Re: Properties of sum of function samples related to its differentiability + integrability

On Nov 6, 5:21 pm, Robert Israel
<isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
eau.de-...@xxxxxxxx writes:
Dear readers,

I have to bound a sum of a function (f) samples in some expression. I
wonder whether i can derive its boundedness from derivability and
integrability properties of the function alone. Here is the problem:
Let f: R-> R be a positive function, N+1 times continuously
differentiable, with [d-th derivative] f^(d) \in L_1 for each d in
{0,.., N+1}.
I wonder if

\sum_{k>0}^{\infty} f^(N)(k) sin(a.k)

(i.e. the sum of N-th derivative of f taken at k times a sine) may be
bounded for sufficiently small a's (not only a = 0)
Apparently, it seems possible to find a pathological function f, non-
zero around smaller and smaller intervals [k-u_k,k+u_k] such that
itself and its derivatives grow as high a needed, while still
satisfying the hypotheses.
Thus, the boundedness of \sum_{k>0}^{\infty} | f^(N)(k) | alone
does not seem reachable.
Dear Robert

Thank you very much for your kind answer. This is definitely what i
was looking for!

But it is reachable.

Let B(k) = int_k^{k+1} |f^(N+1)(t)| dt. Since f^(N+1) is in L^1,
sum_k B(k) <= ||f^(N+1)||_1 < infty.
If k <= x <= k+1,
|f^(N)(k) - f^(N)(x)| <= int_k^x |f^(N+1)(t)| dt <= B(k).
And so | |f^(N)(k)| - int_k^{k+1} |f^(N)(x)| dx| <= B(k).
Now sum_k |f^(N)(k)| <= sum_k B(k) + int_0^infty |f^(N)(x)| dx
<= ||f^(N+1)||_1 + ||f^(N)||_1
--
Robert Israel isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada


.

Relevant Pages