Equioscillation theorem

The equioscillation theorem concerns the approximation of continuous functions using polynomials when the merit function is the maximum difference (uniform norm). Its discovery is attributed to Chebyshev.

Statement

Let $f$ be a continuous function from $[a,b]$ to $\mathbb {R}$ . Among all the polynomials of degree $\leq n$ , the polynomial $g$ minimizes the uniform norm of the difference $||f-g||_{\infty }$ if and only if there are $n+2$ points $a\leq x_{0}<x_{1}<\cdots <x_{n+1}\leq b$ such that $f(x_{i})-g(x_{i})=\sigma (-1)^{i}||f-g||_{\infty }$ where $\sigma =\pm 1$ .

Algorithms

Several minimax approximation algorithms are available, the most common being the Remez algorithm.

gollark: There is no perfect language.

gollark: ```Internet Data Handling email — An email and MIME handling package json — JSON encoder and decoder mailcap — Mailcap file handling mailbox — Manipulate mailboxes in various formats mimetypes — Map filenames to MIME types base64 — Base16, Base32, Base64, Base85 Data Encodings binhex — Encode and decode binhex4 files binascii — Convert between binary and ASCII quopri — Encode and decode MIME quoted-printable data uu — Encode and decode uuencode files```Mostly should be libraries outside of the python core, and why are they not under file formats?

gollark: ```Concurrent Execution threading — Thread-based parallelism multiprocessing — Process-based parallelism The concurrent package concurrent.futures — Launching parallel tasks subprocess — Subprocess management sched — Event scheduler queue — A synchronized queue class _thread — Low-level threading API _dummy_thread — Drop-in replacement for the _thread module dummy_threading — Drop-in replacement for the threading module```Not THAT bad, since they mostly do different things.

gollark: Right beside each other.

gollark: ```argparse — Parser for command-line options, arguments and sub-commandsgetopt — C-style parser for command line options```

References

Notes on how to prove Chebyshev’s equioscillation theorem at the Wayback Machine (archived July 2, 2011)
The Chebyshev Equioscillation Theorem by Robert Mayans

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