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Re: ESSAY ON NOWHERE ANALYTIC CINFINITY FUNCTIONS
Posted:
May 19, 2002 8:29 PM


This is a followup to my earlier essay on nowhere analytic Cinfinity realvalued functions of a real variable. Both the earlier essay and this followup can be found at
http://mathforum.org/epigone/sci.math/brengunsul
CONTENTS
I. REFINEMENTS ON THE TWO WAYS THAT A CINFINITY FUNCTION CAN FAIL TO BE ANALYTIC
II. GENERALIZATIONS OF ZAHORSKI'S 1947 CHARACTERIZATION
III. ZERO SETS OF CINFINITY FUNCTIONS AND ULAM'S PROBLEM
IV. MORE CONCERNING "MOST CINFINITY 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 email access from May 20 to May 25 (inclusive).
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I. REFINEMENTS ON THE TWO WAYS THAT A CINFINITY FUNCTION CAN FAIL TO BE ANALYTIC
We recall the two ways that a Cinfinity 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 nonanalyticity of a Cinfinity function f is called a (C)point of f [a (P)point of f] if it satisfies property (C) [if it satisfies property (P)].
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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 Cinfinity 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 nonanalyticity 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.
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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 = limsup(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 limsup 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 liminf(x > y) of [ r(x) / xy ] 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 JS 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 limsup growth of the n'th derivative, a lower bound on the liminf growth of the n'th derivative, or possibly some intermediate limiting notion (e.g. a limsup with a specified upper asymptotic density condition met for some subsequence of n = 1, 2, 3, ... witnessing the limsup, etc.).
This elaboration on the notion of a (P)point appears in the literature. For instance, it arises in anything having to do with quasianalytic classes of functions, in Bang [36] (see Section II), in BernalGonzález [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 Cinfinity 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 Cinfinity on the reals R, let M_n be a sequence of positive real numbers such that the sequence { 1/n * M_n / M_(n1) } is nondecreasing, and let
1 / r(x) = limsup(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 < liminf(y > x) of r(y).
C(f) is the set of points x such that r(x) > 0 AND 0 = liminf(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 Cinfinity 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!.
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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 2variable version of Z. Zahorski's characterization of the (C)points and the (P)points of a Cinfinity 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 Cinfinity 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 reworking to nvariables of H. Zahorski's 2variable results.
Meres [44] (1982) proved a 2variable version of Z. Zahorski's characterization of the (C)points and the (P)points of a Cinfinity function for the case that C is empty: Given any closed set P in R^2, there exists a Cinfinity 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 nvariable version of Z. Zahorski's characterization of the (C)points and the (P)points of a Cinfinity 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 Bairetypical Cinfinity 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 nvariable version of Zahorski's characterization of the (C)points and the (P)points of a Cinfinity 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 multiindices i whose norm i is at most k.
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III. ZERO SETS OF CINFINITY 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 Cinfinity 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 nonquasianalytic 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 Cinfinity 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 Cinfinity function with the graph of y = x^2? Can every closed set arise from the intersection of some Cinfinity function with the graph of sin(x^3)?
The answer as surprisingly easy as it is strong. Given ANY Cinfinity function g and ANY closed set C, there exists a Cinfinity function f such that {x: f(x) = g(x)} = C. Simply find a Cinfinity function h such that Z(h) = C and put f = gh.
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 nonanalytic 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 Cinfinity function. Zahorski gave a short argument that any Cinfinity 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 nonisolated 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 194041 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 Cinfinity 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 CINFINITY FUNCTIONS ARE NOWHERE ANALYTIC"
BernalGonzález [2] proved the following. Let {a_n} and {b_n} be sequences of positive real numbers. Then the Baire typical Cinfinity function has the property that for each x in R we have
liminf(n > oo) [ a_n *  n'th derivative of f at x  ] = 0 and limsup(n > oo) [ b_n *  n'th derivative of f at x  ] = oo.
