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Arcsine-weighted inner product on the Chebyshev interval
Definition
For continuous real functions on , define
This is the arcsine-weighted pairing on the compact Chebyshev interval: in the variable its weight is for . The angular integral is the endpoint convention, so no endpoint value of the singular density is required.
The displayed integral is defined for every such pair: cosine is continuous and maps into ; composition and multiplication preserve continuity, so the integrand is continuous. Since , it is Riemann integrable by A continuous function on is Riemann integrable, by Heine-Cantor and Riemann's criterion.
Depends on
- Pi as twice the smallest positive zero of cosine
- The derivatives of sine and cosine are cosine and minus sine
- A function differentiable at $c$ is continuous at $c$
- Signs, monotonicity intervals, and ranges of sine and cosine
- A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs
- Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
- A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion
Used by
- Composite-Simpson assembly of the Chebyshev weighted mass matrix Algorithm
- Chebyshev weighted spectral Sobolev scale and projection Definition
- Analytic and nonsmooth data have geometric and algebraic Chebyshev coefficient decay Example
- Gaussian and Simpson mass assembly give distinct exact and approximate checks Example
- Chebyshev polynomials are orthogonal for the arcsine weight Lemma
- Analytic ellipse data give exponential Chebyshev–Lobatto interpolation error Theorem
- Chebyshev–Lobatto interpolation inherits a weighted near-best bound Theorem
- Chebyshev–Lobatto interpolation remainder bound Theorem
- Composite Simpson error bound for Chebyshev weighted mass entries Theorem
Dependency tree · two levels
52 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Jie Shen and Tao Tang, Spectral and High-Order Methods with Applications, §§1.3, 3.3 (standard reference, not scraped)
- Lloyd N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Ch. 8 (standard reference, not scraped)