Real and Complex Analysis

Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s

Boundary value problems

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

As the titel suggests, this is a book dealing with extrapolation methods, and the authors clearly have in mind that a reader, interested in applying these methods, should try this hand in using them. To this end, a convenient floppy disk, containing various programmed extrapolation algorithms, is included with the book, and the final chapter (chapter 7) of this book didactically shows the readers how to use these programs! The material is certainly timely and also self-contained, in the sense that nearly everything on the theory and applications of extrapolation methods is contained here or in numerous references cited. (There are nearly 500 references given here!) Interspersed between theoretical results are interesting historical discussions of various algorithms (who did what and when) and illustrative numerical applications (from a variety of areas), both serving to nicely reinforce the theory presented. Consistent with the aim of stressing applications, proofs of theorems are not given in this book, but clarity is not lost in the process since definitions and notations are unambiguously presented. (By omitting proofs, the density of theorems in this book becomes large: 188 theorems in 395 pages.) Moreover, the book is written in an engaging and lively style. (This reviewer even learned two new English words in reading this book: \"obtention\" and \"limitative\".) For this reviewer, the applications given in chapter 6 were most interesting, as they include integral equations, Monte-Carlo methods, and numerical quadrature, to mention a few topics. There is even a discussion in this chapter of avoiding near-breakdowns of the biconjugate gradient method, which is decidedly of current research interest in numerical analysis today! As a personal user of, and believer in, extrapolation methods, this book is an invaluable reference book on extrapolation methods and their applications, and this book will be part of my regular research library!

Extrapolation methods: theory and practice: C. Brezinski and M. Redivozaglia studies in computational mathematics 2 North-Holland, Amsterdam, 1991

Linear systems and operators in hilbert space

Let m, n be nonnegative integers and B (m+n) be a set of m + n + 1 real interpolation points (not necessarily distinct). Let Rm,n = P m,n/Qm.n be the unique rational function with deg Pm,n ≤ m, deg Qm,n ≤ n, that interpolates ex in the points of B (m+n). If m = mv , n = nv with mv + nv → ∞, and mv / nv → λ as v → ∞, and the sets B (m+n) are uniformly bounded, we show that locally uniformly in the complex plane C, where the normalization Qm,n (0) = 1 has been imposed. Moreover, for any compact set K ⊂ C we obtain sharp estimates for the error |ez — Rm,n (z)| when z ∈ K. These results generalize properties of the classical Pade approximants. Our convergence theorems also apply to best (real) Lp rational approximants to ex on a finite real interval.

Rational Interpolation of the Exponential Function

We consider multipoint Pad\'e approximation to Cauchy transforms of complex measures. We show that if the support of a measure is an analytic Jordan arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that arc with Dini-smooth non-vanishing density, then the diagonal multipoint Pad\'e approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the arc. This asymptotic behavior of Pad\'e approximants is deduced from the analysis of underlying non-Hermitian orthogonal polynomials, for which we use classical properties of Hankel and Toeplitz operators on smooth curves. A construction of the appropriate interpolation schemes is explicit granted the parametrization of the arc.

Convergent Interpolation to Cauchy Integrals over Analytic Arcs

We design convergent multipoint Pade interpolation schemes to Cauchy transforms of non-vanishing complex densities with respect to Jacobi-type weights on analytic arcs, under mild smoothness assumptions on the density. We rely on our earlier work for the choice of the interpolation points, and dwell on the Riemann-Hilbert approach to asymptotics of orthogonal polynomials introduced by Kuijlaars, McLaughlin, Van Assche, and Vanlessen in the case of a segment. We also elaborate on the $\bar\partial$-extension of the Riemann-Hilbert technique, initiated by McLaughlin and Miller on the line to relax analyticity assumptions. This yields strong asymptotics for the denominator polynomials of the multipoint Pade interpolants, from which convergence follows.

Convergent Interpolation to Cauchy Integrals over Analytic Arcs with Jacobi-Type Weights

We study AAK-type meromorphic approximants to functions $F$, where $F$ is a sum of a rational function $R$ and a Cauchy transform of a complex measure $\lambda$ with compact regular support included in $(-1,1)$, whose argument has bounded variation on the support. The approximation is understood in $L^p$-norm of the unit circle, $p\geq2$. We obtain that the counting measures of poles of the approximants converge to the Green equilibrium distribution on the support of $\lambda$ relative to the unit disk, that the approximants themselves converge in capacity to $F$, and that the poles of $R$ attract at least as many poles of the approximants as their multiplicity and not much more.

Meromorphic Approximants to Complex Cauchy Transforms with Polar Singularities

With the aid of Havin’s Lemma (which we generalize) we prove that polynomials orthogonal over the unit disk with respect to certain weighted area measures (Bergman polynomials) cannot satisfy a finite-term recurrence relation unless the weight is radial, in which case the polynomials are simply monomials. For polynomials orthogonal over the unit circle (Szegő polynomials) we provide a simple argument to show that the existence of a finite-term recurrence implies that the weight must be the reciprocal of the square modulus of a polynomial.

/pdf/on-finite-term-recurrence-relations-for-bergman-and-szego-2a5miqgy66.pdf

On Finite-Term Recurrence Relations for Bergman and Szegő Polynomials

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