Personally, I think it is likely to make a difference whether
Riemann or Lebesgue integration is used, but that the difference
will only be noticeable provided that the designers rigorously
adhere to the requirements that that particular integration theory
be used for all the integrals. As soon as they revert to numerical
approximations, instead of using calculus or analysis, they have
muddied the waters so that no direct comparison between one integration
theory and another can be made. Another way to test this might
be to allow numerical approximations of all but one integral
used in the design. For that integral, design the aircraft once
using Riemann integration, and design it again using Lebesgue
integration, but make sure that each of the other integrals approximated
using numerical techniques uses the same techniques and has the
same approximate value in the two different designs. Then try
each aircraft out. An obvious problem with this "experimental
design" is that scientists will be unwilling to part with
the cost of an aircraft - let alone two aircraft - to test the
usefulness of Lebesgue integration in this way. Another obvious
problem is more technical: When the one integral being tested
for its effect on the design of the aircraft is computed, using
Riemann integration, the value so obtained may be used as an
input to some other portion of the design. This would skew the
results of other Unfortunately, there are many complications involved in this
issue: On the one hand, graduates who have studied Lebesgue integration
are more likely to have taken some graduate mathematics or science
or engineering courses, whereas many designers of various things
-- such as aircraft -- Needless to say, I am not enamoured by such anti-mathematical quotations as this one attributed to Hamming.
<< Personally, I think it is likely to make a difference whether Riemann or Lebesgue integration is used, but that the difference will only be noticeable provided that the designers rigorously adhere to the requirements that that particular integration theory be used for all the integrals. As soon as they revert to numerical approximations, instead of using calculus or analysis, they have muddied the waters so that no direct comparison between one integration theory and another can be made. >> This is with reference to Hamming's quip about this difference not being significant for engineering design. Functions that are Riemann integrable are automatically Lebesgue integrable and the values of the integrals will be the same. There are two ways in which a function can be Lebesgue integrable but not Riemann integrable. Any bounded measurable function is L-integrable, but not all of them are R-integrable. Some unbounded measurable functions are also L-integrable, but not Riemann integrable, since all R-integrable functions are bounded. An example: the function 1/sqrt(x) has the L-integral 2/3 on the interval [0,1]. Being unbounded in that interval, it has no R-integral there. Of course, this function does have an "improper integral"
(sometimes called a Riemann-Cauchy integral) in that interval
with the very same value. Only functions having absolutely convergent
improper integrals have Lebesgue integrals. The classical example
is sinx/x on the interval [0,\infnty] which has the value \pi/2
but no Lebesgue integral. This fact is sometimes confusingly
stated as a case where something is R-integrable but not L-integrable.
But one can perfectly well define "improper" Hamming of course hadn't meant to be taken so literally. His aphorism was intended to say that the fine points of mathematical analysis are not relevant to engineering considerations. And, he was perfectly right. In the long-ago days when I had occasion to be on committees administering an oral qualifying exam for the doctorate, I would often ask the hapless student why analysts prefer the L-integral. This was a trap question: students who fell for the trap would tell me that the function on [0,1] that is 1 on the irrationals and 0 on the rationals is L-integrable, but not R-integrable. The right answer is that the L-integral has useful convergence properties not enjoyed by the R-integral. Now whatever did Mat Insall have in mind? sin x/x on [0,infty]? Surely not in any engineering analysis.
Examples of articles I dug up (but have not read) by engineers,
in which the sinc function and its relatives are used are below
[in the bibliography]. I shall keep
an eye open for other places where such functions are used. In
the meantime, remember that the original question had to do with
the
"Enhanced Period-Peak Analysis of the Electroencephalogram Using a Fast Sinc Function,"with M. Ferdjallah, Medical and Biological Engineering and Computing, (in press).
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