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Unformatted text preview: 18.03 Problem Set 7 Fall 2009 Due in boxes in Room 2-106, Friday Nov 6 at 3:00 p.m. You may and should discuss the problems with other students, but you must write solutions entirely on your own. (The main reason for this requirement is that we enforce it so strictly on exams.) Question Zero is required for your problem set to count. 0. On the first page of your homework, list the individuals (other than recitation lead- ers, lecturer, and math department tutors) with whom you discussed the homework; and list the sources (other than the class text, web site, and supplementary notes) that you consulted. This will look like, “Consulted with Jane Doe; consulted text Birkhoff & Rota and web site http://www.clepsydra.ch/ ,” or “No consultation.” Collaboration is good, and finding things to read is good. 1. (6 points) Suppose f and g are 2 π-periodic functions which are piecewise continuous (so that their Fourier coefficients— a n ( f ), a n ( g ), and so on— are all defined). Define a new function ( f * g )( t ) = 1 2 π integraldisplay π- π f ( s ) g ( t- s ) ds. a) Show that f * g is 2 π-periodic. b) Find a formula for the complex Fourier coefficients of c n ( f * g ) in terms of c n ( f ) and c n ( g ). c) Suppose that f and f 1 are the square wave and sawtooth functions defined in Problem Set 6....
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- Spring '09
- Fourier Series, web site, Lecturer, 1.2 meters, complex Fourier coeﬃcients