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Topic: ESSAY ON NOWHERE ANALYTIC C-INFINITY FUNCTIONS
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Dave L. Renfro

Posts: 4,518
Registered: 12/3/04
Re: ESSAY ON NOWHERE ANALYTIC C-INFINITY FUNCTIONS
Posted: May 19, 2002 8:29 PM
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This is a follow-up to my earlier essay on nowhere analytic
C-infinity real-valued functions of a real variable. Both the
earlier essay and this follow-up can be found at

http://mathforum.org/epigone/sci.math/brengunsul


CONTENTS

I. REFINEMENTS ON THE TWO WAYS THAT A C-INFINITY FUNCTION
CAN FAIL TO BE ANALYTIC

II. GENERALIZATIONS OF ZAHORSKI'S 1947 CHARACTERIZATION

III. ZERO SETS OF C-INFINITY FUNCTIONS AND ULAM'S PROBLEM

IV. MORE CONCERNING "MOST C-INFINITY FUNCTIONS ARE NOWHERE
ANALYTIC"

V. REFERENCES

Dave L. Renfro

P.S. In case anyone has comments, corrections, or questions,
I'll have very limited (if any at all) internet and
e-mail access from May 20 to May 25 (inclusive).


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I. REFINEMENTS ON THE TWO WAYS THAT A C-INFINITY FUNCTION
CAN FAIL TO BE ANALYTIC

We recall the two ways that a C-infinity function f can fail
to be analytic at a point x=b:

(C) The Taylor series of f converges in a neighborhood
of x=b, but in no neighborhood of x=b does the
Taylor series of f converge to f.

(P) The Taylor series of f fails to converge in every
neighborhood of x=b (i.e. the Taylor series of f at
x=b has a zero radius of convergence).

A point of non-analyticity of a C-infinity function f is
called a (C)-point of f [a (P)-point of f] if it satisfies
property (C) [if it satisfies property (P)].

***********************************

We first consider a way to classify (C)-points. [See the
discussion of Bang [36] in Section II for another way to
classify (C)-points.]

Let W be any collection of subsets of R such that

(W1) b does not belong to E for each E in W.

(W2) For each epsilon > 0 there exists E in W such that E
is a subset of (b - epsilon, b + epsilon).

We say that x=b is a (C_W)-point of a C-infinity function f if

{ x in (b - epsilon, b + epsilon): TS(f)(x) differs from f(x) }

belongs to W for each epsilon > 0.

The larger the collection W, the stronger the corresponding
notion of non-analyticity at x=b. The usual situation of a
(C)-point arises when we take W to be the collection of all
sequences converging to b (none of whose terms equal b). We can
get stronger types of (C)-points by letting W be collections
such as all sets not containing b having b as a condensation point,
all sets that positive upper Lebesgue density at b, etc.

Other axioms on the collection W might be needed to prove some
nontrivial general theorems, but my first inclination would be
to look for results involving specific natural choices of W before
going down this path of abstraction.

I am not aware of anything in the literature that makes use of
this elaboration on the notion of a (C)-point aside from the
following:

Pringsheim gave the following example in 1892. This is also
example (2) in Section 150 (p. 211) of Hobson [42].

Let f(x) = SUM(n=0 to infinity) of

[ (-1)^n * c^(-n) ] / [ n! * ( c^(-2n) + x^2 ) ],

where c is any fixed real number with c > 1.

Then the Taylor series of f about x=0 is

TS(f)(x) = SUM(n=0 to infinity) of

(-1)^n * exp[-a^(2n+1)] * x^(2n).

This Taylor series converges everywhere. However, for each
epsilon > 0 the set

{ x in (-epsilon, epsilon): TS(f)(x) = f(x) }

is finite.

***********************************

We now consider a way to classify (P)-points.

If the Taylor series of f at x=b converges in a neighborhood
of x=b, then the derivatives of f at x=b can't grow too fast.
More precisely, if f does not have a (P)-point at x=b, then there
exists a number B such that for all positive integers n we have

| n'th derivative of f at b | < B^n * n!,

or equivalently,

| (n'th derivative of f at b) / n! | ^ (1/n)

is bounded.

The proof is straightforward: If r is the radius of convergence
of the Taylor series for f at x=b, then we have

1/r = lim-sup(n --> oo) of

| (n'th derivative of f at b) / n! | ^ (1/n),

where the convention 1/(infinity) = 0 is used.

