Re: Probably haven't seen this one, but...

On May 4, 8:18 am, junoexpress <MTBrenne...@xxxxxxxxx> wrote:
On May 3, 1:12 pm, Robert Israel



<isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
junoexpress <MTBrenne...@xxxxxxxxx> writes:
Hi,

I'm doing some work with a function that is the ratios of (normalized)
sinc functions, having the form:
f:x-> Sinc(Mx)/Sinc(x) for M a natural number and |x| < 1

I'm having to work out some properties of this function, which are not
that bad, but in the process, I keep wondering if this ratio has been
analyzed before (in other words, I hate to present a lot of detailed
derivations only to have someone else say, "Oh yeah, that's just the
Gluckenheimer function" and if it has been looked at before, maybe I
could get some deeper insight into the solution also.)

So I come to the gurus. Is this a function which anyone has seen
analyzed before? To my knowledge, it is not in Abrahamowitz and
Stegan, and the closest I can come to pinning it on anything known is
to say that it's the ratio of 2 spherical Bessel functions (which
doesn't seem like a function that's probably been analyzed).

Since sinc(x) = sin(x)/x (for x <> 0), your function is just
f(x) = sin(Mx)/(M sin(x)). This can also be written as
U_{M-1}(cos(x))/M where U_k is the k'th Chebyshev polynomial of
the second kind.
--
Robert Israel isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

This is a very nice observation. There is one problem I am having in
applying them to my problem. I need to know (or have a decent bounds)
on the absolute value of the first extreme value of U_m(x) after x=0.
So in the case of U_2(x), you could solve for the critical points,
take the critical point closest to zero (which is not equal to zero)
x*, and then compute |U2(x*)|. But of course for M>4, this strategy
will not work, and there is no obvious way I can see to get a (close)
upper bound to this first extreme value. (Could always do this
computationally, but if there was a proven bound, that of course would
be better).

|sin(Mx)/sin(x)| <= |1/sin(x)|

The peaks will be quite close to points where sin(Mx) = (+/-) 1,
especially
when M is large. It should be possible to get good explicit bounds.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada


.

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