Alphabeta MathHow statement and proof provenance work
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Legendre polynomials by Rodrigues' formula and their unweighted pairing
Definition
For each integer , define the Legendre polynomial
with as in The factorial and the falling factorial , defined by recursion in and repeated derivatives as in Higher derivatives and the classes and . Define the unweighted pairing of real polynomials by
The integral is the ordinary integral of a continuous function; polynomial products are continuous and hence integrable by 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 and A continuous function on is Riemann integrable, by Heine-Cantor and Riemann's criterion. The normalization and all polynomial identities needed for the orthogonal Galerkin scale are proved in Legendre polynomials have explicit norms and weighted derivative orthogonality.
Depends on
- The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$
- Higher derivatives and the classes $C^k$ and $C^\infty$
- 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
- Endpoint-preserving integrated Legendre projector Definition
- Legendre weighted spectral-Sobolev scale and projection Definition
- Legendre-Gauss-Lobatto nodes, weights, and tensor quadrature Definition
- Gaussian and Simpson mass assembly give distinct exact and approximate checks Example
- Legendre polynomials have explicit norms and weighted derivative orthogonality Lemma
- Legendre-Gauss-Lobatto nodes are distinct and interior critical points are simple Lemma
- GLL quadrature changes only the top Legendre modes and is p-uniformly equivalent Theorem
Dependency tree · two levels
54 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, Ch. 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)