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algebra arbitrary arithmetic means assume boundary conditions bounded complex numbers consider constant continuous function converges uniformly coordinate cos2 cosine defined definition denote derivative differential equation Dirichlet Dirichlet kernel discussed domain element equal example Exercise exists Figure finite number finite-dimensional formula Fourier series Fourier sine series Fourier transform frequency function g function of period functions of class given graph harmonic function identically zero infinite series inner product integrable function interval Laplace's equation Legendre polynomials Lemma linear combination linear space linearly independent matrix nonzero norm obtain orthogonal orthonormal partial sum period 2ir piecewise continuous piecewise smooth polynomial of degree positive integer problem proj proof prove reader real number region scalar multiple Section 3.3 self-adjoint series converges Show sin2 Sn(x sphere string subspace surface temperature tends to zero term term-by-term Theorem tion trigonometric polynomial uniformly convergent valid values variables vector vibrations zero function