Moment Problem: Unique Solution for Power Series g(x)=∑a(n)(-1)^nx^n

[zetafunction]

given the sequence (power series) $$g(x)= \sum_{n\ge 0}a(n)(-1)^{n}x^{n}$$

if i define $$a(n)=\int_{n=0}^{\infty}dxf(x)x^{n}$$ (1)

if $$f(x)>0$$ on the whole interval $$(0,\infty)$$ , is the solution to (1) unique ?? , this means that the moment problem for a(n) would have only a solution.

 

[HallsofIvy]

How can we tell you if "the solution to (1)" is unique when there is nothing labeled (1)?

And what does an integral of f(x) have to do with a sum of g(x)?

Frankly, nothing here makes any sense.

 

[zetafunction]

i meant

is the solution to

$$a(n)=\int_{n=0}^{\infty}dxf(x)x^{n}$$

where a(n) is given but f(x) is unknown UNIQUE is f(x) is positive on the whole interval (0,oo) ? , i mean if the integral equation

$$a(n)=\int_{n=0}^{\infty}dxf(x)x^{n}$$

has ONLY a solution provided f(x) is always positive, thanks.

 

[g_edgar]

I don't think your integral should start at n=0 should it? Maybe x=0 ...

 

[blue_raver22]

I would first clarify that the moment problem is a mathematical problem that involves finding a function from a sequence of numbers. In this case, the function is g(x) and the sequence is a(n). The question is asking if the solution to the moment problem, given by equation (1), is unique when the function f(x) is positive on the interval (0,∞).

To answer this question, we need to understand the concept of uniqueness in mathematics. A solution is considered unique if there is only one possible solution that satisfies the given conditions. In this case, if there is only one possible function g(x) that can be determined from the sequence a(n) using equation (1), then the solution is considered unique.

In order to determine if the solution is unique, we need to consider the properties of the function f(x) and the interval (0,∞). If f(x) is positive on the entire interval, then the integral in equation (1) will always be positive. This means that the sequence a(n) will always be positive, and therefore, the power series g(x) will always be positive. Since the power series is always positive, there can only be one possible solution for g(x) from the given sequence a(n). Therefore, the solution to the moment problem is unique.

In summary, if f(x) is positive on the interval (0,∞), then the solution to the moment problem, given by equation (1), is unique. This is because the integral in equation (1) will always be positive, leading to a unique sequence a(n) and a unique power series g(x). I would also consider exploring the conditions under which the solution may not be unique, such as when f(x) is not positive on the entire interval.