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

Posts: 4,598
Registered: 12/3/04
ESSAY ON NOWHERE ANALYTIC C-INFINITY FUNCTIONS
Posted: May 9, 2002 6:18 PM
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This is an essay on nowhere analytic C-infinity real-valued
functions of a real variable.

For an essay on power series that could serve as an introduction
to the present essay, see my post at

http://mathforum.org/epigone/ap-calc/kelddyshon


CONTENTS

I. TWO WAYS THAT A C-INFINITY FUNCTION CAN FAIL TO BE ANALYTIC
II. EARLY HISTORY OF NOWHERE ANALYTIC C-INFINITY FUNCTIONS
III. ZAHORSKI'S 1947 CHARACTERIZATION
IV. MOST C-INFINITY FUNCTIONS ARE NOWHERE ANALYTIC
V. UNEXPLORED AREAS AND OTHER RESULTS
VI. REFERENCES

Dave L. Renfro


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

Every C-infinity function has a formal Taylor series expansion
about each point. If this Taylor expansion converges to the
original function in neighborhood of x=b, then the function is
said to be analytic at x=b. There are two ways this can fail:

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

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

Following Zahorski [35], we classify the non-analytic points
of a C-infinity function into the following two categories.
A point of non-analyticity is called a (C)-point (for Cauchy)
if it belongs to (C) above. A point of non-analyticity is called
a (P)-point (for Pringsheim) if it belongs to (P) above.

An example of a (C)-point is given at x=0 by

exp(-1 / x^2) [with f(0) = 0].

An example of a (P)-point is given at x=0 by

Integral(t=0 to infinity) of [exp(-t) * cos(x*t^2)] dt.

The first example is well known. The second example is discussed
on pp. 189-190 of Boas [5]. Cauchy (1823) was the first to give
an example of a (C)-point (the same example given above),
while Du Bois-Reymond (1876) was the first to give an example
of a (P)-point. Du Bois-Reymond's example is

SUM(n=1 to infinity) of (-1)^(n+1) * x^(2n) / [(2n)! * (x^2 + b_n)],

where b_n = (a_n)^2 and {a_n} is any sequence of nonzero real
numbers whose limit is zero.

Du Bois-Reymond's 1876 paper appeared in a journal that was not
very well known [Enter "08.0127.01" (without quotes) in the
"JFM. no." window at .],
and so his example was not generally known at that time.
Du Bois-Reymond rewrote and expanded his 1876 paper and published
the revision in the much better known journal "Mathematische
Annalen". His revised paper, published in 1883, can be obtained
on the internet at

http://134.76.163.65:80/agora_docs/26379TABLE_OF_CONTENTS.html

Du Bois-Reymond's example is given at the bottom of page 111 of
his 1883 paper. Incidentally, there are some minor gaps in his
proof that the Taylor expansion diverges for nonzero values of
x that arise from overzealous manipulations of divergent series.


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II. EARLY HISTORY OF NOWHERE ANALYTIC C-INFINITY FUNCTIONS

Even in the late 1800's it was well known that the set of
non-analytic points of any function is a closed set. This is simply
a restatement of the fact that if a function is analytic at some
point, then it is analytic in a neighborhood of that point
(i.e. the points of analyticity form an open set). Therefore, in
order to obtain a nowhere analytic C-infinity function, it is
only necessary to find a C-infinity function having a dense set
of non-analytic points.

The first published example of such a function was given by
Du Bois-Reymond (1883), who used the method of condensation of
singularities to construct a C-infinity function having a dense
set of (P)-points. [See p. 116 of the Du Bois-Reymond paper
mentioned above.] Cellérier had previously obtained an example
of a C-infinity function having a dense set of (P)-points
(not using the method of condensation of singularities),
possibly as early as 1860, which was only discovered among
his manuscripts after his death and published in 1890. [Enter
"22.0386.01" (without quotes) in the "JFM. no." window at
.] Unlike Du Bois-Reymond's
proof, Cellérier had no gaps in the divergence part of his proof.

Cellérier's function is

SUM(n=1 to infinity) of [ sin(c^n * x) ] / n!,

where c > 1 is an integer. Cellérier's function has a (P)-point at
each of the points (2*Pi*m) * c^(-k), where m is any integer and k
is any nonnegative integer.

