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  "body": "\n## Summary\n\nThe shifted Legendre polynomials are the [[Legendre polynomials|QFYGJVA996]] moved from the interval $[-1, 1]$ to the interval $[0, 1]$.\nThey are given by $\\tilde{P}_n(x) = P_n(2x - 1)$.\nThey are orthogonal on $[0, 1]$ for the constant weight.\nSo they are the natural polynomial basis for least squares and Gaussian quadrature on the unit interval.\n\n## Explanations\n\n### L1 human\n\nLegendre polynomials live on the interval from $-1$ to $1$.\nMany problems live on the interval from $0$ to $1$ instead, for example in probability.\nThe substitution $x \\mapsto 2x - 1$ stretches and moves $[0, 1]$ onto $[-1, 1]$, and composing with it gives the shifted polynomials.\nFor example, $P_1(x) = x$ becomes $\\tilde{P}_1(x) = 2x - 1$, which has average zero on $[0, 1]$.\n",
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