Taylor Series and Fourier Series [closed]

Question

Taylor series expansion of function, f, is a vector in the vector space with basis: {(x-a)^0, (x-a)^1, (x-a)^3, ..., (x-a)^n, ...}. This vector space has a countably infinite dimension. When f is expressed as linear combination of the basis vector the scalar multiple for the n-th basis vector is Diff_n{f}(a)/n!

Fourier series expansion of function, f, is a vector in the vector space with basis: {sin(1x), cos(1x), sin(2x), cos(2x), ..., sin(nx), cos(nx), ...}. This vector space has a countably infinite dimension. When f is expressed as linear combination of the basis vector the scalar multiple for the n-th basis vectors are Int{f.sin(nx)} and Int{f.cos(nx)}.

Questions:

1. The vector space for the Fourier series has an inner product, Int{f.g}, and it's this inner product that provides the above expressions like Int{f.sin(nx)} and Int{f.cos(nx)}. Is there a similar inner product based derivation of the scalar multiples for the vector space of spanned by the polynomial basis in Taylor series?

2. What is the relationship, if any, between the vector space produced by Taylor Series and that of Fourier Series? E.g. is one a subspace of the other?

3. When Fourier series is taught, why isn't Taylor Series re-explained in the vector space framework used for Fourier series? And would this approach not lead the discussion of the implication of the choice of basis (and perhaps the choice of inner product) for function spaces?

4. Just as Fourier series get generalized to Fourier Transform (the summation of the series becomes an integral), is there something equivalent to Taylor series?

5. Are there any recommended resources (books, courses, etc.) available which can help clarify my thinking regarding these issues?

Answer 1

In your question you only discuss the formal analogy, disregarding the questions of convergence, and what exactly a "function" is. This is a reasonable setting indeed, if one restricts oneself to finite sums. Then there is an exact correspondence, if we extend the set of Taylor series to Laurent series. Finite Laurent series $\sum a_n z^n$ are in one-to one correspondence with finite trigonometric sums $\sum a_n\exp(int)$. (These are essentially the same as the series in sines and cosines). The subspace of polynomials consists of those sums for which $n\geq 0$. One can give a condition on the series of sines and cosines to correspond to a Laurent series with $n\geq 0$. If one wants to pass to infinite sums, one has to use some topology. The simplest case is when the topology is defined by the standard scalar product. Completion of the space of polynomials in this topology gives the space $H_2$ which can be thought of as a space of trigonometric (Fourier) series, and as a space of Power series. And for the coefficients of the expansion you have two formulas: one involving integration, another differentiation. These formulas give the same result in view of Cauchy formula.

On your other questions. Fourier series are generalized to Fourier transform. Taylor series are similarly generalized to Laplace transform. (With an intermediate step of Dirichlet series).

For recommended reading... there are a lot of books, the appropriate one depends on your prerequisites. Hoffman's book Banach Spaces of Analytic Functions begins with a very clear explanation of relations between Taylor and Fourier series.