Hence, if r > 0 (i.e. x=b is not a (P)-point of f), then the
lim-sup is finite. Therefore,

| (n'th derivative of f at b) / n! | ^ (1/n)

is bounded.

[[ Textbooks often state this growth restriction on the
derivatives of f at x=b as a consequence of f being
analytic at x=b, but note that it is the positive radius
of convergence of the Taylor series that implies this,
not the full hypothesis of being analytic at x=b. ]]

[[ Incidentally, if this growth restriction on the
derivatives of f exists uniformly throughout a
neighborhood of x=b, then f will be analytic at
x=b (in fact, f will be analytic throughout that same
neighborhood). This was stated and given an incorrect
proof in an 1893 paper by Pringsheim (the same paper cited
in Section II of my previous post). A correct proof of
a stronger result is given on p. 36 of Boas [4]. The
result Boas proves is stronger in this way: Instead of
requiring the radius of convergence r(x) of the Taylor
series at x to be bounded above zero in a neighborhood
of x=b, we only require that r(x) > 0 in a neighborhood
of x=b and that lim-inf(x --> y) of [ r(x) / |x-y| ] be
larger than 1 for all y in some neighborhood of x=b.
Theorem 3.1 in Boghossian/Johnson [37] strengthens
Pringsheim's result in another way: Suppose S is an open
subset of an open interval J and J-S is countable. If r(x)
is uniformly bounded away from zero for x in S, then f will
be analytic on J. ]]

This result suggests that various stronger notions of being
a (P)-point can be defined by requiring that the derivatives at
a point have subsequences exhibiting rapid growth. This can take
the form of a lower bound on the lim-sup growth of the n'th
derivative, a lower bound on the lim-inf growth of the n'th
derivative, or possibly some intermediate limiting notion
(e.g. a lim-sup with a specified upper asymptotic density
condition met for some subsequence of n = 1, 2, 3, ... witnessing
the lim-sup, etc.).

This elaboration on the notion of a (P)-point appears in the
literature. For instance, it arises in anything having to do with
quasi-analytic classes of functions, in Bang [36] (see Section II),
in Bernal-Gonz&aacute;lez [2] (see Section IV), in Cater [8], and in
Schmets/Valdivia [28].


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II. GENERALIZATIONS OF ZAHORSKI'S 1947 CHARACTERIZATION

Bang [36] proved a generalization of Zahorski's 1947 characterization
for the (C)-points and (P)-points of a C-infinity function that,
among other things, makes use of the variations on the notion
of a (P)-point that I discussed in Section I.

Let f be C-infinity on the reals R, let M_n be a sequence of
positive real numbers such that the sequence { 1/n * M_n / M_(n-1) }
is nondecreasing, and let

1 / r(x) = lim-sup(n --> oo) of

| (n'th derivative of f at b) / M_n | ^ (1/n).

In this setting we define the sets A(f), C(f), and P(f):

A(f) is the set of points x such that 0 < lim-inf(y --> x) of r(y).

C(f) is the set of points x such that r(x) > 0
AND 0 = lim-inf(y --> x) of r(y).

P(f) is the set of points x such that r(x) = 0.

THEOREM: Given a partition of R into three pairwise disjoint
subsets A, C, and P, there exists a C-infinity function
f on R such that A = A(f), C = C(f), and P = P(f)
if and only if


(a) C is a first category F_sigma set.
(b) P is a G_delta set.
(c) C union P is closed in R.

This reduces to Zahorski's 1947 result when M_n = n!.

***********************************

There are several papers that generalize Zahorski's characterization
for functions of one real variable to functions of more than one
real variable. What follows is a brief description of the papers
that I'm aware of.

H. Zahorski [33] (1964) proved a partial 2-variable version of
Z. Zahorski's characterization of the (C)-points and the (P)-points
of a C-infinity function. The sets C and P are shown to be first
category F_sigma and G_delta, respectively. The other half of the
characterization is proved for the case that P is empty: Given any
closed nowhere dense set C in R^2, there exists a C-infinity
function f such that (a) f is analytic on (R^2 - C), and (b) C is
the set of (C)-points of f.

Bartczak [1] (1973) gave a re-working to n-variables of
H. Zahorski's 2-variable results.

