BACHELOR OF SCIENCE IN MECHANICAL ENGINEERING Study Guide - Quiz Guide: Self-Adjoint Operator, Hilbert Space, Real Analysis

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However, collaborating with other students or the internet is strictly prohib- ited. Evidence to the contrary will be treated as academic misconduct and will be responded to according to mit institute policy 10. 2: (8 points) determine the fourier coe cients f (n) = Inxdx, n z, where eit = cos t + i sin t, for the function f (x) = k=0 (2k+1)2 . Hint: what is the relationship between kf k2 and use the result to compute. 1 if x [ , 0], if x (0, ], P: (8 points) use the appropriate convergence theorems to compute. Z lim n n sin(x/n)[x(1 + x 2)] 1dx. Hint: a change of variables may be useful in one or both of the integrals before computing the limit: (8 points) let h be a hilbert space, and let w h be a linear subspace.

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