Edwards-Penney, sections 9.1 to 9.4 The textbook topics, definitions and theorems

Edwards-Penney BVP 9.1 to 9.4 (3.7 K, txt, 17 Apr 2015)

Chapter 9 Edwards-Penney BVP textbook Fourier Series Methods 9.1 Periodic Functions and Trigonometric Series DEF. Periodic Function. f(t+p)=f(t) for all t. p=period. Orthogonality Relations DEF. Two functions u, v are said to be othogonal on [a,b] provided integral(u*v,a..b)=0. A list of functions is said to be othogonal on [a,b] provided any two of them are orthgonal on [a,b]. THEOREM. The trigonometric list of sin(nt), cos(mt), n=1..infinity, m=0..infinity, is orthogonal on [-Pi,Pi]. The trigonometric list is independent on [-Pi,Pi], because it is a list of Euler atoms. DEF. A Fourier series is a formal sum of trigonometric terms from the trig list. A Fourier sine series is a Fourier series with no cosine terms. A Fourier cosine series is a Fourier series with no sine terms. Fourier Coefficient Formulas Let f(x) be defined on [-Pi,Pi]. Define a[m] = (1/Pi)*integral(f(t)*cos(mt),-Pi..Pi), m=0..infinity b[n] = (1/Pi)*integral(f(t)*sin(nt),-Pi..Pi), n=1..infinity Classical Fourier Series (1/2)*a[0] + SUM( a[m]*cos(m*x), m=1..infinity) + SUM( b[n]*sin(n*x), n=1..infinity) THEOREM. The formulas for a[m], b[n] need not be memorized. They arise from one idea: (1) Start with f(x) = trigonometric series (2) Multiply the equation in (1) by one trigonometric atom. Integrate over [-Pi,Pi]. (3) Orthogonality implies that the integrated series has exactly one nonzero term! Divide to find the corresponding coefficient a[m] or b[n]. DEF:= integral(u*v,a..b). It has these INNER PRODUCT properties. The vector space V together with these properties is called an INNER PRODUCT SPACE. (1)=+linear in the first argument (2) = c* (3)=symmetry (4) = 0 if and only if u=0 DEF: On inner product space V, the NORM is defined by |u| =sqrt(), or equivalently, |u|^2 =. DEF: A VECTOR is a package in a set V. Set V, called a VECTOR SPACE, is equipped with addition and scalar multiplication, such that the two closure laws hold and the 8 properties are valid (group under addition, scalar disribution laws). THEOREM. Letn=1..infinity be a list of orthogonal functions on [a,b]. Let f = SUM(c[n]*f[n],n=1..infinity). Then c[n] = / = integral(f*f[n],a..b) / integral(f[n]*f[n],a..b) EXAMPLE. Find a[m], b[n] for the square wave f(x) = -1 on (-Pi,0), f(x) = 1 on (0,Pi), f(x)=0 for x=-Pi,0,Pi. Plot the Fourier series F(x) of f(x) on -2Pi to 2Pi. ANSWER. a[m]=0 for all m, because f(x) is odd. b[n] = 4/(n*Pi) for n odd b[n] = 0 for n even GIBB's OVERSHOOT. At discontinuities of f(x), F(x) has a strange behavior, called Gibb's Overshoot. This can be seen by plotting a truncated Fourier series near discontinuities of f(x). EXAMPLE. Find the Fourier series of f(x) on [-Pi,Pi], where f(x) = x*pulse(x,0,Pi) except that f(x)=Pi/2 at x=Pi and x=-Pi. ANSWER. a[0] = Pi/2 a[m] = [(-1)^m-1]/(m^2*Pi^2) for m=1..infinity b[n] = (-1)^n(-1)/n for n=1..infinity 9.2 Fourier Convergence Theorem THEOREM. Let f(x) be smooth on [-Pi,Pi] and F(x) its formal Fourier series, built with the Fourier coefficient formulas. Then f(x) = F(x) for all x in [-Pi,Pi]. THEOREM. The convergence theorem above continues to hold if f(x) is only piecewise smooth, but the equation f(x) = F(x) only holds at points of continuity of f(x). At other points, there is the equation (f(x+)+f(x-))/2 = F(x). THEOREM. The series convergence is uniform if f(x) is smooth. It is not uniform for the Gibb's example.

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