Triangular Dirichlet Kernels and Growth of L~p Lebesgue Constants

摘要

Let P be a polygon in ?~2 and consider the mapping of an L~1(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<∞, then the operator norm will be given by the L~p norm of ∑_((m,n)∈NP)e~(2πi(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))).