Orthogonal functions can be broadly classified into three families: the piecewise constant family, the polynomial family, and the sine-cosine family. Each family is classified into different classes of basis functions. Classes such as block-pulse functions, Haar functions and Walsh functions belong to the piecewise constant family while Chebyshev polynomials of the first kind and the second kind, Legendre polynomials, Laguerre polynomials, Hermite polynomials etc. belong to the polynomial family. The techniques of reducing the (differential, integral) calculus of continuous-time dynamical systems to an attractive algebra, approximate in the sense of least squares, emerged in the early 1970s, first with piecewise constant family functions. ...
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