Lecture 3 Harmonic Analysis on the Cube and Parseval ’ s
نویسنده
چکیده
During the past weeks, we developed the general machinery which we will apply to problems in discrete math and computer science in the following weeks. In the general setting, we can ask how much information can we determine about a function f given its Fourier coefficients f̂ . Or, given f what can we say about f̂? There is some distinction between properties which will hold in the general setting, and those that make sense for the specific spaces we have dealt with. So far, we have looked at
منابع مشابه
Lecture 4 Applications of Harmonic Analysis February 4 , 2005 Lecturer : Nati Linial
Most of our applications of harmonic analysis to computer science will involve only Parseval’s identity. Theorem 4.1 (Parseval’s Identity). ‖f‖2 = ‖f̂‖2 Corollary 4.2. 〈f, g〉 = 〈f̂ , ĝ〉. Proof. Note that 〈f + g, f + g〉 = ‖f + g‖2 = ‖f̂ + g‖2 = ‖f̂ + ĝ‖2. Now as 〈f + g, f + g〉 = ‖f‖2 + ‖g‖2 + 2〈f, g〉, and similarly ‖f̂ + ĝ‖2 = ‖f̂‖2 + ‖ĝ‖2 + 2〈f̂ , ĝ〉, applying Parseval to ‖f‖2 and ‖g‖2 and equating fi...
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