when fn -> f => fn' -> f'
Diego Torquemada
Let (Fn) be a sequence of continuous functions from R to R, defined on
an interval J, that converges uniformly to a continuous and
differentiable function F on J and suppose F'n (the derivative of Fn)
exists on J for any x in J. I want to kindly ask you which additional
conditions are required on Fn and F so that the sequence (F'n) will
converge on J to a function F' that is the derivative of F, the limit
of (Fn).
Thanks for your time,
Diego
José Carlos Santos
The only condition that I know of is: the sequence (F'_n)_n converges
uniformly to some function _g_. When that happens, then g = F' (and you
don't have to assume that F is differentiable; it will follow from these
hypothesis).
Best regards,
Jose Carlos Santos
David C. Ullrich
<diegoto...@gmail.com> wrote:
For functions from R to R really about the most you can say
is this holds if you assume that the derivatives converge
uniformly to _something_.
You might note that if F_n is holomorphic in an opne set in
the plane and F_n -> F uniformly on compact sets then
F_n' -> F' uniformly on compact sets as well - this is one
of the magical things about complex analysis.
>Thanks for your time,
>
>Diego
************************
David C. Ullrich
Dave L. Renfro
> The only condition that I know of is: the sequence (F'_n)_n
> converges uniformly to some function _g_. When that happens,
> then g = F' (and you don't have to assume that F is
> differentiable; it will follow from these hypothesis).
Interestingly, almost no research into refinements of this
result exist, at least for real-valued functions of one
real variable. Sometime around 1989, give or take a year,
Udayan B. Darji asked his Ph.D. advisor Jack B. Brown
(Auburn University) if he knew of any results in the
literature that examined how different g could be from F'.
Brown didn't know of any and passed the question along to
Andrew M. Bruckner. Bruckner, perhaps after asking others
(Solomon Marcus, Jan S. Lipinski, etc.), reported back that
he wasn't aware of anyone having done anything with this topic.
I don't know how widely it was known in 1989 that this obvious
question had apparently never been addressed before, at least
in print, but by 1991 many specialists in real analysis were
aware of Darji's work and found it quite surprising that
apparently no one had previously done any work on this problem.
Incidentally, these remarks are based on my recollection of
what I heard first-hand from all the participants in this story.
(About 5 years ago I posted some general comments about Darji's
results, and it seems that I never returned to the subject
<http://tinyurl.com/y5dq5z>.)
Darji's work on this topic formed a little more than half
of his 1991 Ph.D. Dissertation, and a condensed version
of this part of his Dissertation was published in 1996.
Udayan B. Darji, "Limits of differentiable functions",
Proceedings of the American Mathematical Society 124 #1
(January 1996), 129-134.
http://tinyurl.com/ykwnl3 [.pdf file of the paper]
The Introduction of an earlier version of Darji's paper
(late 1991 or early 1992; text below) gives a much more
leisurely overview of his results than the published
version does.
--------------------------------------------------------------
I. INTRODUCTION
The following is a standard theorem in undergraduate analysis
texts [Rudin's "Principles of Math. Analysis" cited]:
If {f_n} is a sequence of differentiable functions
defined on [a,b] such that {f_n} --> f (uniformly),
{f'_n} --> g (uniformly), then f is differentiable
everywhere and f' = g.
Some texts assume that each f'_n is continuously differentiable
to present a simpler proof. However, if one replaces
{f'_n} --> g (uniformly) with {f'_n} --> g (pointwise), then
the resulting statement is not correct. Consider the following
example:
For each positive integer n, let f_n(x) = -(1/n)*x^(-n)
for each x in [1,2]. Then, d/dx [lim(n --> oo) f_n] does
not equal to lim(n -->oo) (d f_n / dx) at x = 1.
Two natural questions arise at this point:
1. Given that {f_n} --> f (uniformly), f is differentiable,
and {f'_n} --> g (pointwise), how large can the set
M = {x: f'(x) \= g(x)} be?
