Approximation by Polynomials

Real Analysis · Axiom Academy

LESSON Approximation by Polynomials The Weierstrass theorem: every continuous function on a closed interval can be uniformly approximated by polynomials — to any accuracy you name. 1. The Theorem: Staying Inside the ε-Tube Let f be continuous on a closed interval [a,b] . Draw a band of vertical half-width around its graph — an -tube . Weierstrass guarantees that no matter how thin you make the tube, some polynomial p fits entirely inside it, hugging f at every single point of the interval. The sup norm measures the single worst gap, anywhere on [a,b] . 2. Building One: Bernstein Polynomials Weierstrass's existence claim has a beautifully concrete recipe. On [0,1] , sample f at the n+1 evenly spaced points k/n , then take a weighted average of those heights — the weight on sample k is the Bernstein basis polynomial . Each weight peaks near x = k/n and the weights sum to 1 at every x , so B_n(f) is a genuine running average of the sampled heights f(k/n) — a true polynomial that leans toward whichever sample is nearest. 3. Raising the Degree Tightens the Fit Push n up and the Bernstein polynomial hugs f ever more closely. Here the target is a smooth wave; watch B_n(f) for peel toward it while the worst-case error — printed live and computed , not faked — shrinks toward 0 . Few samples — B_n is a loose, rounded sketch of the target. The sup-gap is large. Many samples — B_n tracks every wiggle. The single worst gap collapses toward zero.

This is the written version of the interactive lesson above. See the full Real Analysis course.