PMATH450_S2015.pdf

# 1 x n 1 a n b n 2 i n c e 2 inx ˆ v n v x so for

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1 X n = -1 ( A n - B n )2 i n c e 2 inx = ˆ v ( n ) = v ( x ) So for each n , A n + B n = ˆ g ( n ), and A n - B n = ˆ v ( n ) 2 i n c for n 6 = 0. We can solve for n 6 = 0 to get A n = 1 2 ˆ g ( n ) + 1 2 ˆ v ( n ) c 2 in , B n = 1 2 ˆ g ( n ) - 1 2 ˆ v ( n ) c 2 in Thus u ( x, t ) = ˆ g (0) X n 6 =0 1 2 ˆ g ( n ) e 2 in ( x + t c ) + e 2 in ( x - t c ) + X n 6 =0 1 2 ˆ v ( n ) c 2 in e 2 in ( x + t c ) - e 2 in ( x - t c ) = 1 2 g x + t c + 1 2 g x - t c + X n 6 =0 1 2 ˆ v ( n ) c 2 in e 2 in ( x + t c ) - e 2 in ( x - t c ) Lecture 21: June 22 Now look at Z t 0 v x + s c ds = Z t 0 X n 6 =0 ˆ v ( n ) e 2 in ( x + s c ) ds =? X n 6 =0 ˆ v ( n ) Z t 0 e 2 in ( x + s c ) ds = X n 6 =0 ˆ v ( n ) e 2 in ( x + s c ) 2 in/c s = t s =0 = X n 6 =0 ˆ v ( n ) c 2 in e 2 in ( x + t c ) - e 2 inx Similarly, Z t 0 v x - s c ds = X n 6 =0 ˆ v ( n ) c 2 in - e 2 in ( x - t c ) + e 2 inx Thus 1 2 Z t 0 v x + s c + v x - s c ⌘⌘ dx = 1 2 0 @ X n 6 =0 ˆ v ( n ) c 2 in e 2 in ( x + t c ) - e 2 in ( x - t c ) 1 A Then Z t 0 v x ± s c ds = ± Z x + t c z = x cv ( z ) dz so 1 2 Z t 0 v x + s c + v x - s c ds = c 2 Z x + t c x - t c v ( z ) dz. Ideas Requiring Justification. 1. What kind of functions can be represented as f = P ˆ f ( n ) e 2 inx ? 2. What kind of convergence? 3. Is the representation unique? 4. When can we switch an integral and infinite sum? 5. Can we di erentiate an infinite sum term-by-term?

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3 FOURIER ANALYSIS 33 3.5 Introduction to Fourier Analysis on the Circle Definition. The unit circle in the Complex plane is T = { z 2 C : | z | = 1 } = { z = e i : 2 [0 , 2 ) } . We may also parameterize it as T = [0 , 2 ] with 0 , 2 identified, or T = [ - , ] with ± identified, or T = [0 , 1], where we identify angle with 2 2 [0 , 1]. Topology. We get the usual topology from the circle in the Complex plane. That is, it is compact. Functions. A function defined on T = [0 , 2 ) can be viewed as defined on [0 , 2 ] with f (0) = f (2 ) or defined on R that is 2 -periodic. Note. T is a group under multiplication (on C ) or addition mod 2 (on [0 , 2 ]). Notation. C ( T ) is the set of continuous functions on T , that is, the continuous 2 -periodic functions on R . m denotes Lebesgue measure, restricted to [0 , 2 ] and normalized so that m [0 , 2 ] = 1. This maintains all of the good properties of Lebesgue measure. Thus R 2 0 f dm = 1 2 R 2 0 f ( x ) dx . L p ( T ) is the set of equivalence classes of Lebesgue measurable functions on [0 , 2 ) with k f k p = ( 1 2 R 2 0 | f | p dx ) 1 /p < 1 . L 1 ( T ) is the set of essentially bounded functions on T . L 2 ( T ) is the Hilbert space with inner product h f, g i = 1 2 R 2 0 f g dx . Note. We have C ( T ) ( L 1 ( T ) ( L p ( T ). Also C ( T ) are dense in L p ( T ) for p < 1 because the continuous functions on [0 , 2 ] with f (0) = f (2 ) are dense in C [0 , 2 ] under the L p -norm. Holder’s Inequality. For 1 p + 1 q = 1.
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