estimation of the zeroes of Sobolev orthogonal polynomials.     

A note on the Geronimus transformation and Sobolev orthogonal polynomials

In this note we recast the Geronimus transformation in the framework of polynomials orthogonal with respect to symmetric bilinear forms. We also show that the double Geronimus transformations lead to non-diagonal Sobolev type inner products.