COROLLARY: The Bairetypical Cinfinity 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 BernalGonzález's result shows is that not only is every point a (P)point for the Baire typical Cinfinity 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 Cinfinity 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 Cinfinity function is nowhere analytic: Given a compact interval J with midpoint b, the collection of Cinfinity functions f whose Taylor series about x=b converge to f on J is a closed proper (vector) subspace of all Cinfinity 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 Cinfinity function has a dense set of points of nonanalyticity. Note that this proof doesn't tell us anything about the (C)points and (P)points of the prevalent Cinfinity 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. Lódzk. Nauki Mat. Przyrodn. Ser. II Zeszyt 52 Mat. (1973), 85108. [MR 48 #11428; Zbl 264.26016]
[2] Luis BernalGonzález, "Funciones con derivadas sucesivas grandes y pequeñas por doquier" [Functions with successive derivatives everywhere large or small], Collect. Math. 38 (1987), 117122. [MR 90c:26013; Zbl 661.26009]
[4] Ralph P. Boas, "When is a Cinfinity function analytic?", Mathematical Intelligencer 11(4) (1989), 3437. [MR 91k:26023; Zbl 704.41021]
[8] Frank S. Cater, "Most Cinfinity functions are nowhere Gevrey differentiable of any order", Real Analysis Exchange 27 (200102), 7779.
[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. 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), 149166. [MR 98a:26026; Zbl 877.26014]
[29] Józef Siciak, "Punkty regularne i osobliwe funkcji klasy Cinfinity" [Regular and singular points of Cinfinity functions], Zesz. Nauk. Politech. Sl., Mat.Fiz. 48(853) (1986), 127146. [Zbl 777.26020]
[33] Helene Zahorska, "Über die singulären punkte einer funktion der klasse Cinfinity" [On the singular points of functions of class Cinfinity], Acta Math. Acad. Sci. Hungar. 15 (1964), 7794. [MR 28 #3129; Zbl 127.28502]
[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), 183245. [A correction to a remark Zahorski had made about a result proved by Ganapathy Iyer is given in Fund. Math. 36, (1949), 319320 (MR 11, 718a; Zbl 38.04201).] [MR 10, 23c; Zbl 33.25504]
ADDITIONAL REFERENCES FOR THIS ESSAY
[36] Thøger Bang,"Sur les points singuliers (dans un sens généralisé) des fonctions indéfiniment dérivables" [On the singular points (in a generalized sense) of infinitely differentiable functions], pp. 259263 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), 301309. [MR 684.26009; Zbl 90e:26040]
[38] Jack B. Brown, "Restriction theorems in real analysis", Real Analysis Exchange 20 (199495), 510526. 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 (199394), 414432. [MR 95h:26002; Zbl 822.26003]
[40] William Burnside, "Note on functions of a real variable", Messenger of Mathematics 23 (189394), 3942. [Burnside gives an example of a nowhere analytic Cinfinity function.] [JFM 25.0731.02]
[41] Richard B. Darst, "Zero sets of functions from nonquasianalytic classes", Proc. Amer. Math. Soc. 40 (1973), 543544. [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 nonquasianalytic classes", Proc. Amer. Math. Soc. 27 (1971), 539542. [Erratum in Proc. AMS 39 (1973), 651.] [MR 42 #7846; Zbl 208.07401]
[44] Lucjan Meres, "On the singular points of PringsheimDu BoisReymond of a function of two variables", Demonstratio Math. 15 (1982), 297310. [MR 84m:26025; Zbl 511.26010]
[45] Alexander M. Olevskii, "UlamZahorski problem on free interpolation by smooth functions", Trans. Amer. Math. Soc. 342 (1994), 713127. [MR 94f:26007; Zbl 801.26006]
[46] Hongjian Shi, "Prevalence of some known typical properties", Acta Math. Univ. Comenianae 70 (2001), 185192.
http://www.emis.de/journals/AMUC/_vol70/_no_2/_shi/shi.pdf http://www.maths.soton.ac.uk/EMIS/journals/AMUC/_vol70/_no_2/_shi/shi.ps.gz
[47] Pawel Walczak, "A proof of some theorem on the Cinfinity functions of one variable which are not analytic", Demonstratio Math. 4 (1972), 209213. [MR 49 #504; Zbl 253.26011]
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