I am quite confused with some ideas regarding the Real analytic functions.

Just to introduce my questions:

A function $f$ is real analytic on an open set $D$ of the real line if for any $x_0\in D$ there exist an interval $I=(x_0-\epsilon,x_0+\epsilon)$ such that locally in $I,$ $f(x)=\sum_{k=0}^{\infty}a_k(x-x_0)^k.$

Call $A=A(D)$ the set of real analytic functions on $D.$

Call $R^{*}$ the set of finite sequences of distinct real numbers $(x_1,x_2,\ldots,x_n),$ $x_i\leq x_{i+1},$ and call $\mathbb{R},\mathbb{N}$ the set of real and natural numbers, respectively.

1)This implies that there exists a function between $A$ and $R^{*}\times \mathbb{R}^{\mathbb{N}}$ that associated to $f\in A$ the element $\left((x_1,x_2,\ldots,x_n),(a_0^1,a_0^2,\ldots,a_1^n,a_1^1,a_1^2,\ldots,a_1^n,\ldots)\right)$ in the natural way in which locally around $x_i$ the map has series with coefficients $(a_0^i,a_1^i,\ldots),$ $n$ is minimal such that $x_1$ is in the boundary of $D$ and the pairwise separation between $x_{i}$ and $x_{i+1}$ is always maximal and constructed inductively from $i=1.$ (Of course it is not surjective, but it should be a bijection when we consider $BV$ instead of $A$ like in 6) ).

2)Analytic properties of elements in $A$ depends on $R^{*}\times \mathbb{R}^{\mathbb{N}}.$ For example, define $R^{2}$ the set sequences of real numbers $(x_1\leq x_2), x_1\neq x_2.$ I am interested in properties of $R^{2}\times \mathbb{R}^{\mathbb{N}},$ like, it seems that we can integrate very accurately maps on $R^{2}\times \mathbb{R}^{\mathbb{N}},$ because we just need to find the 2 points for our Taylor expansions, and we can truncate the integration of each polynomials to the neighborhood of each point to approximate the integral. Another example is the case in which $R^{1}$ the set of real numbers $(x_1)$ and then to integrate a map in $R^{1}\times \mathbb{R}^{\mathbb{N}},$ we just need to approximate by truncate the polynomial.

3)It seems that we can prove some degree of accuracy when integrating maps on $R^{n}\times \mathbb{R}^{\mathbb{N}}.$ It seems also that there is a huge difference when passing from $n<\infty$ to the case $R^{*}.$

4)I have not seen anyone trying to study $R^{n}\times \mathbb{R}^{\mathbb{N}}$ instead of $A.$ I have not seen anyone trying to study the convergence of numerical algorithm to integrate by looking at $R^{n}\times \mathbb{R}^{\mathbb{N}}.$ I have not seen anyone interested in the following algorithm: Given $f\in A$ approximate $(x_1,x_2,\ldots,x_n).$ This is a surjection, but it is interesting, because it is getting us a lot of information about $f,$ thereafter we only need to care some accuracy on finitely many derivatives on each point and we could prove some bounds for the error when computing properties of $f.$

5)Could you give me an example or way to construct maps in $A$ for which $(x_1,x_2,\ldots,x_n)$ has a very big $n?$

6)I guess that $f\in BV$ (Bounded variation) means that when we try to do the same for $f$ we get finite sequence $(x_0,x_1,\ldots)$ for which in a similar way we can associate f to an unique element in $R^{*}\times \mathbb{R}^{\mathbb{N}}$ for which however we have a formal and not absolute convergent series in each neighborhood. So in that way $BV$ are formal $A$ maps.

What I want to ask is: Am I correct in what I am thinking? and If someone knows any paper I could find where someone had elaborate some similar sort of ideas.

I have been unable to find anything, but for me it is a very natural approach to integrate real analytic functions. Maybe I am completely lost, in that case, I apologize because of having taken your time.

I will give you an example: Suppose we want to compute numerically $\int_{0}^{1} 5\pi/2(e^{\pi}-2)^{-1}e^{\pi x} \cos(\pi x/2)dx.$

My ideas comes from a paper in Ergodic theory, that you can find on the web with the name: COMPUTING ENTROPY RATES FOR HIDDEN MARKOV PROCESES.

In this paper they approximate this integral:

N = 1 0.18575506891852380346423780644

N = 2 −0.841124284383205603881801616792 (of course +)

N = 3 0.40608775333324283787989060678

N = 4 1.09233276774560006235284301088

N = 5 0.99681510352795871656533973381

N = 6 1.00004673478255155995818417282

N = 7 0.99999977398633338017700990934

N = 8 0.99999999982780643498244012823

N = 9 1.00000000000326837809455213367

N = 10 0.99999999999999360196233983786

N = 11 1.00000000000000000436159826786

N = 12 0.9999999999999999999989026376489

N = 13 1.0000000000000000000000000742107229

N = 14 1.00000000000000000000000000000513617960

N = 15 0.9999999999999999999999999999999993061845

N = 16 0.9999999999999999999999999999999993061845283932

N = 17 0.9999999999999999999999999999999999999999997039178592326

Now, using the ideas before, what we can do is to consider a sequence of 2^N equidistributed points for N=0,1,..,5, a fixed maximum number T of coefficients to carry on the Taylor expansion, for example T=10.

We obtain

N=0 0.9999997

N=1 0.99999999992

N=2 0.99999999999998

N=3 0.999999999999999995

N=4 0.999999999999999999998

N=5 0.9999999999999999999999997

When T=20:

N=0 1.00000000000000000009

N=1 0.999999999999999999999998

N=2 0.9999999999999999999999999999995

N=3 0.9999999999999999999999999999999999998

N=4 0.99999999999999999999999999999999999999999997

N=5 0.999999999999999999999999999999999999999999999999993

When T=30:

N=0 1.00000000000000000000000000001

N=1 1.000000000000000000000000000000000000004

N=2 1.0000000000000000000000000000000000000000000000009

N=3 1.0000000000000000000000000000000000000000000000000000000002

N=4 1.00000000000000000000000000000000000000000000000000000000000000000005

N=5 1.00000000000000000000000000000000000000000000000000000000000000000000000000001

When T=40:

N=0 0.99999999999999999999999999999999999999999991

N=1 1.00000000000000000000000000000000000000000000000000000001

N=2 1.000000000000000000000000000000000000000000000000000000000000000000006

N=3 1.000000000000000000000000000000000000000000000000000000000000000000000000000000001

N=4 1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004

N=5 1.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000009

I will stop here, however, it is obvious to come with thousand of conjectures at this point.

My question here is, where someone has done this before? I would like to say that this is the most efficient possible way to integrate numerically real analytic maps on the interval $[0,1].$

My deepest question (maybe it is absurd) says this: Is there an optimal* way to compute numerically the integral of a real analytic map on the interval?

Optimal in the sense that in finite time it gives a better approximation of the real integral than any other algorithm.