Measure, Integration & Real Analysis · Problem 8C.1

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · orthonormal-families-trigonometric

Problem

Verify that the family {ek}k∈Z\{e_k\}_{k\in\mathbf{Z}} as defined in the third bullet point of Example 8.51 is an orthonormal family in L2((−π,π])L^2\bigl((-\pi, \pi]\bigr). The following formulas should help: (sin⁡x)(cos⁡y)=sin⁡(x−y)+sin⁡(x+y)2,(\sin x)(\cos y) = \frac{\sin(x-y)+\sin(x+y)}{2}, (sin⁡x)(sin⁡y)=cos⁡(x−y)−cos⁡(x+y)2,(\sin x)(\sin y) = \frac{\cos(x-y)-\cos(x+y)}{2}, (cos⁡x)(cos⁡y)=cos⁡(x−y)+cos⁡(x+y)2.(\cos x)(\cos y) = \frac{\cos(x-y)+\cos(x+y)}{2}.

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