From high oscillation to rapid approximation V: the equilateral triangle

We address in this paper the approximation of functions in an equilateral triangle by a linear combination of Laplace–Neumann eigenfunctions. The Laplace–Neumann basis exhibits a number of advantages. The approximations converge fairly fast and their speed of convergence can be much improved by using techniques familiar in Fourier analysis and spectral methods, in particular, the hyperbolic cross and polynomial subtraction. Moreover, expansion coefficients can be computed rapidly by a mixture of asymptotic methods and Birkhoff–Hermite quadratures.

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