Maths 361 Fourier Series Notes 2

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1 Today s topics: Even and odd functions Real trigonometric Fourier series Section 1. : Odd and even functions Consider a function f : [, ] R. Maths 361 Fourier Series Notes f is odd if f( x) = f(x) for all x [, ]. f is even if f( x) = f(x) for all x [, ]. Some useful properties of odd and even functions : 1. If f is even If f is odd f(x) dx = f(x) dx 0 f(x) dx = 0. If f, g are even functions and q, r are odd functions then fg and qr are even functions, fq is an odd function. A function defined on the interval [0, ] can be extended to [, ] as an even function or an odd function: The odd extension of f is defined by f odd (x) = The even extension of f is defined by f even (x) = { f(x), x [0, ], f( x), x [, 0) { f(x), x [0, ], f( x), x [, 0) 1

2 Section 1.3 : Real trig Fourier series Recall from last lecture : If {φ n } is a complete orthogonal set in P S[a, b] then f P S[a, b] can be represented as a sum of φ n : f(x), φ n (x) f(x) = c n φ n = φ φ n n (x) Equality will hold except possibly for a finite number of points x [a, b]. We would like to express a function f(x) as a sum of cosine and sine terms, i.e., would like for some a 0, a n, b n, n = 1,... f(x) = a 0 + a n cos + b n sin The right side of equation (1) is the called the real trig Fourier representation of f(x). Assuming (for now) no problems with convergence of the infinite series, we can calculate a 0, a n, b n using earlier results. In particular, { ( ) ( )} S = 1, cos, sin is an orthogonal set in the IPS P S[, ] with inner product f, g, = f(x)g(x) dx, and is in fact complete (for pointwise convergence). So if f P S[, ] then (1) a 0 = f, 1 1 for n = 1,.... a n = f, cos ( ) cos ( ) b n = f, sin ( ) sin ( ) We can calculate 1 =, cos ( ) = and sin ( ) = for n = 1,,.... Summary : A function f P S[, ] has the (classical or real trig) Fourier representation where f(x) a 0 + a n cos + b n sin a 0 = 1 f(x) dx a n = 1 f(x) cos dx

3 for n = 1,... b n = 1 f(x) sin dx Note that in () means has the Fourier representation. We do not (yet) know whether f is equal to its Fourier representation at any particular point x although we expect this to be the case at all but finitely many points of [, ]. Example : Calculate the real trig Fourier representation of f(x) = 1 + x for x [, ] We find that i.e., a 0 = 1, all other a n = 0, b n = nπ ( 1)n+1 ( ) f(x) 1 + nπ ( 1)n+1 sin We can use Matlab to plot the first few terms of the Fourier series: 3

4 Example : Calculate the real trig Fourier representation of f(x) = { 1, x < 0 1, 0 x We find that f(x),3,5... ( ) 4 nπ sin i.e., all a n = 0, b n = 4 nπ for odd n and b n = 0 for even n. We can use Matlab to plot the first few terms of the Fourier series: 4

5 Today s topics: Convergence of Fourier series Sketching Fourier series Fourier cosine and sine series Recommended reading: Haberman 3. and 3.3 (excluding Gibb s phenomenon) Recommended exercises: Haberman 3..1(b),(d),(f); 3..(a),(c),(f); 3.3.1(d); 3.3.(d); 3.3.4; Section 1.4 Convergence of Fourier series Consider a function f : [a, b] R. We define f per, the periodic extension of f, by f per (x + n(b a)) = f(x) for each integer n and each x [a, b). Theorem (Dirichlet) : et f P S[, ]. et for n = 1,.... a n = 1 b n = 1 a 0 = 1 f(x) dx f(x) cos dx f(x) sin dx Define N N S N (x) = a 0 + a n cos + b n sin, i.e., S N is the sum of the first N + 1 terms in the real trig Fourier representation of f for x [, ]. Then for x R. lim S N(x) = f per(x + ) + f per (x ) N 5

6 Dirichlet s theorem says that for f P S[, ], the Fourier series of f converges to f per whenever f per is continuous; the average of the right and left limits of f per whenever f per has a jump discontinuity. See section 5.5 of PDE s : an introduction by W. A. Strauss for a proof of this theorem. Sketching Fourier series We can now sketch the Fourier representation of a function without first calculating a 0, a n, b n. For f P S[a, b], let S (x) = lim N S N be the real trig Fourier representation of f in [, ]. To sketch S (x): 1. Sketch f per (x) without marking value at points of discontinuity.. Mark at points of discontinuity. f per (x + ) + f per (x ) 6

7 1.5 Fourier sine and cosine series The sets { ( )} S 1 = sin and { ( )} S = 1, cos are orthogonal and complete for pointwise convergence in P S[0, ] with inner product f, g = Define the Fourier sine series of f to be 0 f(x)g(x) dx b n sin where b n = 0 f(x) sin dx Define the Fourier cosine series of f to be a 0 + a n cos where a 0 = 1 f(x) dx 0 a n = f(x) cos dx 0 Theorem : If f P S[0, ], then at each x R the Fourier sine series of f converges to f per (x + ) + f per (x ) where f per is the periodic extension of f odd and f odd is the odd extension of f to [, ] as defined in the last lecture. Similarly, if f P S[0, ], then at each x R the Fourier cosine series of f converges to f per (x + ) + f per (x ) where f per is the periodic extension of f even and f even is the even extension of f to [, ] as defined in the last lecture. 7