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)…
Weierstrass approximation theorem
Every continuous function on [a,b] is uniformly approximable by polynomials. Stone-Weierstrass generalisation: any subalgebra of C(K, ℝ)…
Taylor series approximation
Smooth function f admits Taylor expansion f(x) = Σ f^(n)(a)(x-a)^n/n! within radius of convergence. Lagrange remainder bounds error. …
Fourier series approximation
Periodic L²-function decomposes into orthogonal trigonometric basis: f(x) = Σ c_n e^{inx}. Carleson 1966 a.e. convergence for L². Gibbs…
Spline interpolation (piecewise polynomial)
Schoenberg 1946: piecewise-polynomial interpolant with continuity at knots. B-splines basis; cubic-splines minimise integrated curvature. …
Minimax uniform approximation
Chebyshev: best uniform approximation in C[a,b] from finite-dim subspace exists, is unique, and equioscillates at n+2 points. Remez…
Lebesgue constant for interpolation
Λ_n = ||L_n|| operator-norm of n-point interpolation projection bounds approximation error: ||f - p_n|| ≤ (1+Λ_n) E_n(f). Equispaced…
Weierstrass approximation (1885)
Weierstrass 1885: continuous f on [a, b] uniformly approximated by polynomial; Bernstein 1912 explicit formula via Bernstein polynomials;…
Chebyshev equioscillation
Chebyshev 1854: best polynomial approximation p* of f in C[a,b] characterized by equioscillation: f - p* attains +/- |f-p*|_inf at >= n+2…
Muntz-Szasz theorem (1914)
Muntz 1914 + Szasz 1916: span of {x^{lambda_n}} dense in C[0,1] iff sum 1/lambda_n = +inf; density of monomial subseries; basis of…
Jackson theorem (1911)
Jackson 1911: best polynomial approximation E_n(f) <= C omega(f, 1/n) where omega is modulus of continuity; quantitative rate of polynomial…
Fejer kernel (Cesaro summation 1900)
Fejer 1900: Cesaro mean F_n = sum_{k<=n} (1 - k/n) e^{ikx} converges uniformly to f for f continuous + 2pi-periodic; salvages divergent…
Nyquist-Shannon sampling theorem
Whittaker 1915 + Nyquist 1928 + Shannon 1949 + Kotelnikov: bandlimited signal with cutoff B reconstructed exactly from samples at f_s >=…
Weierstrass theorem (1885)
K Weierstrass 1885 polynomial-approximation + Stone 1937 generalization; modern Bernstein + neural-network universal-approx.
Chebyshev (1854)
P Chebyshev 1854 equi-oscillation; modern minimax-approximation + Remez algorithm + Chebfun numerical computing toolbox.
KAR theorem (1957)
A Kolmogorov-V Arnold 1957 13th-Hilbert-problem solution; modern KAN Liu 2024 Kolmogorov-Arnold-Networks deep-learning.
Padé (1892)
H Padé 1892 [m/n]-rational-approximation; modern function-extrapolation + numerical-physics + matrix-Padé exponentials.
Splines (Schoenberg 1946)
I Schoenberg 1946 B-splines; modern NURBS CAD/graphics + finite-element + isogeometric-analysis Hughes-Cottrell-Bazilevs 2009.
Wavelets (Mallat 1989)
S Mallat 1989 multiresolution-analysis + I Daubechies 1988 orthogonal-wavelets; modern JPEG2000 + signal-denoising MR / EEG.