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From: Golabi Doon on 3 Jul 2010 11:11
Thank you a lot Stephen and Rob for your valuable clarification. I am
convinced that you are absolutely right.

Regards

Golabi

On Jul 3, 10:06 am, Rob Johnson <r...(a)trash.whim.org> wrote:
> In article <279cf805-79d9-4ebb-9f0a-b23b7d062...(a)z15g2000prn.googlegroups..com>,
>
>
>
>
>
> Golabi Doon <golabid...(a)gmail.com> wrote:
> >Consider a function f_n(x) defined on [-pi,pi] that is represented
> >with n frequency components:
> >f_n(x)=sum_{k=1}^n a_k cos(kx) + b_k sin(kx)
>
> >The set {a_k,b_k} for k=1,...n is given.
>
> >The function f_n may have multiple peaks (I think at most 2n). By a
> >peak I mean a point that is a local minimum or local maximum.
>
> >Now, does the fact that the basis functions have bounded frequency
> >(due to finite n) imply a bound on how close two peaks of f_n can be?
>
> >In other words, can we define a quantity d, that is a function of n,
> >such that we can say "For any two peaks of f_n(x), whose location is
> >denoted by x_1 and x_2, we always have |x_1-x_2| >= d(n)"?
>
> >If so, what that d(n) is in terms of n?
>
> Consider the function for n = 2
>
>     f(x) = sin(2x) - 2(1-t) sin(x)
>
> Take the derivative of f
>
>     f'(x) = 2 cos(2x) - 2(1-t) cos(x)
>
>           = 4 cos^2(x) - 2(1-t) cos(x) - 2
>
> Solving this equation for f'(x) = 0, we get
>
>              1-t + 3 sqrt(1 - 2t/9 + t^2/9)
>     cos(x) = ------------------------------
>                            4
>
>                  t + 3(1-sqrt(1 - 2t/9 + t^2/9))
>            = 1 - -------------------------------
>                                 4
>
>            ~ 1 - t/3
>
> Therefore, f'(x) = 0 when x^2 ~ 2t/3.  By making t small, we can
> bring the two local extrema of f(x) near x = 0 as close as we want.
> Thus, there is no lower bound, based on n, for the distance between
> two local extrema of such a function.
>
> Rob Johnson <r...(a)trash.whim.org>
> take out the trash before replying
> to view any ASCII art, display article in a monospaced font- Hide quoted text -
>
> - Show quoted text -

From: W^3 on 3 Jul 2010 16:41
Another approach: Set p(t) = |e^(it)-1|^2|e^(it)-e^(it_0)|^2. Then p
is a trigonometric polynomial with p(0) = 0 = p(t_0); clearly 0 and
t_0 give absolute minima, and they are as close together as we want.
(This p may have a constant term; just subtract it off if desired.)
From: Rob Johnson on 3 Jul 2010 20:20
In article <279cf805-79d9-4ebb-9f0a-b23b7d0628c2(a)z15g2000prn.googlegroups.com>,
Golabi Doon <golabidoon(a)gmail.com> wrote:
>Consider a function f_n(x) defined on [-pi,pi] that is represented
>with n frequency components:
>f_n(x)=sum_{k=1}^n a_k cos(kx) + b_k sin(kx)
>
>The set {a_k,b_k} for k=1,...n is given.
>
>The function f_n may have multiple peaks (I think at most 2n). By a
>peak I mean a point that is a local minimum or local maximum.
>
>Now, does the fact that the basis functions have bounded frequency
>(due to finite n) imply a bound on how close two peaks of f_n can be?
>
>In other words, can we define a quantity d, that is a function of n,
>such that we can say "For any two peaks of f_n(x), whose location is
>denoted by x_1 and x_2, we always have |x_1-x_2| >= d(n)"?
>
>If so, what that d(n) is in terms of n?

The preceding was quoted simply for context.

In article <aderamey.addw-B1FB3C.13412603072010(a)News.Individual.NET>,
W^3 <aderamey.addw(a)comcast.net> wrote:
>Another approach: Set p(t) = |e^(it)-1|^2|e^(it)-e^(it_0)|^2. Then p
>is a trigonometric polynomial with p(0) = 0 = p(t_0); clearly 0 and
>t_0 give absolute minima, and they are as close together as we want.
>(This p may have a constant term; just subtract it off if desired.)

Using the Law of Cosines, we get

p(t) = 4 (1 - cos(t)) (1 - cos(t-t_0))

With a bit of manipulation and getting rid of the k = 0 term (the
constant term) we get, in the form requested,

2 sin(t_0) sin(2t) + 2 cos(t_0) cos(2t)

- 4 sin(t_0) sin(t) - 4 (1 + cos(t_0)) cos(t)

This actually has 3 extrema within a distance of t_0, and still only
needs n = 2. Cool.

Rob Johnson <rob(a)trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font
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