Finite Legendre transform

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.

Legendre filter

The fLT of a noisy experimental outcome s(t) and the subsequent application of the inverse fLT (ifLT) on an appropriately truncated Legendre spectrum of s(t) gives a smoothed version of s(t). The fLT and incomplete ifLT thus act as a filter. In contrast to the common Fourier low-pass filter which transmits low frequency harmonics and filters out high frequency harmonics, the Legendre lowpass transmits signal components proportional to low degree Legendre polynomials, while signal components proportional to higher degree Legendre polynomials are filtered out. [3]

gollark: Oh, so they're harder to cool despite the same total heat output because of the greater density of 7nm? That does make more sense.
gollark: I guess that you could maybe, I don't know, have differences in *measured* temperature depending on where the thermal sensors are, or have different fan control. But that couldn't really change total heat output.
gollark: The *only way* it can heat up is by converting electricity to heat when operating.
gollark: That makes absolutely no sense.
gollark: You are obviously not seeing the people without issues by looking there though.

References
  1. Jerri, A.J. (1992). Integral and discrete transforms with applications and error analysis. Pure and Applied Mathematics. 162. New York: Marcel Dekker Inc. Zbl 0753.44001.
  2. 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.
  3. Guobin Bao and Detlev Schild, Fast and accurate fitting and filtering of noisy exponentials in legendre space, 2014. PLoS ONE, 9(3), e90500

Further reading
  • Butzer, Paul L. (1983). "Legendre transform methods in the solution of basic problems in algebraic approximation". Functions, series, operators, Proc. int. Conf., Budapest 1980, Vol. I,. Colloq. Math. Soc. János Bolyai. 35. pp. 277–301. Zbl 0567.41010.
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