Approximation by polynomials on a triangle is studied in the Sobolev space
W r
2 that consists of functions whose derivatives of up to r-th order have bounded L2
norm. The first part aims at understanding the orthogonal structure in the Sobolev
space on the triangle, which requires explicit construction of an inner product that
involves derivatives and its associated orthogonal polynomials, so that the projection
operators of the corresponding Fourier orthogonal expansion commute with partial
derivatives. The second part establishes the sharp estimate for the error of polynomial
approximation in W2 r, when r = 1 and r = 2, where the polynomials of approximation
are the partial sums of the Fourier expansions in orthogonal polynomials of the Sobolev
space.

• Approximation and Orthogonality in Sobolev Spaces on a Triangle