What does "*the best approximation $A_{a}(x)$ of a function $f(x)$*" mean?

Question

I have a general question. What does "the best approximation $A_{a}(x)$ of a function $f(x)$" mean? $A_{a}(x)$ stands for the approximation of $f$ at $a$ and then you get a function of $x$ which you evaluate at $x$. Does it mean that the difference between $A(a+\Delta x)$ and $f(x+\Delta x)$ for a very small $\Delta x$ is least? Or does it mean that the average error between $A_{a}(x)$ and $f(x)$ for $-\infty <x<\infty$ and $-\infty <a<\infty$ is least?

Answer 1

Some more context might be helpful. There are lots of different kinds of approximation, and usually one specifies more details that determine in what sense this is the best approximation. For example, it could be that the Taylor series of $A_a(x)$ and $f(x)$ about $x=a$ agree to as many terms as possible, or it could be that some particular norm of $A_a - f$ is minimized, over some specified class of functions.

Answer 2

First, the phrase "the best approximation" doesn't have a whole lot of meaning, in and of itself. The "best approximation" to a function is the function itself. No other approximation comes close! So a more specific question is usually asked, often implicity: "What is the best polynomial approximation to this function?" "What is the best power law approximation to this function?" "What is the best trigonometric approximation to this function?" etc.

Often, the function is defined -- or its values have importance -- over some finite range, and the point is to find a function of a given type (polynomial, power law, etc.) that has the best fit over that range. This is usually but not always determined by "least squares" analysis, which I am sure you can look up on innumerable web sites.

Often, however, for instance when we are learning about Taylor series, the "best fit" is taken to be the polynomial that comes closest to the function over some very small $\delta x$.

Moral of the story: The definition of "best fit" is the definition used by the person speaking. Be sure and ask lots of questions and clarify.