2. Characterize the class of all such sets M.
These started out as seemingly simple looking problems,
but the solutions use some nontrivial ideas. Some theorems
on approximate continuity developed by Zahorski [Trans. AMS
69 (1950), 1-54], the notion of density topology developed
by Goffman, Neugebauer, Nishiura and Waterman [Duke Math. J.
28 (1961), 497-505; Proc. AMS 12 (1961), 116-121], and some
techniques similar to those of Preiss [Czech. Math. J. 21
(1971), 371-372] were used to obtain solutions of these
problems.
We state our main theorems and some of their implications
here. But before we do that, we state the following definition.
DEFINITION: The statement that ( {f_n}, f, {g_n}, g ) is
*appropriate* means that each of {f_n} and {g_n}
is a sequence of real functions defined on [0,1],
{f_n} --> f (uniformly), {g_n} --> g (pointwise),
f'_n = g_n for each n, and f is differentiable.
THEOREM (General Dominated Case): A set M subset of [0,1] is
G_delta_sigma and of measure zero if and only if
there exists appropriate ( {f_n}, f, {g_n}, g )
where {g_n} is dominated by some L^1 function
and M = {x: f'(x) /= g(x)}.
Note that in the General Dominated Case, M has to be small
in the measure theoretic sense, but it may be big in the
categorical sense (i.e. it may be a dense G_delta set).
The following is the solution to Question 2.
THEOREM (General Non-Dominated Case): A set M subset of [0,1]
is G_delta_sigma if and only if there exists appropriate
( {f_n}, f, {g_n}, g ) such that M = {x: f'(x) /= g(x)}.
Note that it follows from this theorem that M can be the
entire interval. So, this theorem also answers Question 1.
Our proof of this theorem is rather involved. But, one may
ask whether a simpler solution may be obtained for the
specific case M = [0,1]. In some sense, no simple solution
exists for this. It will follow from Theorem 19 that, in
order to achieve M = [0,1], the derivatives will not only
have to be discontinuous, but also the set of discontinuities
of each derivative will have to be of positive measure.
The following is a characterization for the C^1 Dominated
Case. Here, C^1 denotes the class of continuously differentiable
functions.
THEOREM (C^1 Dominated Case): A set M subset of [0,1] is
F_sigma and of measure zero if and only if there
exists appropriate ( {f_n}, f, {g_n}, g ) where
each f_n is C^1, {g_n} is dominated by an L^1 function,
and M = {x: f'(x) /= g(x)}.
So, it follows that if the derivatives are continuous and
dominated by an L^1 function, then M has to be small in the
sense of measure and category. [Renfro -- In fact, even smaller
than just this. See my post at <http://tinyurl.com/wx89j>.]
The C^1 Non-Dominated Case took an interesting turn. Before we
state the theorem, we state the following definition.
DEFINITION: The statement that M is measure dense in the
interval J means that if J' is any subinterval
of J, then (J' intersect M) has positive measure.
The statement that M is nowhere measure dense
means that M is measure dense in no interval.
THEOREM (C^1 Non-Dominated Case): A set M subset of [0,1] is
F_sigma and nowhere measure dense if and only if there
exists appropriate ( {f_n}, f, {g_n}, g ) where each
f_n is C^1 and M = {x: f'(x) /= g(x)}.
--------------------------------------------------------------
If anyone is interested, a possibly worthwhile research
project would be to see what happens if g_n --> f pointwise
is replaced with g_n --> f relatively uniformly (in the
sense of E. H. Moore and E. W. Chittenden -- google the
phrase "relatively uniform convergence"). This is a scale
of uniform convergence notions that forms a bridge between
pointwise convergence and uniform convergence, much like
modulus of continuity conditions form a bridge between
uniform continuity and Lipschitz continuity.
Dave L. Renfro
Dave L. Renfro
> (About 5 years ago I posted some general comments about Darji's
> results, and it seems that I never returned to the subject
> <http://tinyurl.com/y5dq5z>.)
Actually, "about 6 years ago" is a lot more accurate!
Dave L. Renfro