About: Finite Legendre transform

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The finite Legendre transform (fLT) transforms a mathematical function defined on the finite interval into its Legendre spectrum.Conversely, the inverse fLT (ifLT) reconstructs the original function from the components of the Legendre spectrum and the Legendre polynomials, which are orthogonal on the interval [−1,1]. Specifically, assume a function x(t) to be defined on an interval [−1,1] and discretized into N equidistant points on this interval. The fLT then yields the decomposition of x(t) into its spectral Legendre components,

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  • The finite Legendre transform (fLT) transforms a mathematical function defined on the finite interval into its Legendre spectrum.Conversely, the inverse fLT (ifLT) reconstructs the original function from the components of the Legendre spectrum and the Legendre polynomials, which are orthogonal on the interval [−1,1]. Specifically, assume a function x(t) to be defined on an interval [−1,1] and discretized into N equidistant points on this interval. The fLT then yields the decomposition of x(t) into its spectral Legendre components, where the factor (2k + 1)/N serves as normalization factor and Lx(k) gives the contribution of the k-th Legendre polynomial to x(t) such that (ifLT) The fLT should not be confused with the Legendre transform or Legendre transformation used in thermodynamics and quantum physics. (en)
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  • The finite Legendre transform (fLT) transforms a mathematical function defined on the finite interval into its Legendre spectrum.Conversely, the inverse fLT (ifLT) reconstructs the original function from the components of the Legendre spectrum and the Legendre polynomials, which are orthogonal on the interval [−1,1]. Specifically, assume a function x(t) to be defined on an interval [−1,1] and discretized into N equidistant points on this interval. The fLT then yields the decomposition of x(t) into its spectral Legendre components, (en)
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  • Finite Legendre transform (en)
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