Book contents
- Frontmatter
- Contents
- Preface
- List of contributors
- Acknowlegements
- Generating expanders from two permutations
- Sum-free subsets
- Is there a different proof of the Erdős-Rado theorem?
- Almost collinear triples among N points on the plane
- Hamilton cycles in random graphs of minimal degree at least k
- The circumference of a graph with a given minimal degree
- On arithmetic progressions in sums of sets of integers
- On graphs not containing prescribed induced subgraphs
- Partitions sans petits sommants
- A compact sequential space
- The critical parameter for connectedness of some random graphs
- Multiplicative functions on arithmetic progressions: III. The large moduli
- Locally finite groups of permutations of ℕ acting on l∞
- Hypergraph games and the chromatic number
- On arithmetic graphs associated with integral domains
- On the number of certain subgraphs of graphs without large cliques and independent subsets
- Sets of multiples of Behrend sequences
- A functional equation arising from mortality tables
- The differences between consecutive primes, IV
- On the cofinality of countable products of cardinal numbers
- On σ-centered posets
- A Galvin–Hajnal conjecture on uncountably chromatic graphs
- Necessary conditions for mean convergence of Hermite–Fejér interpolation
- On the Erdős–Fuchs theorems
- A tournament which is not finitely representable
- On the volume of the spheres covered by a random walk
- Special Lucas sequences, including the Fibonacci sequence, modulo a prime
- A remark on heights of subspaces
- Incompactness for chromatic numbers of graphs
- Graphs with no unfriendly partitions
- On the greatest prime factor of an arithmetical progression
- The probabilistic lens: Sperner, Turan and Bregman revisited
- On the mean convergence of derivatives of Lagrange interpolation
- Sur une question d'Erdős et Schinzel
- Large α-preserving sets in infinite α-connected graphs
- Some recent results on interpolation
- Partitioning the quadruples of topological spaces
- Some of my favourite unsolved problems
Necessary conditions for mean convergence of Hermite–Fejér interpolation
Published online by Cambridge University Press: 05 March 2012
- Frontmatter
- Contents
- Preface
- List of contributors
- Acknowlegements
- Generating expanders from two permutations
- Sum-free subsets
- Is there a different proof of the Erdős-Rado theorem?
- Almost collinear triples among N points on the plane
- Hamilton cycles in random graphs of minimal degree at least k
- The circumference of a graph with a given minimal degree
- On arithmetic progressions in sums of sets of integers
- On graphs not containing prescribed induced subgraphs
- Partitions sans petits sommants
- A compact sequential space
- The critical parameter for connectedness of some random graphs
- Multiplicative functions on arithmetic progressions: III. The large moduli
- Locally finite groups of permutations of ℕ acting on l∞
- Hypergraph games and the chromatic number
- On arithmetic graphs associated with integral domains
- On the number of certain subgraphs of graphs without large cliques and independent subsets
- Sets of multiples of Behrend sequences
- A functional equation arising from mortality tables
- The differences between consecutive primes, IV
- On the cofinality of countable products of cardinal numbers
- On σ-centered posets
- A Galvin–Hajnal conjecture on uncountably chromatic graphs
- Necessary conditions for mean convergence of Hermite–Fejér interpolation
- On the Erdős–Fuchs theorems
- A tournament which is not finitely representable
- On the volume of the spheres covered by a random walk
- Special Lucas sequences, including the Fibonacci sequence, modulo a prime
- A remark on heights of subspaces
- Incompactness for chromatic numbers of graphs
- Graphs with no unfriendly partitions
- On the greatest prime factor of an arithmetical progression
- The probabilistic lens: Sperner, Turan and Bregman revisited
- On the mean convergence of derivatives of Lagrange interpolation
- Sur une question d'Erdős et Schinzel
- Large α-preserving sets in infinite α-connected graphs
- Some recent results on interpolation
- Partitioning the quadruples of topological spaces
- Some of my favourite unsolved problems
Summary
Abstract
Necessary conditions are given for the Hermite–Fejér interpolation polynomials based at the zeros of orthogonal polynomials to converge in weighted Lp spaces at the Jackson rate. These conditions are known to be sufficient in the case of the generalized Jacobi polynomials.
Introduction
The first detailed study of weighted mean convergence of Hermite–Fejér interpolation based at the zeros of orthogonal polynomials was accomplished in [13] and [14], where it was shown that some of the most delicate problems associated with mean convergence of Hermite–Fejér interpolation can be approached through the general theory of orthogonal polynomials; in particular, a distinguished role is played by Christoffel functions. As opposed to Lagrange interpolation operators, Hermite–Fejér interpolation operators are not projectors, and thus in general the rate of convergence cannot be expected to equal the rate of the best approximation. Nevertheless, Jackson rates can be obtained.
Unaware of the general theory in [13] and [14] and of a variety of technical tools developed in [6], [9] and [10] (see [11] for a survey), A. K. Varma & J. Prasad in [22] investigated mean convergence of Hermite–Fejér interpolation in a particular case, namely in the case of interpolation based at the zeros of the Chebyshev polynomials. Subsequently, P. Vértesi & Y. Xu [23] wrote a paper dealing with the case of generalized Jacobi polynomials. However, their results left a significant gap between the necessary and the sufficient conditions.
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- A Tribute to Paul Erdos , pp. 317 - 330Publisher: Cambridge University PressPrint publication year: 1990