Analyze the asymptotic growth of polynomial degree needed to approximate a continuous function with decreasing error tolerance | Step-by-Step Solution

What We're Solving:

We want to understand how the degree of polynomials must grow as we demand increasingly accurate approximations to continuous functions. Specifically, we're investigating the relationship between error tolerance ε and the minimal polynomial degree n needed to achieve that tolerance.

The Approach:

If we want to trace a curved line with straight-line segments, how many segments do we need for a given accuracy? We're doing something similar but with polynomials approximating functions. We'll examine this through different lenses - the smoothness of our function dramatically affects the answer!

Step-by-Step Solution:

Step 1: Set Up the Framework For a continuous function f on [a,b], let E_n(f) be the best polynomial approximation error of degree n: E_n(f) = min_{p∈P_n} ||f - p||_∞

We want to find how n must grow as E_n(f) → 0.

Step 2: The Basic Weierstrass Result The original theorem tells us E_n(f) → 0 as n → ∞, but doesn't specify the rate. This is where the fun begins - the rate depends entirely on how "nice" our function is!

Step 3: Case 1 - Just Continuous Functions For merely continuous functions, we can have arbitrarily slow convergence!

• Key insight: There exist continuous functions where E_n(f) decreases slower than any polynomial rate
• Example: Functions that are continuous but highly oscillatory near points
• Practical bound: E_n(f) ≤ ω(1/n) where ω is the modulus of continuity

Step 4: Case 2 - Smooth Functions (The Game Changer!) Here's where things get exciting! If f has k continuous derivatives:

• Jackson's Theorem: E_n(f) ≤ C ω(f^{(k)}, 1/n)/n^k
• Translation: More derivatives = faster convergence = lower degree needed for same accuracy

Step 5: Case 3 - Analytic Functions (The Best Case) For analytic functions that extend to the complex plane:

• Bernstein's Theorem: E_n(f) ≤ C r^{-n} for some r > 1
• Amazing result: Exponential convergence! We need only n ≈ log(1/ε) for error ε

Step 6: The Inversion - Degree vs. Error To achieve error ε, we need approximately:

• Just continuous: n could be as large as we want (no general bound)
• k times differentiable: n ≈ ε^{-1/k}
• Analytic: n ≈ log(1/ε)

The Answer:

The asymptotic behavior follows this beautiful hierarchy:

• 1. Continuous only: No universal rate - can be arbitrarily slow
• 2. C^k smooth: n ~ ε^{-1/k} (polynomial growth in 1/ε)
• 3. Analytic: n ~ log(1/ε) (logarithmic growth - amazing!)

Key Formula: For C^k functions, to achieve error ε, we need degree n ≈ (C/ε)^{1/k} where C depends on the function's k-th derivative.

Memory Tip:

Remember the "Smoothness Ladder":

• Rough functions (just continuous) → Rough convergence (slow)
• Smooth functions (many derivatives) → Polynomial rate
• Super-smooth functions (analytic) → Exponential rate

The more derivatives your function has, the fewer polynomial terms you need - it's like having better building blocks to construct your approximation!

This connects beautifully to Fourier analysis and shows why understanding function regularity is so crucial in numerical analysis. Keep exploring - this rabbit hole goes deep and connects to some of the most beautiful mathematics!