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J Fourier Anal Appl (2010) 16: 10531069 DOI 10.1007/s00041-009-9115-8
 

Summary: J Fourier Anal Appl (2010) 16: 10531069
DOI 10.1007/s00041-009-9115-8
Triangular Dirichlet Kernels and Growth of Lp
Lebesgue Constants
Marshall Ash
Received: 30 September 2009 / Published online: 24 December 2009
Springer Science+Business Media, LLC 2009
Abstract Let P be a polygon in Z2 and consider the mapping of an L1(T2) function
into the partial sum of its Fourier series determined by the dilate of P by the integer
N. If the image space is endowed with the Lp norm, 1 < p < , then the operator
norm will be given by the Lp norm of (m,n)NP e2i(mx+ny). The size of this oper-
ator norm is shown to be O(N2(1-1/p)) when the polygon is a triangle. The estimate
is independent of the shape of the triangle. For a k sided polygon the corresponding
estimate is O(kN2(1-1/p)).
Keywords Lebesgue constant Dirichlet kernels in two dimensions Dirichlet
kernels for polygons
Mathematics Subject Classification (2000) Primary 42B15 42A05 Secondary
42A45
1 Introduction
In one dimension there is a very well known estimate [9]

  

Source: Ash, J. Marshall - Department of Mathematical Sciences, DePaul University

 

Collections: Mathematics