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 ...