Sum and integral

In summary, the conversation discusses the properties and definitions of the Stieljes integral and the Riemann integral. It also poses a question about finding a function w that satisfies a certain equation. The answer is that such a function exists if w is a distribution.
  • #1
eljose
492
0
let be the function w(x) that only takes discrete values in the sense that is only defined for x=n being n an integer..then my question is if the integral:

[tex]\int_{-\infty}^{\infty}dxw(x)f(x)[/tex]

would be equal to the sum of the serie:

[tex]\sum_{n=-\infty}^{\infty}w(n)f(n) [/tex]


if the sum and integral would be equal imply that the function

[tex]w(x)=g(x)\sum_{-\infty}^{\infty}\delta(x-n) [/tex]



thanks...
 
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  • #2
Yes, if you mean the Stieljes integral.

Where the Riemann integral [itex]\int_a^b f(x)dx[/itex] is defined by partitioning the interval from a to b into many small intervals, {xi}, choosing x* in each interval, forming the sum [itex]\Sigma f(x*)(x_{n+1}- x{n})[/itex] and taking the limit as the interval is partioned into more and more intervals, the Stieljes (or Riemann-Stieljes) integral, [itex]\int_a^b f(x)d\alpha(x)[/itex] does the same thing but uses the sum [itex]\Sigma f(x*)(\alpha(x_{n+1})- \alpha(x{n}))[/itex] where [itex]\alpha(x)[/itex] can be any increasing function. If [itex]\alpha(x)[/itex] is differentiable, that gives simply the Riemann integral [itex]\int_a^bf(x)\frac{d\alpha}{dx}dx[/itex] but if [itex]\alpha[/itex] is a step function, say the greatest integer function, then it gives the sum [itex]\Sigma_{n=a}^{b} f(n)[/itex].
 
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  • #3
It sounds like your initial sentence is saying:

Let w be a function whose domain is the integers. I.E. w(x) is defined only when x is an integer.

In that case, the integral

[tex]\int_{-\infty}^{+\infty} w(x) f(x) \, dx[/tex]

is not even defined.

---------------------------------------------------

Let me pose another question, the one I think you meant to ask:

If we're given a function g that is defined on the integers (and not always zero), can we find a function w such that:

[tex]\int_{-\infty}^{+\infty} w(x) f(x) \, dx
=
\sum_{n = -\infty}^{+\infty} g(n) f(n)
[/tex]

is true for all functions f? The answer is no.

However, if we let w be a distribution (or generalized function), then we can find such a w, and it can be given by the sum

[tex]
w(x) = \sum_{n=-\infty}^{+\infty} g(n) \delta(x - n)
[/tex]

If g is also defined at every real number, the above expression is indeed the same as

[tex]
w(x) = g(x) \sum_{n=-\infty}^{+\infty} \delta(x - n)
[/tex]
 
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What is the difference between a sum and an integral?

A sum is the result of adding together a finite number of terms, while an integral is a mathematical concept that represents the area under a curve. In other words, a sum is a discrete operation, while an integral is a continuous operation.

What is the purpose of using a sum or integral in science?

Sums and integrals are used in science to model and analyze various phenomena in the natural world. They are particularly useful in physics, engineering, and other fields that involve measuring and quantifying continuous quantities.

How do you solve a sum or integral?

To solve a sum, you simply add together all the terms. To solve an integral, you can use various techniques such as substitution, integration by parts, or using a table of integrals. The method you use will depend on the complexity of the integral and the techniques you are comfortable with.

What is the difference between a definite and indefinite sum or integral?

A definite sum or integral has specific limits or boundaries, while an indefinite sum or integral does not. In other words, a definite sum or integral will give you a specific numerical value, while an indefinite sum or integral will give you a function that represents the sum or integral.

How are sums and integrals related to each other?

Sums and integrals are closely related to each other. In fact, an integral can be thought of as a continuous version of a sum. The sum of a function can be approximated by taking smaller and smaller intervals and adding up the values within those intervals, which is essentially what an integral does.

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