Triangular Dirichlet Kernels and Growth of L p Lebesgue Constants |
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Authors: | Marshall Ash |
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Affiliation: | 1. DePaul University, Chicago, IL, USA
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Abstract: | Let P be a polygon in ℤ2 and consider the mapping of an L1(mathbbT2)L^{1}(mathbb{T}^{2}) 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 L p norm, 1<p<∞, then the operator norm will be given by the L p norm of ∑(m,n)∈NP e 2π i(mx+ny). The size of this operator norm is shown to be O(N 2(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(kN 2(1−1/p)). |
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