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

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In summary, the conversation discusses the uniqueness of the solution to the integral equation a(n)=∫f(x)dx, where a(n) is defined as the sum of a power series g(x) and f(x) is unknown. The question is whether the solution is unique if f(x) is positive on the interval (0,∞). There is also a discussion about the starting point of the integral, with one person suggesting it should start at x=0.
  • #1
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given the sequence (power series) [tex] g(x)= \sum_{n\ge 0}a(n)(-1)^{n}x^{n} [/tex]

if i define [tex] a(n)=\int_{n=0}^{\infty}dxf(x)x^{n} [/tex] (1)

if [tex] f(x)>0 [/tex] on the whole interval [tex] (0,\infty) [/tex] , is the solution to (1) unique ?? , this means that the moment problem for a(n) would have only a solution.
 
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  • #2
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.
 
  • #3
i meant

is the solution to

[tex]
a(n)=\int_{n=0}^{\infty}dxf(x)x^{n}
[/tex]

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

[tex]
a(n)=\int_{n=0}^{\infty}dxf(x)x^{n}
[/tex]

has ONLY a solution provided f(x) is always positive, thanks.
 
  • #4
I don't think your integral should start at n=0 should it? Maybe x=0 ...
 
  • #5


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.
 

1. What is the Moment Problem?

The Moment Problem is a mathematical problem that involves finding a unique solution for a given function or power series.

2. What is a Power Series?

A Power Series is a mathematical series in which each term is a constant multiplied by a variable raised to a certain power. It is commonly used to represent functions in the form of an infinite polynomial.

3. What is the significance of (-1)^n in the given power series?

The (-1)^n term is known as the alternating sign and it ensures that the series converges. Without this term, the series may diverge or have multiple solutions.

4. How is the unique solution for the Moment Problem found?

The unique solution for the Moment Problem can be found using mathematical techniques such as the method of moments, which involves setting up a system of equations using the moments (defined as the integrals of the function or power series) and solving for the coefficients.

5. What are the applications of the Moment Problem?

The Moment Problem has various applications in mathematics, physics, and engineering. It can be used to approximate functions, solve differential equations, and analyze data from experiments or simulations.

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