Mathematics / advanced
Approximation, interpolation, and compression
Compare polynomial, spline, rational, and nonlinear approximations using error bounds and stable representations.
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Inside this module
Module 01 advanced
Approximation, interpolation, and compression
Compare polynomial, spline, rational, and nonlinear approximations using error bounds and stable representations.
- 01
Best approximation and Chebyshev equioscillation
Choose a norm before optimizing, prove a minimax certificate from alternating extrema, and distinguish optimality from interpolation.
27 min - 02
Orthogonal polynomials and projection error
Derive weighted least-squares coefficients from orthogonality and account for the discarded energy without claiming uniform optimality.
27 min - 03
Chebyshev nodes and interpolation stability
Use endpoint-clustered nodes to control the interpolation product while keeping derivative error bounds separate from data conditioning.
28 min - 04
Barycentric interpolation and Lebesgue constants
Evaluate the interpolant without solving for monomial coefficients and quantify the sensitivity that no formula can remove.
29 min - 05
Cubic splines and variational characterization
Build a natural cubic interpolant and prove why it minimizes squared curvature under fixed interpolation constraints.
29 min - 06
Rational approximation and Padé approximants
Match a local power series with a quotient while distinguishing local accuracy, genuine singularities, and unstable pole-zero cancellations.
28 min - 07
Jackson estimates and smoothness
Translate a modulus of continuity into a constructive approximation rate and keep existence bounds distinct from a chosen Fourier truncation.
28 min - 08
Wavelet bases and multiresolution
Build Haar scaling and detail spaces, derive energy-preserving coefficients, and distinguish localized resolution from global Fourier modes.
29 min - 09
Sparse approximation and n-term rates
Select the largest orthonormal coefficients, derive an explicit tail rate, and account for the extra information needed by adaptive compression.
28 min - 10
Quadrature as functional approximation
Approximate an integral as a bounded linear functional, derive exactness-based error bounds, and connect Gaussian rules to orthogonal polynomials.
29 min