The finite Legendre transform (fLT) transforms a mathematical function defined on the finite interval into its Legendre spectrum.[1][2] 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.
^Jerri, A.J. (1992). Integral and discrete transforms with applications and error analysis. Pure and Applied Mathematics. Vol. 162. New York: Marcel Dekker Inc. Zbl 0753.44001.
^Méndez-Pérez, J.M.R.; Miquel Morales, G. (1997). "On the convolution of the generalized finite Legendre transform". Math. Nachr. 188: 219–236. doi:10.1002/mana.19971880113. Zbl 0915.46038.
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