MATB42H3 Lecture Notes - Lecture 4: Piecewise, Trigonometric Polynomial, Fourier Series
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Note: on the ith subinterval, i = 1, , n, f (x) coincides with some fi(x) which is continuous on [ti 1, ti]. If fi(x), i = 1, , n, have continuous rst derivatives f (x) is said to be piecewise smooth. If fi(x), i = 1, , n, have continuous second derivatives f (x) is said to be piecewise very smooth. De nition: the fourier series obtained from f (x) converges to f (x) if f (x) = lim. Xk=1 (cid:0)ak cos kx + bk sin kx(cid:1)(cid:21) assuming period 2 (and can be adjusted for other periods). The ak and bk are the fourier coe cients. Theorem: let f (x) be continuous and piecewise very smooth for all x and let f (x) have period 2 . Then the fourier series of f (x) converges uniformly to f (x) for all x.
