Are Taylor Polynomials Optimal for Polynomial Approximation?

Taylor polynomials have long been a cornerstone in mathematics for approximating functions around specific points. A Taylor polynomial of degree n centered at x a is defined as:

[ P_n(x) f(a) f'(a)(x - a) frac{f''(a)}{2!}(x - a)^2 ldots frac{f^{(n)}(a)}{n!}(x - a)^n ]

Key Points to Consider

While Taylor polynomials are undoubtedly powerful, they are not always the most optimal polynomial approximation of a given degree. Here are several important points to consider:

Local Approximation

Taylor polynomials offer accurate approximations near the center point a. However, as we move away from a, the approximation may become less precise. This is due to the nature of polynomial approximations, which are local in their accuracy.

Error Terms

The error in a Taylor polynomial approximation can be calculated using the remainder term from Taylor's theorem. For a function f and a Taylor polynomial of degree n, the error R_n(x) is given by:

[ R_n(x) frac{f^{(n-1)}(c)}{(n-1)!}(x - a)^{n-1} ]

where c is a point between a and x. As the distance from a increases, so does the error.

Best Approximation

The best polynomial approximation over a broader interval, in terms of minimizing the maximum error (uniform approximation), is provided by Chebyshev polynomials rather than Taylor polynomials. Chebyshev polynomials are renowned for minimizing the worst-case error across the entire interval, which is especially valuable in approximation theory.

Convergence

Taylor series for infinitely differentiable and well-behaved functions can converge to the function in some interval around a. However, for functions with singularities or discontinuities, Taylor series may not converge to the function or may converge to a different value.

Conclusion

In conclusion, while Taylor polynomials are excellent for local approximations, they are not always the optimal approximation for a given degree over a broader interval. For global approximation, other methods like Chebyshev polynomials may be more effective.

It is worth noting that the concept of 'best' can vary. For computer algorithms, 'best' might mean the least number of multiplications and divisions to achieve the desired accuracy. In the 1970s, a different approach was explored by using rational functions, which often provided better approximations due to the error function difference between the exact and the approximation being more manageable and bounded.