Linear Algebra Done Right · Problem 6C.18

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · minimization-problems

Problem

Find p∈P5(R)p \in \mathcal{P}_5(\mathbf{R}) that makes ∫−ππ∣sin⁡x−p(x)∣2 dx\int_{-\pi}^{\pi} |\sin x - p(x)|^2 \, dx as small as possible. The polynomial 6.65 is an excellent approximation to the answer to this exercise, but here you are asked to find the exact solution, which involves powers of π\pi. A computer that can perform symbolic integration should help.

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