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I'm looking into integrals of the form:

$$\int_a^b {f(x)g(x)dx}$$ Where $f(x)$ is unknown, but it's integral is: $$\int_a^b {f(x)dx}=F$$ It's been suggested to me that one could approximate this integral by: $$\int_a^b {f(x)g(x)dx}\sim F \cdot\overline{g(x)}=\frac{F}{b-a}\int_a^b {g(x)dx}$$ This of course being just a "good guess" form of the integral mean value theorem.

My question is the following:

How would one go about calculating the error bounds on this approximation? are there any conditions on $f(x)$ that would tighten the said error bounds?

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For simplicity's sake, let's just say that $a=0,b=1$. Then you want to estimate $$ \int_0^1 f(x) g(x) dx - \left(\int_0^1 f(x) dx \right) \left(\int_0^1 g(x) dx\right) $$ from above and below. Furthermore, let me consider two cases:

  • If $g(x)$ has mean value zero, i.e., $\int_0^1 g(x) dx = 0$, then you know a priori that your approximation isn't a good one. Let's not consider this case any more then.

  • If $g(x)$ has a mean value different than zero. Let's only consider this case.

Since in this second case it doesn't make a difference in the problem, let us assume for simplicity that $g(x)$ has been scaled in such a way that $\int_0^1 g(x) dx = 1$. Then you are looking for estimates of the term $$ \int_0^1 f(x) g(x) dx - \int_0^1 f(x) dx $$ from above and below where $g(x)$ has mean value one.

Now consider a sequence of functions $f_n(x)=(n+1)x^n$. Then, assuming that $g(x)$ is bounded, $$ \int_0^1 f_n(x) g(x) dx \rightarrow 0, \qquad \int_0^1 f_n(x) dx =1 $$ so your relative error can become arbitrarily large, and by tweaking the sign of $f_n,g$ you can make this an error that can be positive or negative.

Of course, all this shows is that you need to say in which class (or function space) you are looking for your function $f(x)$. Each function $f_n$ above is $C^\infty$, and its limit is in $L_1$, so you can't expect an error bound to hold in $L_1$. But the limit function is not in in $L_p,p>1$ and so it may be that you can find, for example, an estimate in $L_\infty$. In other words, more information is necessary to answer your question.

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  • $\begingroup$ What about calculating the variance using $var = \int{(f(x)-F)^2}dx$? Shouldn't this give a reasonable estimate for the squared error? $\endgroup$
    – Nathaniel Bubis
    Commented Jul 9, 2012 at 12:11
  • $\begingroup$ No, there you have the same problem. Take the function $f_n(x)=n^{2/3} x^n$, for example. On $[0,1]$ you have that the mean value of $f_n$ goes to zero but $\int f_n(x)^2 dx \rightarrow \infty$. You will not go anywhere if you don't try to restrict the class of functions within which you try to find $f(x)$. $\endgroup$
    – Wolfgang Bangerth
    Commented Jul 9, 2012 at 13:57
  • $\begingroup$ The functions $g(x), f(x)$ are "physical" in the sense that although mathematically I understand that you can always find an extreme case, It's intuitively obvious to me that this is a reasonable approximation. I'm just trying to understand what restrictions I would need to apply to the functions in order for the error to be calculable, and then see if I can reasonably assume the said restrictions. Thanks! $\endgroup$
    – Nathaniel Bubis
    Commented Jul 9, 2012 at 15:33
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    $\begingroup$ But it's a good question how you translate "physical" into mathematical terms. For example, if you have an electric field, then it's definitely in $L_2$. A "physical" velocity field would also be in $L_\infty \cap C^0$. Probability distributions must be in $L_1$. What you need to do is determine what the function space is that corresponds to the "physical" functions in your particular case. That involves some mathematical thinking and I understand that you'd like to avoid that, but there's no way around it if you want to answer the question you pose in this thread. $\endgroup$
    – Wolfgang Bangerth
    Commented Jul 9, 2012 at 15:57

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