MATH-M 344 Chapter Notes - Chapter 10: Fourier Series, Periodic Function, Orthogonality
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M344 section 10. 2 notes- introduction to fourier series. Good way to solve pdes generalization of orthogonality of vectors (see 7. 2) When interval becomes infinite, generalize sum to integral. Fourier series has form 0(cid:2870)+ (cid:3040) (cid:2869: on set of point where series converges, this formula defines the function, recall definition of periodic function- : with period >(cid:882) such that (cid:4666)(cid:1876)+(cid:4667)= (cid:4666)(cid:1876)(cid:4667),(cid:1876) ; series contains 2 periodic functions. Sine, cosine are (cid:884)-periodic cos(cid:4672)(cid:3040) (cid:1876)(cid:4673) and sin(cid:4672)(cid:3040) (cid:1876)(cid:4673: for (cid:1865)=(cid:883),(cid:884), , these are (cid:2870)(cid:3040) periodic, cos((cid:3040) (cid:4672)(cid:1876)+(cid:2870)(cid:3040)(cid:4673))=cos(cid:4672)(cid:3040) +(cid:884)(cid:4673)=cos(cid:4672)(cid:3040) (cid:1876)(cid:4673, (cid:2870)(cid:3040) is a fundamental period for both of these functions (can be proven for sine too) Also (cid:884)-periodic, though it is not fundamental period unless (cid:1865)=(cid:883) [(cid:1853)(cid:3040)cos(cid:4672)(cid:3040) (cid:1876)(cid:4673)+(cid:1854)(cid:3040)sin(cid:4672)(cid:3040) (cid:1876)(cid:4673)] that exists for all (cid:1876) is a (cid:884)-periodic function (cid:3040) (cid:2869: can also write sum with . (cid:3040)=(cid:2869: for real functions (cid:1873) ,(cid:1874) defined on [(cid:2009),(cid:2010)], can define an inner product (cid:4666)(cid:1873),(cid:1874)(cid:4667)= (cid:1873)(cid:4666)(cid:1876)(cid:4667)(cid:1874)(cid:4666)(cid:1876)(cid:4667) (cid:1876); (cid:3081)(cid:3080)
