week6 - Week 6 Lectures, Math 716, Tanveer 1 Fourier Series...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Week 6 Lectures, Math 716, Tanveer 1 Fourier Series In the context of separation of variable to find solutions of PDEs, we encountered f ( x ) = summationdisplay n =1 b n sin nx l for x (0 ,l ) (1) or f ( x ) = a 2 + summationdisplay n =1 a n cos nx l for x (0 ,l ) (2) and in other cases f ( x ) = a 2 + summationdisplay n =1 braceleftBig a n cos nx l + b n sin nx l bracerightBig for x ( l,l ) (3) The general representation (3) is called the Fourier representation of f ( x ) in the ( l,l ) interval, while (1) and (2) are the Fourier sine and cosine representations of f ( x ) in the interval (0 ,l ). Our discussions revolve around some basic questions 1. When are represenations (1)-(3) valid and in what sense? 2. How do we determine coefficients a n , b n in terms of f ( x ). 3. What conditions allow term by term differentiation of the Fourier-Series. 2 General L 2 theory We need enough generality to be able to lay the framework for discussion of more general series representations of L 2 , i.e. square integrable functions than (1)-(3), since they arise in other PDE problems. In the space L 2 ( a,b ) of generally complex valued functions in the interval ( a,b ), we introduce the L 2 inner-product: ( f,g ) = integraldisplay b a f ( x ) g ( x ) dx (4) We note that the L 2 norm is related through bardbl f bardbl = ( f,f ) 1 / 2 = bracketleftBigg integraldisplay b a | f | 2 ( x ) dx bracketrightBigg 1 / 2 (5) This may be generalized to any number of dimension, with x replaced by x and integration over the interval ( a,b ), replaced by integration over appropriate n-dimensional rectangle. 1 Definition 1 A sequence { X n } n =1 L 2 ( a,b ) is orthogonal if ( X n ,X m ) = 0 iff m negationslash = n (6) This sequence is said to be orthonormal, if in addition ( X n ,X n ) = bardbl X n bardbl 2 = 1 . Theorem 2 Let { X n } n =1 L 2 ( a,b ) be a orthogonal set of functions. Let bardbl f bardbl < . Let N be a fixed positive integer. The choice of A n that minimizes mean square error E N = bardbl f N n =1 A n X n bardbl 2 is given by A n = ( f,X n ) bardbl X n bardbl 2 (7) Further, we have bardbl f bardbl 2 summationdisplay n =1 | ( f,X n ) | 2 bardbl X n bardbl 2 Bessel inequality Proof. Define the square of the E N = bardbl f N summationdisplay n =1 A n X n bardbl 2 = parenleftBigg f N summationdisplay n =1 A n X n ,f N summationdisplay n =1 A n X n parenrightBigg Expanding the above using properties of inner product and the orthogonality of X n , we get E N = ( f,f ) N summationdisplay n =1 A n ( X n ,f ) n summationdisplay n =1 A n ( f,X n ) + N summationdisplay n =1 A n A n ( X n ,X n ) (8) We minimize E n as a function of 2 N real variables, { ( c n ,d n ) } N n =1 , where A n = c n + id n . On taking partial derivatives, we get E N c n = ( X n ,f ) ( f,X n ) + 2 c n ( X n ,X n ) = 0 implying c n = { ( X n ,f ) } bardbl X n bardbl 2 Also,...
View Full Document

This note was uploaded on 07/26/2011 for the course MATH 716 taught by Professor Tanveer,s during the Spring '08 term at Ohio State.

Page1 / 10

week6 - Week 6 Lectures, Math 716, Tanveer 1 Fourier Series...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online