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As far as I understand, there is no "natural" inner product on the set of monomials $x^n$, and so we can't really interpret a Taylor series as an expansion in an orthonormal basis. However, we can$^3$ interpret a Taylor series as an expansion in a non-orthogonal basis, and so we get back a little of the interpretational power.

As far as I understand, there is no "natural" inner product on the set of polynomials that makes the set of monomials $x^n$ orthonormal, and so we can't really interpret a Taylor series as an expansion in an orthonormal basis. However, we can$^3$ interpret a Taylor series as an expansion in a non-orthogonal basis, and so we get back a little of the interpretational power.

• The expansion coefficients in orthonormal basis have a direct physical interpretation. In the context of basic mechanics, for instance, the $v_x$ attached to $\hat{x}$ in a velocity vector really is the velocity in the $x$ direction, in the following sense. Imagine moving along the $x$-axis at velocity $v_x$ (of the object) and watching the object as it moves: the object will recede from you but keep pace with you along that direction. So, importantly, the components of a vector in an orthonormal basis reveal physically relevant information about the system described by that vector.

• The type of physical information that's revealed depends on the orthonormal basis that's chosen. If you choose to represent a velocity vector in the basis $\{\hat{x},\hat{y},\hat{z}\}$, then you get the velocities in these directions (in the sense explained in the previous bullet point). If instead you choose the basis where one of the basis vectors lies along $\vec{v}$, then the component along this basis vector is the speed of the particle. Different bases reveal different information and conceal different information.

• Finally, the expansion coefficients in an orthonormal basis are unique, which is what really allows us to interpret the quantities physically. (What would the $x$-component of velocity really mean if it could take on two different values?)

• The expansion coefficients tell us how much the function looks like a particular sine or cosine function. Let's consider one of the terms in the expansion, given by $$a_n\cos\left(\omega_n t\right) + b_n\sin\left(\omega_n t\right),$$ where $\omega_n=2\pi n/T$. If we re-write this as $$A_n\sin\left(\omega_n t+\phi_n\right),$$ where $A_n=\sqrt{a_n^2+b_n^2}$ and $\phi_n=\tan^{-1}(a_n/b_n)$, then we can interpret the coefficients as giving us two pieces of information. The amplitude $A_n$ tells us how much this particular function comes into the expansion, and therefore how much the original function looks like this particular sine function. In other words, $A_n$ tells us to what extent the original function oscillates with frequency $T/n$. In other words, the expansion coefficients are giving us the frequency content of the original signal, where the expansion coefficients quantify this relationship. (The phases $\phi_n$ would be relevant in, for instance, interference experiments, but for the purposes of this discussion, it's the amplitude that really matters.)

• A Fourier series (and a Fourier transform) yields the frequency (or wavelength, depending on the context) content of the function that is describing some physical quantity in some physical system. One can use other sets of orthogonal functions, too. For instance, in electrostatics, we can expand the electrostatic potential using Legendre polynomials, and these functions yield the$^1$ multipole (monopole, dipole, quadrupole, etc.) structure of the electrostatic potential. Different orthonormal bases reveal different types of physical information about the system.

• Uniqueness again allows us to do the above. If the Fourier series was actually a Fourier series$^2$, then this wouldn't really work.

_{1. All but the first non-zero term in the expansion actually depend on the origin chosen, so using "the" here is not correct, and there's a little sophistry here in the sense that I'm stressing the importance of the uniqueness of the expansion matters. The point is still a good one, though.}

_{3. In physics we can, anyway, to get an intuitive feel for what we're doing. And perhaps we can in mathematics, too, to the extent that they allow us. I don't know to what extend mathematicians think of Taylor series in this way, though.}

• The expansion coefficients in an orthonormal basis have a direct physical interpretation. In the context of basic mechanics, for instance, the $v_x$ attached to $\hat{x}$ in a velocity vector really is the velocity in the $x$ direction, in the following sense. Imagine moving along the $x$-axis at velocity $v_x$ (of the object) and watching the object as it moves: the object will recede from you but keep pace with you along that direction. So, importantly, the components of a vector in an orthonormal basis reveal physically relevant information about the system described by that vector.

• The type of physical information that's revealed depends on the orthonormal basis that's chosen. If you choose to represent a velocity vector in the basis $\{\hat{x},\hat{y},\hat{z}\}$, then you get the velocities in these directions (in the sense explained in the previous bullet point). If instead you choose the basis where one of the basis vectors lies along $\vec{v}$, then the component along this basis vector is the speed of the particle. Different bases reveal different information and conceal different information.

• Finally, the expansion coefficients in an orthonormal basis are unique, which is what really allows us to interpret the quantities physically. (What would the $x$-component of velocity really mean if it could take on two different values?)

• The expansion coefficients tell us how much the function looks like a particular sine or cosine function. Let's consider one of the terms in the expansion, given by $$a_n\cos\left(\omega_n t\right) + b_n\sin\left(\omega_n t\right),$$ where $\omega_n=2\pi n/T$. If we re-write this as $$A_n\sin\left(\omega_n t+\phi_n\right),$$ where $A_n=\sqrt{a_n^2+b_n^2}$ and $\phi_n=\tan^{-1}(a_n/b_n)$, then we can interpret the coefficients as giving us two pieces of information. The amplitude $A_n$ tells us how much this particular function comes into the expansion, and therefore how much the original function looks like this particular sine function. In other words, $A_n$ tells us to what extent the original function oscillates with frequency $n/T$. In other words, the expansion coefficients are giving us the frequency content of the original signal, where the expansion coefficients quantify this relationship. (The phases $\phi_n$ would be relevant in, for instance, interference experiments, but for the purposes of this discussion, it's the amplitude that really matters.)

• A Fourier series (and a Fourier transform) yields the frequency (or wavelength, depending on the context) content of the function that is describing some physical quantity in some physical system. One can use other sets of orthogonal functions, too. For instance, in electrostatics, we can expand the electrostatic potential using Legendre polynomials, and these functions yield the$^1$ multipole (monopole, dipole, quadrupole, etc.) structure of the electrostatic potential. Different orthonormal bases reveal different types of physical information about the system.

• Uniqueness again allows us to do the above. If the Fourier series was actually a Fourier series$^2$, then this wouldn't really work.

_{1. All but the first non-zero term in the expansion actually depend on the origin chosen, so using "the" here is not correct, and there's a little sophistry here in the sense that I'm stressing the importance of the uniqueness of the expansion. The point is still a good one, though.}

_{3. In physics we can, anyway, to get an intuitive feel for what we're doing. And perhaps we can in mathematics, too, to the extent that they allow us. I don't know to what extent mathematicians think of Taylor series in this way, though.}

Numerical approximation

_{3. In physics we can, anyway, to get an intuitive feel for what we're doing. And perhaps we can in mathematics, too, to the extent that they allow us. I don't know to what extend mathematicians think of Taylor series in this way, though.}

This is not the type of information that is revealed via a Fourier series$^4$, and hence different bases reveal different information.

Numerical approximation

_{3. In physics we can, anyway, to get an intuitive feel for what we're doing. And perhaps we can in mathematics, too, to the extent that they allow us. I don't know to what extend mathematicians think of Taylor series in this way, though.}

_{4. Except that, as mentioned in another answer, there is a relationship between Fourier series and Taylor series that is revealed in the context of complex analysis. So there's another bit of sophistry in the name of getting my point across.}