I’m always on the lookout for interesting problems, particularly ones that I might assign my students at some point. The content of this post comes from Exercise 19, Chapter 4 of Rudin’s Real and Complex Analysis.
Fix a positive integer , and set . Then we have the orthogonality relations
Proof. It is clear that , for . For the second assertion, observe that and by the geometric series formula,
In the case , we can use the orthogonality relations to derive the following identity for inner product spaces:
Similarly, for any inner product space, we have the identity
We can also obtain the preceding identity from the preceding identity and the dominated convergence theorem. Indeed,