Chebyshev Series (numpy.polynomial.chebyshev) This module provides a number of objects (mostly functions) useful for dealing with Chebyshev series, including a Chebyshev class that encapsulates the usual arithmetic operations. (General information on how this module represents and works with such polynomials is in the docstring for its “parent” sub-package, numpy.polynomial).  Classes  
Chebyshev(coef[, domain, window]) A Chebyshev series class.     Constants  
chebdomain An array object represents a multidimensional, homogeneous array of fixed-size items.  
chebzero An array object represents a multidimensional, homogeneous array of fixed-size items.  
chebone An array object represents a multidimensional, homogeneous array of fixed-size items.  
chebx An array object represents a multidimensional, homogeneous array of fixed-size items.     Arithmetic  
chebadd(c1, c2) Add one Chebyshev series to another.  
chebsub(c1, c2) Subtract one Chebyshev series from another.  
chebmulx(c) Multiply a Chebyshev series by x.  
chebmul(c1, c2) Multiply one Chebyshev series by another.  
chebdiv(c1, c2) Divide one Chebyshev series by another.  
chebpow(c, pow[, maxpower]) Raise a Chebyshev series to a power.  
chebval(x, c[, tensor]) Evaluate a Chebyshev series at points x.  
chebval2d(x, y, c) Evaluate a 2-D Chebyshev series at points (x, y).  
chebval3d(x, y, z, c) Evaluate a 3-D Chebyshev series at points (x, y, z).  
chebgrid2d(x, y, c) Evaluate a 2-D Chebyshev series on the Cartesian product of x and y.  
chebgrid3d(x, y, z, c) Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z.     Calculus  
chebder(c[, m, scl, axis]) Differentiate a Chebyshev series.  
chebint(c[, m, k, lbnd, scl, axis]) Integrate a Chebyshev series.     Misc Functions  
chebfromroots(roots) Generate a Chebyshev series with given roots.  
chebroots(c) Compute the roots of a Chebyshev series.  
chebvander(x, deg) Pseudo-Vandermonde matrix of given degree.  
chebvander2d(x, y, deg) Pseudo-Vandermonde matrix of given degrees.  
chebvander3d(x, y, z, deg) Pseudo-Vandermonde matrix of given degrees.  
chebgauss(deg) Gauss-Chebyshev quadrature.  
chebweight(x) The weight function of the Chebyshev polynomials.  
chebcompanion(c) Return the scaled companion matrix of c.  
chebfit(x, y, deg[, rcond, full, w]) Least squares fit of Chebyshev series to data.  
chebpts1(npts) Chebyshev points of the first kind.  
chebpts2(npts) Chebyshev points of the second kind.  
chebtrim(c[, tol]) Remove "small" "trailing" coefficients from a polynomial.  
chebline(off, scl) Chebyshev series whose graph is a straight line.  
cheb2poly(c) Convert a Chebyshev series to a polynomial.  
poly2cheb(pol) Convert a polynomial to a Chebyshev series.  
chebinterpolate(func, deg[, args]) Interpolate a function at the Chebyshev points of the first kind.     See also numpy.polynomial   Notes The implementations of multiplication, division, integration, and differentiation use the algebraic identities [1]:  \[\begin{split}T_n(x) = \frac{z^n + z^{-n}}{2} \\ z\frac{dx}{dz} = \frac{z - z^{-1}}{2}.\end{split}\] where  \[x = \frac{z + z^{-1}}{2}.\] These identities allow a Chebyshev series to be expressed as a finite, symmetric Laurent series. In this module, this sort of Laurent series is referred to as a “z-series.”   References  1 
A. T. Benjamin, et al., “Combinatorial Trigonometry with Chebyshev Polynomials,” Journal of Statistical Planning and Inference 14, 2008 (https://web.archive.org/web/20080221202153/https://www.math.hmc.edu/~benjamin/papers/CombTrig.pdf, pg. 4)