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I am aware that this is a very simple question, but your opinion would help me a lot. In a wider context, in a text, I have to distinguish between polynomials written in two different forms. For example, linear function can be written as:

(a) $$f(x) = \frac{x-x_{i+1}}{x_i-x_{i+1}} f_i +\frac{x-x_{i}}{x_{i+1}-x_{i}} f_{i+1}$$

or as

(b) $$f(x) = f_i + \frac{f_{i+1} - f_i}{x_{i+1} - x_i}(x - x_i)$$

i.e. $f(x) = a_0 + a_1 (x - x_0)$ or $f(x) = a_0 + a_1 (x - x_0) + a_2 (x - x_0)^2$ for parabola and so on. The former, (a), I'm calling the Lagrange form. How would you call the latter? Is it fair to call it "Taylor's form"?

CFDIAC
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    This is not an elementary question. There is great debate about how to define and classify polynomials. See https://math.stackexchange.com/questions/2185587/what-actually-is-a-polynomial

    In General, I think its best to distinguish based on behavior. What information do the forms accentuate differently?

     – Travis Mar 02 '23 at 16:39
  • I'll take a look of the other discussion. Perhaps one way to distinguish between these forms is to call them "symmetric" (a) and "asymmetric" (b). Or, a bit jokingly, poly-binomial (a) and mono-binimial (b), as the former includes binomials $x-x_i$, $x-x_{i+1}$, ... and the latter only $x-x_i$. But that sounds ugly. In my text I'm referring to these forms many times, so it would be handy to give them nicknames. While I found Lagrange undisputable for (a), it is true that (b) must be very old and difficult to attribute to a single person. – CFDIAC Mar 03 '23 at 12:38

1 Answers1

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I think ascribing people's names to these forms is actually a negative, not a positive. Mathematics is already full of objects and facts that are named after Europeans who were not the first ones to discover and examine them; this is not a trend that needs exacerbating.

If you must give names to them at all, I recommend giving names that reflects their function. So the first one could be the "interpolation form" of the polynomial, for example (as opposed to the "slope-intercept form", which is an existing example of the paradigm I'm suggestion); and the second one might be the "centered-at-$x_0$" form.

Greg Martin
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  • I see your point, but Lagrange's contribution to the theory of polynomials is not, I guess, controversial? Same goes for Taylor's series.

    I agree that wrongfully assigned names are not good, but I also don't like too descriptive names, especially if they are used in the text. Or if they are not precise enough. For example, I cannot call the form (a) "interpolation form form (b) can be used for interpolation, too. Making distinction between the two is actually essential for clarifying difference between Hermitean and Lagrangian interpolation.

     – CFDIAC Mar 03 '23 at 12:30
  • @CFDIAC - what makes the first an "interpolation form" is that the sum of the coefficients of the $f$ values is always $1$. One coefficient moves from $0$ to $1$ as you move from one support point to the next, while the other moves from $1$ to $0$. This is a highly convenient property for interpolating. The other (which is the point-slope form of a line, not slope-intercept) does not offer anything similar. – Paul Sinclair Mar 03 '23 at 17:52
  • Yes, that's the way the Lagrange interpolation works. The Hermite linear interpolation requires one function value ($f_i$) and the slope ($(f_{i+1} - f_i)/(x_{i+1} - x_i)$). In that sense both expressions are "interpolation forms". I would be so nitpicking if the topic I am writing about wasn't related exactly to that - different interpolation/approximation techniques. – CFDIAC Mar 03 '23 at 22:51