In 1888 Lerch published an example similar to Cellérier's function.
[Enter "20.0380.01" into the "JFM. no." window, as above. Also,
see Výborný [31] and p. 119 of Bilodeau [3].]

In 1893 Pringsheim gave various examples whose verifications
are easier, as well as examples of C-infinity functions belonging
to each of the following categories:

(i) C is dense in R and P is empty.
(ii) C is empty and P is dense in R.
(iii) Both C and P are dense in R.

Pringsheim's paper can be obtained on the internet at

http://134.76.163.65:80/agora_docs/35981TABLE_OF_CONTENTS.html

In 1895 Borel described a simple way to obtain nowhere analytic
C-infinity functions. Let f(z) be a complex-valued function of a
complex variable z that is analytic in an open disk and C-infinity
on the closure of this disk, and which cannot be continued
analytically across the boundary of this disk. Then f(ix), where
x is a real variable, is a complex-valued function that is
C-infinity and nowhere analytic in the interval -Pi < x < Pi.
[If you want a real-valued example, take either the real part or
the imaginary part of this function.]

A simple example of such a function f(z) was given in 1891 by
Fredholm:

SUM(n=0 to infinity) of c^n * z^(n^2),

where c is any nonzero complex number with |c| < 1.

By the mid 1890's examples of nowhere analytic C-infinity were
well known.

For more information on the early history of nowhere analytic
C-infinity functions, see Bilodeau [3]. For a general survey of
nowhere analytic C-infinity functions, including issues relating
to Zahorski's theorem discussed below, see Salzmann/Zeller [26].
An elementary verification of a nowhere analytic C-infinity function
is given in Merryfield [21], a paper that you can download at

http://www.math-cs.cmsu.edu/~mjms/1992-3p.html [48 K .ps file]
http://www.math-cs.cmsu.edu/~mjms/1992-3d.html [18 K .dvi file].


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III. ZAHORSKI'S 1947 CHARACTERIZATION

None of these early examples was shown to have a (C)-point
everywhere or to have a (P)-point everywhere. Boas proved
in 1935 (see [4] or p. 192 of [5]) that the (P)-points
of a nowhere analytic C-infinity function are dense in R.
Therefore, it is not possible to have an example in which every
point is a (C)-point. However, the possibility that every point
could be a (P)-point remained open. The first example of a
C-infinity function such that every point is a (P)-point was
given by Cartan in 1940 ([6], pp. 20-22). In 1947 Zahorski [35]
gave the following characterization for the non-analytic points
of a C-infinity function. [An announcement [34] of this result was
made in 1946.]

THEOREM (Zahorski): A necessary and sufficient condition for
two sets of real numbers C and P to be the (C)-points
and the (P)-points, respectively, of some C-infinity
function is that the following four properties hold:

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

[[ Zahorski died on May 8, 1998 at the age of 84. I
believe his work on the singularities of C-infinity
functions was part of his Ph.D. research. ]]

We mention three corollaries.

** If P is empty in some interval, then the set of (C)-points in
that interval is a relatively closed first category subset of
that interval. Since every closed first category set is nowhere
dense, no such function can be nowhere analytic in that interval.
Hence, if a function is nowhere analytic and C-infinity, then
every interval must contain some (P)-points, and we get the
result that Boas proved in 1935.

** Another corollary is the existence of a C-infinity function
having a (P)-point everywhere: Choose C to be empty and P = R.

** Still another corollary is that there exist C-infinity functions
belonging to each of the following categories (recall Pringsheim's
examples in Section II):

(i) C is c-dense in R and P is empty.
(ii) C is empty and P is c-dense in R.
(iii) Both C and P are c-dense in R.

For multivariable versions of Zahorski's theorem, see Bartczak [1],
Schmets/Valdivia [27] [28], Siciak [29], and H. Zahorska [33].


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IV. MOST C-INFINITY FUNCTIONS ARE NOWHERE ANALYTIC

The Baire category theorem can be used to prove the existence of
nowhere analytic C-infinity functions, and this has been
re-discovered several times. Using the standard metric on
C-infinity (more precisely, any metric that generates the topology
of uniform convergence for all orders of derivatives on compact
sets), Morgenstern [22] (1954) gave a concise proof that the
Baire-typical C-infinity function is nowhere analytic. An outline
of a proof can be found on pp. 301-302 of Dugundji [12], with
Morgenstern's name mentioned. However, Dungundji states that
the particular proof he presents is due to Salzmann and Zeller,
apparently from a personal communication. For an expanded version
in English of Morgenstern's original proof, see pp. 95-97 of
Jones [16].

