Thomson’s Theorem on Mean Square Polynomial Approximation
نویسنده
چکیده
In 1991, J. E. Thomson determined completely the structure of H2(μ), the closed subspace of L2(μ) that is spanned by the polynomials, whenever μ is a compactly supported measure in the complex plane. As a consequence he was able to show that if H2(μ) = L2(μ), then every function f ∈ H2(μ) admits an analytic extension to a fixed open set Ω, thereby confirming in this context a phenomenon noted earlier in various situations by S. N. Bernštein, S. N. Mergelyan, and others. Here we present a new proof of Thomson’s results, based on Tolsa’s recent work on the semiadditivity of analytic capacity, which gives more information and is applicable to other problems as well.
منابع مشابه
Factorization of multivariate positive Laurent polynomials
Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Féjer-Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. F...
متن کاملMultivariate Positive Laurent Polynomials
Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Féjer-Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. F...
متن کاملExtensions of D. Jackson's Theorem on Best Complex Polynomial Mean Approximation1)
Sufficient conditions for the uniform convergence of polynomials Pniz) of best qth power approximation to a given function /(z) of a complex variable on a smooth curve Y were presented in an early paper by Jackson [1]. Although his theorem has found frequent application in approximation theory (see [2]), only slight extensions of the result are to be found in the literature. The present paper g...
متن کاملThe Expected Number of Real Roots of a Multihomogeneous System of Polynomial Equations
The methods of Shub and Smale [SS93] are extended to the class of multihomogeneous systems of polynomial equations, yielding Theorem 1, which is a formula expressing the mean (with respect to a particular distribution on the space of coefficient vectors) number of real roots as a multiple of the mean absolute value of the determinant of a random matrix. Theorem 2 derives closed form expressions...
متن کاملSpectral Theorem for Normal Operators: Applications
In these notes, we present a number of interesting and diverse applications of the Spectral Theorem for Normal Operators. These include a Spectral Mapping Theorem for Normals Operators, a Spectral Characterization of Algebraic Operators, von Neumann’s Mean Ergodic Theorem, Pathconnectivity of the Group of Invertible Operators, and some Polynomial Approximation Results for Operators.
متن کامل