Branch of analysis that studies how functions may be approximated by simpler ones — polynomials, rational functions, splines, wavelets — and quantifies the error. Foundational theorems: Weierstrass 1885 uniform approximation (continuous f…
approximation-theory
Chebyshev polynomials T_n: orthogonal basis + minimax optimality
Chebyshev polynomials of the first kind T_n are defined by the trigonometric identity T_n(cos θ) = cos(n θ), with closed forms T_0 = 1, T_1…
Bernstein polynomials B_{k,n}(x): partition of unity + Weierstrass proof
Bernstein polynomials of degree n are B_{k,n}(x) = C(n,k) x^k (1 − x)^{n-k} for 0 ≤ k ≤ n, forming a partition of unity (Σ_k B_{k,n} ≡ 1)…
Padé approximant [m/n]: rational P_m/Q_n matching Taylor to order m+n
Padé 1892 rational approximants [m/n]_f(x) = P_m(x)/Q_n(x) are the unique rational function of numerator degree ≤ m and denominator degree…
T_2(x) = 2x² − 1: alternation (−1,0,1)→(1,−1,1); roots at ±√2/2
Sympy-exact minimax witness for the smallest non-trivial Chebyshev case. sp.chebyshevt(2, x) expands to 2x² − 1 identically. Evaluation…
Bernstein partition-of-unity n=2: Σ_k B_{k,2}(x) ≡ 1; midpoint (1/4,1/2,1/4)
Sympy-exact symbolic verification of the Bernstein partition-of-unity identity at n = 2: the sum (1 − x)² + 2x(1 − x) + x² = 1 − 2x + x² +…
[1/1] Padé of exp: (1+x/2)/(1-x/2); leading O(x³) residual 1/12
Sympy-exact Taylor-to-order-3 comparison of the [1/1] Padé approximant of e^x against its Taylor series. Step 1 — the Padé rational: P(x)…