>>7767349 the more terms you add to the taylor sequence, the broader your approximation gets. the term of order 0 is just the value of the function at 0 for example. if you include the term of order 1, you take into account the neighborhood of 0 as well. This will be a line that approximates best f around 0. if you include the term of order 2, you start also taking into account the curvature, so your approximation gets more precise further from 0. And so on and so forth.
(all of this works well only for functions that are equal to their taylor series of course)
>>7767309 I would imagine you could do so because you can assume that a taylor series converges to your function for neigborhood around x, and apply Taylor's formula to deduce that a partial series will adequately represent the data that you have. Then it's a matter of simply finding the coefficients. But really, Taylor Series aren't used for curve fitting.
>>7767421 and also I guess this is its main difference from fourier transform (apart from the fact that you use cos/sin only now); in this case you don't "start" from a single point but you're already looking from -oo to +oo
Power series expansions only work around the points you calculate them at.
And Fourier transforms only work for continuous functions and smooth functions.
If you want a goal for power series expansion, it allows you approximate the behaviour of a complex function with a few easy ones. In engineering this is usually done to simplify the equations around the points that physics allows.
Resistance for example is not actually R= V/I
It is a sigmoidal function, but a linear function approximates it well enough for usage.
>>7768218 >And Fourier transforms only work for continuous functions and smooth functions. Fourier transform works for every L^1 function and can be extended to work with L^2 functions too. You can even further extend it for all tempered distributions.
>>7767309 >curve fitting You're thinking of polynomial interpolation. https://en.wikipedia.org/wiki/Polynomial_interpolation Also, Taylor series just require knowing the function and all its derivatives at a point, and there's different methods for different functions/situations.
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