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Math. Z. 236,643–676 (2001) Digital Object Identifier (DOI) 10.1007/s002090000196 Riesz means associated with convex domains in the plane 1, 2, Andreas Seeger , Sarah Ziesler Department of Mathematics,University of Wisconsin-Madison,WI 53706,USA (e-mail: seeger@math.wisc.edu) Dominican University,River Forest,IL 60305,USA (e-mail: ziessara@email.dom.edu) Received August 25,1999; in final form February 10,2000 / Published online December 8,2000 – Springer-Verlag 2000 1. Introduction Let Ω be a bounded open convex setR which contains the origin. Let ρ be the associated Minkowski functional defined by −1 ρ(ξ) = inf t> 0: t ξ ∈ Ω }. We shall investigate the Riesz means of the inverse Fourier integral associ- ated with Ω 1 ρ(ξ) iξ,x (1.1) R f(x)= 1 − f(ξ)e dξ; λ,t (2π) t ρ(ξ)≤t −iy,ξ here our definition of the Fourier transform is f(ξ)= f(y)e dy.For t =1 we also setR = R and refer toR as the Bochner-Riesz operator λ λ,1 λ associated with Ω . Note that for λ =0 the Riesz means R are just the 0,t partial sum operators associated with the sets tΩ , t> 0,while for λ =1 −1 one recovers the Fejer ´ means,namely the averages R = t R ds. 1,t 0,s p 2 The objective
Mathematische Zeitschrift – Springer Journals
Published: Apr 1, 2001
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