Meres [44] (1982) proved a 2-variable version of Z. Zahorski's
characterization of the (C)-points and the (P)-points of a
C-infinity function for the case that C is empty: Given any
closed set P in R^2, there exists a C-infinity function f such
that (a) f is analytic on (R^2 - P), and (b) P is the set of
(P)-points of f.

Siciak [29] (1986) gave a proof of the full n-variable version of
Z. Zahorski's characterization of the (C)-points and the (P)-points
of a C-infinity function. [The Zbl review inadvertently left out
the hypothesis that the (C)-points form a first category set.]
Siciak also proved (at the end of his paper) that the Baire-typical
C-infinity function on R^n has a Taylor series with zero radius of
convergence at each point (i.e. the set of (P)-points equals R^n).

Schmets/Valdivia [27] (1995) proved an n-variable version of
Zahorski's characterization of the (C)-points and the (P)-points
of a C-infinity function that does not use the sets

(M_k)(f,x) = sup { [ | (D^i)f(x) | / i! ] ^ [1 / (|i| + 2)] },

where the supremum is taken over all multi-indices i whose norm |i|
is at most k.


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III. ZERO SETS OF C-INFINITY FUNCTIONS AND ULAM'S PROBLEM

Let Z(f) = {x: f(x) = 0} be the zero set of f. If f is continuous,
then Z(f) is closed. Conversely, given any closed set C, there
exists a continuous function f such that Z(f) = C. For instance,
the function defined by dist(C,x) will work. This converse can be
strengthened: Given any closed set C there exists a C-infinity
function f such that Z(f) = C. Moreover, the orders of the zeros at
the isolated points of C can be arbitrarily specified. [This even holds
in any non-quasi-analytic class -- see Hughes [43]. (A shorter proof
of Hughes' result is given in Darst [41].)]

Since the zero set of f is where f intersects the horizontal line
y=0, it is natural to investigate the sets that arise when C-infinity
functions intersect other curves. If we restrict ourselves to graphs
of continuous functions, then every such intersection will be a
closed set. What about the converse, for various specific continuous
functions? Can every closed set arise from the intersection of some
C-infinity function with the graph of y = x^2? Can every closed set
arise from the intersection of some C-infinity function with the
graph of sin(x^3)?

The answer as surprisingly easy as it is strong. Given ANY C-infinity
function g and ANY closed set C, there exists a C-infinity function f
such that {x: f(x) = g(x)} = C. Simply find a C-infinity function
h such that Z(h) = C and put f = g-h.

All of this fails rather spectacularly for analytic functions.
The only sets that can arise as the zero set of a nonzero analytic
function are closed isolated sets. [Not just countable, as there
are countable sets that are far from being isolated. Not just
isolated, as there are isolated sets whose closures even have
positive measure.] But maybe we can salvage matters and come up
with something interesting to pursue if we turn things around a bit.

Note that I had to say "nonzero analytic function" above. Rather than
casting aside the zero function in our search for an interesting
result involving analytic functions, let's embrace it in this way:
There exists an analytic function that agrees with the zero function
on a "large" set, namely the zero function itself. Since we can do
the same thing when "zero function" is replaced by any analytic
function, the situation doesn't become interesting until we begin
allowing the use of non-analytic functions. In the mid 1930's Ulam
asked the following version of this problem: Given a continuous
function g, does there exist an analytic function f such that
{x: f(x) = g(x)} is uncountable? [Or equivalently, has cardinality
2^(aleph_0), since the sets involved are closed.]

Ulam conjectured that the answer is YES. However, Zahorski showed
that the answer is NO even if we restrict g to be a C-infinity
function. Zahorski gave a short argument that any C-infinity function
whose Taylor series has a zero radius of convergence about each point
would serve as an example (p. 238 of [35]: If the intersection contained
a non-isolated point x, then all the derivatives of f and g would agree
at x, which means that they have the same Taylor series expansion at x,
and therefore the same radius of convergence at x) and then Zahorski
observed that his characterization of the (C)-points and (P)-points
immediately provides for such an example. Indeed, Ulam's problem
appears to have been one of the primary motivating factors behind
Zahorski's research in this area, at least this seems to be the case
from the organization of his paper. [The results in Zahorski's paper
were obtained in 1940-41 but WW II caused a delay in its writing and
publication. Incidentally, Zahorski appears not to have been aware of
Cartan's 1940 example of such a C-infinity function. Moreover, I don't
believe that Cartan was aware of Ulam's question, nor was Ulam aware
of Cartan's example.] Zahorski raised some questions in his paper that
are related to the Ulam problem he solved, and these have given rise
to a number of results in this vein. Brown [38], Bruckner/Jones [39],
and Olevskii [45] are good places to begin if you want to investigate
these issues further. [In the references I give a URL for a .pdf file
of Brown's paper.]


