Zeros of orthogonal polynomials

THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Laplace Transform

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

Preface 1. Fourier series and Fourier transforms 2. Distributions 3. Elliptic equations (fundamental theory) 4. Initial value problems (Cauchy problems) 5. Evolution equations 6. Hyperbolic equations 7. Semi-linear hyperbolic equations 8. Green's functions and spectra Supplementary remarks Guide to the literature Bibliography Symbols Index.

The theory of partial differential equations

. Dependencies between extreme events (extremal dependencies) are attracting an increasing attention in modern risk management. In practice, the concept of tail dependence represents the current standard to describe the amount of extremal dependence. In theory, multi-variate extreme-value theory turns out to be the natural choice to model the latter dependencies. The present paper embeds tail dependence into the concept of tail copulae which describes the dependence structure in the tail of multivariate distributions but works more generally. Various non-parametric estimators for tail copulae and tail dependence are discussed, and weak convergence, asymptotic normality, and strong consistency of these estimators are shown by means of a functional delta method. Further, weak convergence of a general upper-order rank-statistics for extreme events is investigated and the relationship to tail dependence is provided. A simulation study compares the introduced estimators and two financial data sets were analysed by our methods.

Non‐parametric Estimation of Tail Dependence

We prove that the Hausdorff operator generated by a function φ ∈ L1(R) is bounded on the real Hardy space H1(R). The proof is based on the closed graph theorem and on the fact that if a function f in L1(R) is such that its Fourier transform f̂(t) equals 0 for t < 0 (or for t > 0), then f ∈ H1(R). 1. Preliminaries We recall that the Fourier transform f̂ of a function f in L(R) is defined by letting f̂(t) := 1 √ 2π ∫ R f(x)e−itx dx, t ∈ R; while its Hilbert transform f̃ is defined by letting f̃(x) := 1 π (P.V.) ∫ ∞ 0 f(x− u)− f(x+ u) u du, x ∈ R, where the principal value integral is defined to be limε↓0 ∫∞ ε . It is well known that this limit exists for almost every x ∈ R. The real Hardy space H(R) is defined to be H(R) := {f ∈ L(R) : f̃ ∈ L(R)}, endowed with the norm ‖f‖H1 := ‖f‖L1 + ‖f̃‖L1 , (1.1) where ‖f‖L1 := ∫

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The Hausdorff operator is bounded on the real Hardy space ¹(ℝ)

For a wide family of multivariate Hausdorff operators, the boundedness of an operator from this family is proved on the real Hardy space. By this we extend and strengthen previous results due to Andersen and Moricz.

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Multidimensional hausdorff operators on the real hardy space

In this survey, results on the representation of a function as an absolutely convergent Fourier integral are collected, classified and discussed. Certain applications are also given.

The Wiener algebra of absolutely convergent Fourier integrals: an overview

Sufficient conditions for the boundedness of the Hausdorff operators in the Hardy spaces H, 0 < p < 1, on the real line are proved. Two related negative results are also given.

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Boundedness of the Hausdorff operators in $H^{p}$ spaces, 0 < p < 1

In this paper we investigate properties of the general monotone sequences and functions, a generalization of monotone sequences and functions as well as of those of bounded variation. Some applications to various problems of analysis are given. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Elijah Liflyand

Papers

The Hausdorff operator is bounded on the real Hardy space ¹(ℝ)

Multidimensional hausdorff operators on the real hardy space

The Wiener algebra of absolutely convergent Fourier integrals: an overview

Boundedness of the Hausdorff operators in $H^{p}$ spaces, 0 < p < 1

A concept of general monotonicity and applications