6.6-08 Best polynomial approximation

Best polynomial approximation

Best polynomial approximation projects a function onto a finite polynomial space. These questions combine concrete low-degree approximations with residual orthogonality, error calculations and the distinction between integral fitting and interpolation.

What you will practise

• Form projection coefficients and function-space Gram equations.
• Approximate polynomial targets by lower-degree spaces.
• Calculate minimum squared integral error.
• Reason about nested spaces and uniqueness.

Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail.

Before you start

Review orthogonal polynomials and the closest-point theorem. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports finite Fourier projection and broader approximation methods within Module 6 of the Complete Linear Algebra Course.