Next up, we have Christensen [9] (1972), who proves the same
result, unaware of Morgenstern. Then we have Darst [10] (1973),
who was apparently unaware of both Morgenstern's and Christensen's
papers. However, Darst followed this up with [11] (1974), where a far
stronger Baire-typical result is proved. [See page 26. The result
I am alluding to only shows up in the proof of a certain theorem,
not in any of his theorem statements.] Darst shows that certain
quasi-analytic classes have a Baire-typical set of functions that
are nowhere quasi-analytic relative to other quasi-analytic notions.
Next, we have Cater [7] (1984), who was aware of both Christensen's
and Darst's papers, and probably also of Morgenstern's paper.
[Darst writes (p. 618): "Two references in English are ...".]
Cater's paper is well written and his proof has all the details
worked out. Siciak [29] (1986), who was aware of Morgenstern's
paper, proves a Baire-typical multi-dimensional analog of
Morgenstern's result (theorem 10 on p. 144). After this, there
is Bernal [2] (1987). Bernal states that a corollary of the main
result in his paper is that the Baire-typical C-infinity function
is nowhere analytic. [Bernal cites Cater, Christensen, and Darst
in his bibliography.] Finally, Ramsamujh [23] (1991) proves that
the Baire-typical C-infinity function is nowhere analytic, unaware
at that time of any of the preceding papers.

Aside from simply being nowhere analytic, can we say anything about
the (C)-points and the (P)-points of the Baire-typical C-infinity
function? As far as I can tell, it appears that all the sources I
mentioned in the previous two paragraphs, except for Bernal [2],
Ramsamujh [23], and Siciak [29], prove only that the Baire-typical
C-infinity function has a dense set of (P)-points. From Zahorski's
theorem we know that the set of (P)-points is a G_delta set
(actually, this is immediate from the definition), and so we have
the interesting observation that the Baire-typical C-infinity
function has a Baire-typical set of (P)-points. In fact,
Christensen [9] explicitly states his result in this way.

However, Bernal [2], Ramsamujh [23], and Siciak [29] manage to
prove more. They actually prove that EVERY point is a (P)-point
for the Baire-typical C-infinity function. Thus, the Baire-typical
C-infinity function has no (C)-points at all. Although this
completely settles the matter, the result is unfortunate because
it closes the door on the possibility of investigating the Lebesgue
measure (or Hausdorff dimension, if measure zero) of the sets of
(C)-points and (P)-points of the Baire-typical C-infinity function.


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V. UNEXPLORED AREAS AND OTHER RESULTS

If we let f(x) = exp(-1 / x^2) for x > 0 and zero elsewhere, then f
is a C-infinity function whose Taylor series at x=0 converges to f
in a left neighborhood of x=0 but doesn't converge to f in any
right neighborhood of x=0. Let's call such a point for a function
a "right (P)-point" of that function, and define "left (P)-point"
of a function in the obvious way. Finally, we say a point is a
"bilateral (P)-point" of a function if the Taylor series for the
function about that point has a positive radius of convergence,
but it fails to converge to the function in every left neighborhood
and every right neighborhood of that point.

It is easy to see that the set of (P)-points of a function is the
disjoint union of its left (P)-points, its right (P)-points, and
its bilateral (P)-points. What more can we say about these sets?
For example, can the set of right (P)-points be uncountable?

Note that the notion of a unilateral (C)-point doesn't arise, since
Taylor series converge equally far on both sides of their expansion
point.

To my knowledge, the results in the previous Section have not been
investigated for the porosity-typical C-infinity function or for the
prevalent C-infinity function (i.e. the complement of a Christensen
null set; see
).

An investigation for the porosity-typical C-infinity function
would probably be a good Ph.D. topic, since both the results
and the proofs will likely be similar to the Baire category
results. Also, I've looked into this enough to know that the
strengthening from Baire category to porosity isn't immediate.
The Baire category proofs I've seen would have to be restructured
so that you can explicitly get your hands on a decomposition of
nowhere dense sets. After this, you need to have these nowhere
dense sets sufficiently defined in a constructive way that will
allow you to make the necessary porosity computations.