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IV. MORE CONCERNING "MOST C-INFINITY FUNCTIONS ARE NOWHERE
ANALYTIC"

Bernal-Gonz&aacute;lez [2] proved the following. Let {a_n} and {b_n} be
sequences of positive real numbers. Then the Baire typical
C-infinity function has the property that for each x in R we have

lim-inf(n --> oo) [ a_n * | n'th derivative of f at x | ] = 0
and
lim-sup(n --> oo) [ b_n * | n'th derivative of f at x | ] = oo.

COROLLARY: The Baire-typical C-infinity function has a Taylor series
with zero radius of convergence about each point.

PROOF: Let b_n = (n^n * n!)^(-1) and use the fact that x is a
(P)-point of f if for each B > 0 there exists a positive
integer n such that

| n'th derivative of f at x | > B^n * n!.

What Bernal-Gonz&aacute;lez's result shows is that not only is every point a
(P)-point for the Baire typical C-infinity function (i.e. the n'th
derivatives at each point are not bounded in their growth by B^n * n!
for some B > 0, where B is not required to be uniform in x), but we
can require the n'th derivatives at each point to have a subsequence
whose growth rate exceeds any preassigned growth rate. Simultaneously,
the n'th derivatives at each point have a subsequence whose growth
rate is slower than any preassigned growth rate.

In Section V of my previous essay I said that I did not know if
the analyticity properties of the prevalent C-infinity function
had been investigated. In Section 3 of Shi [46] (URL's for
.pdf and .ps files of this paper are given in the references) there is
a short and elementary proof that the prevalent C-infinity function
is nowhere analytic: Given a compact interval J with midpoint b,
the collection of C-infinity functions f whose Taylor series about
x=b converge to f on J is a closed proper (vector) subspace of
all C-infinity functions, and hence is shy (a Haar null set). [See
the middle of p. 222 of Hunt/Sauer/Yorke, Bull. AMS 27 (1992).] By
taking unions over all such intervals J with rational endpoints, it
follows that the prevalent C-infinity function has a dense set of
points of non-analyticity. Note that this proof doesn't tell us
anything about the (C)-points and (P)-points of the prevalent
C-infinity function that we don't already know from Zahorski's
results.


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V. REFERENCES


REFERENCES FROM MY PREVIOUS ESSAY

[1] Teresa Bartczak, "The Cauchy singular points of a function of
several variables" (Polish), Zeszyty Nauk. Uniw. L&oacute;dzk. Nauki
Mat. Przyrodn. Ser. II Zeszyt 52 Mat. (1973), 85-108.
[MR 48 #11428; Zbl 264.26016]

[2] Luis Bernal-Gonz&aacute;lez, "Funciones con derivadas sucesivas
grandes y peque&ntilde;as por doquier" [Functions with successive
derivatives everywhere large or small], Collect. Math. 38
(1987), 117-122. [MR 90c:26013; Zbl 661.26009]

[4] Ralph P. Boas, "When is a C-infinity function analytic?",
Mathematical Intelligencer 11(4) (1989), 34-37.
[MR 91k:26023; Zbl 704.41021]

[8] Frank S. Cater, "Most C-infinity functions are nowhere Gevrey
differentiable of any order", Real Analysis Exchange 27
(2001-02), 77-79.

[27] Jean Schmets and Manuel Valdivia, "A short proof of the Zahorski
theorem in R^n", Publication 95.013, Institut de Math&eacute;matique
Universit&eacute; de Li&egrave;ge, 1995, 8 pages.
http://www.ulg.ac.be/sectmath/prep.html

[28] Jean Schmets and Manuel Valdivia, "The Zahorski theorem is valid
in Gevrey classes", Fund. Math 151 (1996), 149-166.
[MR 98a:26026; Zbl 877.26014]

[29] J&oacute;zef Siciak, "Punkty regularne i osobliwe funkcji klasy
C-infinity" [Regular and singular points of C-infinity
functions], Zesz. Nauk. Politech. Sl., Mat.-Fiz. 48(853)
(1986), 127-146. [Zbl 777.26020]

