Chebyshev Polynomials

  For the Chebyshev polynomials of the first kind are defined by

In particular, for . A Chebyshev series in x,

is a Fourier series in . Terms , etc. can be ignored (for ) as long as is smaller than the error one can tolerate.

The truncated series

can be computed by the recursion formula ( Horner's Rule)

which is numerically stable for .

The Chebyshev series converges faster (if convergence is measured in terms of the maximum error for ) than the Taylor series for the same function,

The two series are approximately related by c_j = 2^j-1 a_j, if the sequence is rapidly decreasing. Rearrangement of a Taylor series into a Chebyshev series is called economization. The Chebyshev series is optimal in the sense that S_m (x) is approximately equal to the polynomial of degree m that minimizes the maximum of the error |S(x) - S_m(x)| for (the assumption is again that the absolute values |a_j| decrease rapidly).

If the function S(x) is known for , the coefficients in its Chebyshev series are

This follows from the orthogonality relation for the Chebyshev polynomials.

For a rapidly converging series the truncation error is approximately equal to the first neglected term, and the approximation

implies that

where for are the m+1 zeros of T_m+1(x). This follows from the orthogonality relation

for . (Note an error in [NBS52], where the term is omitted.)

These results may be useful if a polynomial interpolation of measured values of S(x) is wanted. One may choose to measure and use the above formula to determine . Then S_m(x) is the best polynomial approximation to S(x) for in the sense that the maximal error is (nearly) minimized.

Moreover, if the measurement error is the same for all S(x_l), then for any r<m, S_r(x) determined in this way is the polynomial of degree r which gives the least squares approximation to the measured values. also [Abramowitz74], [NBS52], [Press95].