As to whether the prevalent C-infinity function is nowhere
analytic, one way of trying to prove that it is would be to
follow the method of proof given in Christensen [9]. This
would require a Fubini/Kuratowski-Ulam type result for Haar
null sets. The exact analog doesn't hold (an example is given
in the paper for MR 48 #4637), but the weaker version that's
given in the paper for MR 99g:49013 might suffice.

Other papers that show the existence of a lot of nowhere analytic
C-infinity functions are Fabius [14], Kabaya/Iri [17], and
Fournier/Gauthier [15]. In the first two papers certain distribution
functions are shown to be C-infinity and nowhere analytic for any
sequence of independent and uniformly distributed random variables.
The last paper proves that the Baire-typical complex-valued sequence
gives rise to a Maclaurin series whose radius of convergence is 0.
Results related to this last paper can also be found in
Salát/Tóth [24] and Sálat/Taylor/Tóth [25].


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VI. REFERENCES

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

[2] Luis G. Bernal, "Functions with successive derivatives everywhere
large or small" (Spanish), Collect. Math. 38 (1987), 117-122.
[MR 90c:26013]

[3] Gerald G. Bilodeau, The origin and early development of
non-analytic infinitely differentiable functions, Arch. Hist.
Exact Sci. 27 (1982), 115-135. [MR 84g:26017; Zbl 503.01005]

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

[5] Ralph P. Boas, A PRIMER OF REAL FUNCTIONS, 4'th edition (revised
and updated by Harold P. Boas), Carus Mathematical Monographs
#13, Mathematical Association of America, 1996, xiv + 305 pages.
[MR 97f:26001; Zbl 865.26001]

[6] Henri Cartan, "Sur les classes de fonctions définies par des
inégalités portant sur leurs derivées successives" [On classes
of functions defined by inequalities holding on their successive
derivatives], Actualités Scientifiques et Industrielles 867
(1940), 36 pages. [MR 3, 292b; Zbl 61.11701; JFM 66 (p. 1231)]

[7] Frank S. Cater, "Differentiable, nowhere analytic functions",
Amer. Math. Monthly 91 (1984), 618-624.
[MR 86b:26034; Zbl 598.26035]

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

[9] Jens Peter Reus Christensen, "A topological analogue of the
Fubini theorem and some applications", Various Publications
Series, Aarhus University, No. 21 (July 1972) [papers from the
"Open House for Probabilists", July-August 1971, Aarhus
University, Denmark], 26-31. [MR 52 #6685; Zbl 241.28008]

[10] Richard B. Darst, "Most infinitely differentiable functions
are nowhere analytic", Canad. Math. Bull. 16 (1973), 597-598.
[MR 49 #10838; Zbl 285.26015]

[11] Richard B. Darst, "Localization properties of basic classes
of C-infinity functions", Proc. Amer. Math. Soc. 46 (1974),
24-28. [MR 49 #10839; Zbl 294.26019]

[12] James Dugundji, TOPOLOGY, Allyn and Bacon, 1966,
xvi + 447 pages. [MR 33 #1824; Zbl 144.21501]

[13] Ricardo Estrada, "Existence of smooth functions that are
nowhere analytic" (Spanish), Mat. Costarricense 1 (1984), 1-5.
[MR 85h:26028]

[14] J. Fabius, "A probabilistic example of a nowhere analytic
C-infinity--function", Z. Wahrscheinlichkeitstheorie Verw.
Geb. 5 (1966), 173-174. [Fabius's example is given in Chapter 5
of Stromberg (1994).] [MR 33 #5820; Zbl 139.35601]

[15] John J. F. Fournier and Paul M. Gauthier, "Most power series
have radius of convergence 0 or 1", Canad. Math. Bull. 18
(1975), 39-40. [MR 52 #5947; Zbl 315.60004]

[16] Sara H. Jones, THE BAIRE CATEGORY THEOREM: ITS SCOPE, Ph.D.
Dissertation (under Andrew M. Bruckner), The University of
California at Santa Barbara, 1995, v + 113 pages.
[DAI 57, #1B (July 1996), p. 378-B]

[17] Keiko Kabaya and Masao Iri, "Sum of uniformly distributed random
variables and a family of nonanalytic C-infinity--functions",
Japan J. Appl. Math. 4(1) (1987), 1-22.
[MR 89d:26023; Zbl 625.60054]

[18] Sung S. Kim and Kil H. Kwon, "Smooth (C-infinity) but nowhere
analytic functions", Amer. Math. Monthly 107 (2000), 264-266.
[In equation (2) on p. 264, and also in the first line on
p. 265, the expression phi((2^j)*x - [(2^j)*x]) should be
phi((2^j)*(x - [x])) according to errata given on p. 968 of
Amer. Math. Monthly 107 (2000).]

