Abstract. An object-oriented MATLAB system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices. About eighty MATLAB functions from plot and sum to svd and cond have been overloaded so that one can work with our “chebfun ” objects using almost exactly the usual MATLAB syntax. All functions live on [−1, 1] and are represented by values at sufficiently many Chebyshev points for the polynomial interpolant to be accurate to close to machine precision. Each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation, differentiation, integration, zerofinding, or transforms. Applications in approximation theory and numerical analysis are explored, and possible extensions for more substantial problems of scientific computing are mentioned.

Jacobi polynomials are solutions of the differential equation (z 2 − 1)f ′ ′ (z)+(az + b)f ′ (z)+cf(z) =0, (1) where a, b, c are constants satisfying a>b, a + b>0 and c = n(1 − a − n) for

The equilibrium measure in the presence of an external field plays a role in a number of areas in analysis, for example in random matrix theory: the limiting mean density of eigenvalues is precisely the density of the equilibrium measure. Typical behavior for the equilibrium measure is: 1. it is positive on the interior of a finite number of intervals, 2. it vanishes like a square root at endpoints, and 3. outside the support, there is strict inequality in the Euler-Lagrange variational conditions. If these conditions hold, then the limiting local eigenvalue statistics is loosely described by a "bulk" in which there is universal behavior involving the sine kernel, and "edge effects" in which there is a universal behavior involving the Airy kernel. Through techniques from potential theory and integrable systems, we show that this "regular" behavior is generic for equilibrium measures associated with real analytic external fields. In particular, we show that for any one-parameter family of external fields V=c the equilibrium measure exhibits this regular behavior, except for an at most countable number of values of c. We discuss applications of our results to random matrices, orthogonal polynomials and integrable systems.

We give a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the conjugate gradient method or other Krylov subspace methods. We present a new bound on the relative error after n iterations. This bound is valid in an asymptotic sense when the size N of the system grows together with the number of iterations. The bound depends on the asymptotic eigenvalue distribution and on the ratio n/N. Under appropriate conditions we show that the bound is asymptotically sharp. Our findings are related to some recent results concerning asymptotics of discrete orthogonal polynomials. An important tool in our investigations is a constrained energy problem in logarithmic potential theory. The new asymptotic bounds for the rate of convergence are illustrated by discussing Toeplitz systems as well as a model problem stemming from the discretization of the Poisson equation.

Abstract. Let L(f) = � log �Df � dµf denote the Lyapunov exponent of a rational map, f: P 1 → P 1. In this paper, we show that for any holomorphic family of rational maps {fλ: λ ∈ X} of degree d> 1, T (f) = dd c L(fλ) defines a natural, positive (1,1)-current on X supported exactly on the bifurcation locus of the family. The proof is based on the following potential-theoretic formula for the Lyapunov exponent: L(f) = � GF (cj) − log d + (2d − 2) log(cap KF). Here F: C 2 → C 2 is a homogeneous polynomial lift of f; | det DF (z) | = � |z ∧ cj|; GF is the escape rate function of F; and cap KF is the homogeneous capacity of the filled Julia set of F. We show, in particular, that the capacity of KF is given explicitly by the formula cap KF = | Res(F) | −1/d(d−1), where Res(F) is the resultant of the polynomial coordinate functions of F. We introduce the homogeneous capacity of compact, circled and pseudoconvex sets K ⊂ C 2 and show that the Levi measure (determined by the geometry of ∂K) is the unique equilibrium measure. Such K ⊂ C 2 correspond to metrics of non-negative curvature on P 1, and we obtain a variational characterization of curvature.

Abstract. When discussing the convergence properties of the Lanczos iteration method for the real symmetric eigenvalue problem, Trefethen and Bau noted that the Lanczos method tends to find eigenvalues in regions that have too little charge when compared to an equilibrium distribution. In this paper a quantitative version of this rule of thumbis presented. We describe, in an asymptotic sense, the region containing those eigenvalues that are well approximated by the Ritz values. The region depends on the distribution of eigenvalues and on the ratio between the size of the matrix and the number of iterations, and it is characterized by an extremal problem in potential theory which was first considered by Rakhmanov. We give examples showing the connection with the equilibrium distribution. Key words. Ritz values, equilibrium distribution, potential theory

Abstract. We prove that Birkhoff normal form of hamiltonian flows at a nonresonant singular point with given quadratic part are always convergent or generically divergent. The same result is proved for the normalization mapping and any formal first integral.

We study families of quantum eld theories of free bosons on a compact Riemann surface of genus g. For the case g > 0 these theories are parameterized by holomorphic line bundles of degree g 1, and for the case g = 0 | by smooth closed Jordan curves on the complex plane. In both cases we dene a notion of -function as a partition function of the theory and evaluate it explicitly. For the case g > 0 the -function is an analytic torsion [3], and for the case g = 0 | the regularized energy of a certain natural pseudo-measure on the interior domain of a closed curve. For these cases we rigorously prove the Ward identities for the current correlation functions and determine them explicitly. For the case g > 0 these functions coincide with those obtained in [21, 36] using bosonization. For the case g = 0 the -function we have dened coincides with the -function introduced in [29, 44, 24] as a dispersionless limit of the Sato's -function for the two-dimensional Toda hierarchy. As a corollary of the Ward identities, we obtain recent results [44, 24] on relations between conformal maps of exterior domains and -functions. For this case we also dene a Hermitian metric on the space of all contours of given area. As another corollary of the Ward identities we prove that the introduced metric is Kahler and the logarithm of the -function is its Kahler potential. Contents

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The continuum limit of the Toda lattice was studied by Deift and McLaughlin in the spirit of the Lax-Levermore theory for the zero dispersion limit of the Kortewegde Vries equation. An important role is played by a quadratic minimization problem arising from an asymptotic analysis of the inverse spectral transform. The minimum is taken over density functions ψ in the spectral variable satisfying the constraints 0 ≤ ψ ≤ φ where φ is a function determined by the initial conditions. The finite gap ansatz is said to hold if the set where the two constraints are not effective consists of a finite union of intervals. If the finite gap ansatz hold, weak limits are described in terms of the endpoints of the intervals. Using techniques from logarithmic potential theory, we show that the finite gap ansatz holds for real analytic spectral data. This extends a previous result of Deift, Kriecherbauer and McLaughlin for the situation without upper constraint φ. 1