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The Success of the Approximation
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a+2L a0+a1cosx+a2cos2x+ approximating function approximation to f Assumption boundary conditions calculate Chapter consider constant cos2 cosine functions Cosine Series cosm and sinn define a scalar differential equations differential operator distance eigenfunctions eigenvalue Example Exercise expression extended f and g f with domain F(x)cosnxdx Find the Fourier finding coefficients finite number form an orthogonal formula Fourier coefficients Fourier Series approximates Fourier Sine Series function f function with domain Gibbs Phenomenon give given function heat equation Hermite polynomials infinite inner product integral Lemma linearly independent look mathematical method obtained odd function orthogonal set particular period 2n periodic with period point a e problem properties proved real number result scalar product sequence Series for f Series of f sets of functions Show sin2 sin3x sinn functions sinx solve space of functions specified string symmetric temperature Theorem 4.1 trigonometric functions values vector space