[19] Hans Lewy, "An example of a smooth linear partial differential
equation without solution", Annals of Math. (2) 66 (1957),
155-158. [Errata in Annals 68 (1958), 202]
[MR 19, 551d; Zbl 78.08104]

[20] Lawrence E. May, "On C-infinity functions analytic on sets of
small measure", Canad. Math. Bull 12 (1969), 25-30.
[MR 39 #2933; Zbl 176.01201]

[21] Kent G. Merryfield, "A nowhere analytic C-infinity function",
Missouri J. Math. Sci. 4(3) (1992), 132-138.

[22] Dietrich Morgenstern, "Unendlich oft differenzierbare
nicht-analytische funktionen" [Infinitely differentiable
non-analytic functions], Math. Nachrichten 12 (1954), page 74.
[MR 16, 342b; Zbl 58.28902]

[23] Taje I. Ramsamujh, "Nowhere analytic C-infinity functions",
J. Math. Analysis Appl. 160 (1991), 263-266.
[MR 92j:26014; Zbl 751.26014]

At the very end of the paper Ramsamujh concludes that g does
not belong to F(M) from the fact that
|g^{(n+1)}(x_0)| > [M^(n+1)]*(n+1)!, where x_0 is a specific
point obtained earlier. However, F(M) was defined to be the
collection of all C-infinity functions f having the property
that there exists an x_0 such that for all n we have
|f^{(n+1)}(x_0)| > [M^(n+1)]*(n+1)!. Thus, to show g doesn't
belong to F(M) requires a failure at every point x_0 for
some n, not the failure at some point x_0 for some n. However,
I discussed this problem with Ramsamujh in 1998 and we believe
his proof is repairable.

[24] Tibor Salát and János T. Tóth, "On radii of convergence of
power series", Bull. Math. Soc. Sci. Math. Roum. 38 (1994-95),
183-198. [Zbl 886.40002]

[25] Tibor Salát, S. James Taylor, and János T. Tóth, "Radii of
convergence of power series", Real Analysis Exchange 24
(1998-99), 263-273. [MR 2000h:40004]

[26] Helmut Salzmann and Karl Zeller, "Singularitäten unendlich oft
differenzierbarer funktionen" [Singularities of infinitely
differentiable functions], Math. Zeitschrift 62 (1955), 354-367.
[MR 17, 134b; Zbl 64.29903]

[27] Jean Schmets and Manuel Valdivia, "A short proof of the Zahorski
theorem in R^n", Publication 95.013, Institut de Mathématique
Université de Liège, 1995, 8 pages.

[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] Józef Siciak, "Regular and singular points of C-infinity
functions" (Polish), Zesz. Nauk. Politech. Sl., Mat.-Fiz.
48(853) (1986), 127-146. [Zbl 777.26020]

[30] Karl R. Stromberg, PROBABILITY FOR ANALYSTS, Lecture notes
prepared by Kuppusamy Ravindran, Chapman & Hall, 1994,
xiv + 330 pages. [The Fabius (1966) nowhere analytic C-infinity
function is given in Chapter 5.] [MR 95h:60001; Zbl 803.60001]

[31] Rudolf Výborný, "A C-infinity nowhere analytic function. A
forgotten example from 1888", Austral. Math. Soc. Gaz. 15(1)
(1988), 7-8. [MR 89d:26025]

[32] Yohei Yamazaki, "A simple example of a C-infinity function that
is nowhere analytic" (Japanese), Sugaku 27 (1975), 366-367.
[MR 58 #31185]

[33] Helene Zahorska, "&Uuml;ber die singulären 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]

[34] Zygmunt Zahorski, "Problèmes de la théorie des ensembles et des
fonctions" [Problems in the theory of sets and of functions],
C. R. Acad. Sci. Paris 223 (1946), 449-451.
[MR 8, 141c; Zbl 60.13705]

[35] Zygmunt Zahorski, "Sur l'ensemble des points singuliers d'une
fonction d'une variable réelle admettant les dérivées 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]

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