[33] Helene Zahorska, "&Uuml;ber die singul&auml;ren punkte einer funktion
der klasse C-infinity" [On the singular points of functions
of class C-infinity], Acta Math. Acad. Sci. Hungar. 15 (1964),
77-94. [MR 28 #3129; Zbl 127.28502]

[35] Zygmunt Zahorski, "Sur l'ensemble des points singuliers d'une
fonction d'une variable r&eacute;elle admettant les d&eacute;riv&eacute;es de tous
les ordres" [On the set of singular points of a function of
one real variable that has derivatives of all orders],
Fund. Math. 34 (1947), 183-245. [A correction to a remark
Zahorski had made about a result proved by Ganapathy Iyer
is given in Fund. Math. 36, (1949), 319-320 (MR 11, 718a;
Zbl 38.04201).] [MR 10, 23c; Zbl 33.25504]

ADDITIONAL REFERENCES FOR THIS ESSAY

[36] Th&oslash;ger Bang,"Sur les points singuliers (dans un sens g&eacute;n&eacute;ralis&eacute;)
des fonctions ind&eacute;finiment d&eacute;rivables" [On the singular points
(in a generalized sense) of infinitely differentiable functions],
pp. 259-263 in Den 11'th Skandinaviske Matematikerkongress
(1949), Trondheim, Johan Grundt Tanums Forlag, Oslo, 1952.
[MR 14, 626d; Zbl 48.04002]

[37] Artin Boghossian and Peter D. Johnson, "Pointwise conditions
for analyticity and polynomiality of infinitely differentiable
functions", J. Math. Anal. Appl. 140 (1989), 301-309.
[MR 684.26009; Zbl 90e:26040]

[38] Jack B. Brown, "Restriction theorems in real analysis", Real
Analysis Exchange 20 (1994-95), 510-526.
For a 197 K .pdf file of this paper, see
http://web6.duc.auburn.edu/~brownj4/restthm.pdf
[MR 96f:46059; Zbl 841.26002]

[39] Andrew M. Bruckner and Sara H. Jones, "Behavior of continuous
functions with respect to intersection patterns", Real Analysis
Exchange 19 (1993-94), 414-432. [MR 95h:26002; Zbl 822.26003]

[40] William Burnside, "Note on functions of a real variable",
Messenger of Mathematics 23 (1893-94), 39-42. [Burnside gives
an example of a nowhere analytic C-infinity function.]
[JFM 25.0731.02]

[41] Richard B. Darst, "Zero sets of functions from non-quasi-analytic
classes", Proc. Amer. Math. Soc. 40 (1973), 543-544.
[MR 48 #2330; Zbl 268.26018]

[42] Ernest W. Hobson, THE THEORY OF FUNCTIONS OF A REAL VARIABLE
AND THE THEORY OF FOURIER'S SERIES, Volume II, Dover
Publications, 1957 (reprint of 2'nd edition, 1926).
[MR 19, 1166b; Zbl 81.27702; JFM 53.0226.01]

[43] Robert B. Hughes, "Zero sets of functions from non-quasi-analytic
classes", Proc. Amer. Math. Soc. 27 (1971), 539-542. [Erratum
in Proc. AMS 39 (1973), 651.] [MR 42 #7846; Zbl 208.07401]

[44] Lucjan Meres, "On the singular points of Pringsheim-Du
Bois-Reymond of a function of two variables", Demonstratio
Math. 15 (1982), 297-310. [MR 84m:26025; Zbl 511.26010]

[45] Alexander M. Olevskii, "Ulam-Zahorski problem on free
interpolation by smooth functions", Trans. Amer. Math. Soc.
342 (1994), 713-127. [MR 94f:26007; Zbl 801.26006]

[46] Hongjian Shi, "Prevalence of some known typical properties",
Acta Math. Univ. Comenianae 70 (2001), 185-192.

http://www.emis.de/journals/AMUC/_vol-70/_no_2/_shi/shi.pdf
http://www.maths.soton.ac.uk/EMIS/journals/AMUC/_vol-70/_no_2/_shi/shi.ps.gz

[47] Pawel Walczak, "A proof of some theorem on the C-infinity
functions of one variable which are not analytic", Demonstratio
Math. 4 (1972), 209-213. [MR 49 #504; Zbl 253.26011]


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