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ESSAY ON NOWHERE ANALYTIC CINFINITY FUNCTIONS
Posted:
May 9, 2002 6:18 PM


This is an essay on nowhere analytic Cinfinity realvalued 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/apcalc/kelddyshon
CONTENTS
I. TWO WAYS THAT A CINFINITY FUNCTION CAN FAIL TO BE ANALYTIC II. EARLY HISTORY OF NOWHERE ANALYTIC CINFINITY FUNCTIONS III. ZAHORSKI'S 1947 CHARACTERIZATION IV. MOST CINFINITY FUNCTIONS ARE NOWHERE ANALYTIC V. UNEXPLORED AREAS AND OTHER RESULTS VI. REFERENCES
Dave L. Renfro
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I. TWO WAYS THAT A CINFINITY FUNCTION CAN FAIL TO BE ANALYTIC
Every Cinfinity 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 nonanalytic points of a Cinfinity function into the following two categories. A point of nonanalyticity is called a (C)point (for Cauchy) if it belongs to (C) above. A point of nonanalyticity 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. 189190 of Boas [5]. Cauchy (1823) was the first to give an example of a (C)point (the same example given above), while Du BoisReymond (1876) was the first to give an example of a (P)point. Du BoisReymond'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 BoisReymond'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 BoisReymond 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 BoisReymond'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 CINFINITY FUNCTIONS
Even in the late 1800's it was well known that the set of nonanalytic 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 Cinfinity function, it is only necessary to find a Cinfinity function having a dense set of nonanalytic points.
The first published example of such a function was given by Du BoisReymond (1883), who used the method of condensation of singularities to construct a Cinfinity function having a dense set of (P)points. [See p. 116 of the Du BoisReymond paper mentioned above.] CellÃÂÃÂ©rier had previously obtained an example of a Cinfinity 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 BoisReymond's proof, CellÃÂÃÂ©rier had no gaps in the divergence part of his proof.
CellÃÂÃÂ©rier's function is
SUM(n=1 to infinity) of [ sin(c^n * x) ] / n!,
where c > 1 is an integer. CellÃÂÃÂ©rier'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 CellÃÂÃÂ©rier's function. [Enter "20.0380.01" into the "JFM. no." window, as above. Also, see VÃÂÃÂ½bornÃÂÃÂ½ [31] and p. 119 of Bilodeau [3].]
In 1893 Pringsheim gave various examples whose verifications are easier, as well as examples of Cinfinity 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 Cinfinity functions. Let f(z) be a complexvalued function of a complex variable z that is analytic in an open disk and Cinfinity 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 complexvalued function that is Cinfinity and nowhere analytic in the interval Pi < x < Pi. [If you want a realvalued 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 Cinfinity were well known.
For more information on the early history of nowhere analytic Cinfinity functions, see Bilodeau [3]. For a general survey of nowhere analytic Cinfinity functions, including issues relating to Zahorski's theorem discussed below, see Salzmann/Zeller [26]. An elementary verification of a nowhere analytic Cinfinity function is given in Merryfield [21], a paper that you can download at
http://www.mathcs.cmsu.edu/~mjms/19923p.html [48 K .ps file] http://www.mathcs.cmsu.edu/~mjms/19923d.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 Cinfinity 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 Cinfinity function such that every point is a (P)point was given by Cartan in 1940 ([6], pp. 2022). In 1947 Zahorski [35] gave the following characterization for the nonanalytic points of a Cinfinity 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 Cinfinity 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 Cinfinity 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 Cinfinity, 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 Cinfinity function having a (P)point everywhere: Choose C to be empty and P = R.
** Still another corollary is that there exist Cinfinity functions belonging to each of the following categories (recall Pringsheim's examples in Section II):
(i) C is cdense in R and P is empty. (ii) C is empty and P is cdense in R. (iii) Both C and P are cdense 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 CINFINITY FUNCTIONS ARE NOWHERE ANALYTIC
The Baire category theorem can be used to prove the existence of nowhere analytic Cinfinity functions, and this has been rediscovered several times. Using the standard metric on Cinfinity (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 Bairetypical Cinfinity function is nowhere analytic. An outline of a proof can be found on pp. 301302 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. 9597 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 Bairetypical 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 quasianalytic classes have a Bairetypical set of functions that are nowhere quasianalytic relative to other quasianalytic 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 Bairetypical multidimensional 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 Bairetypical Cinfinity function is nowhere analytic. [Bernal cites Cater, Christensen, and Darst in his bibliography.] Finally, Ramsamujh [23] (1991) proves that the Bairetypical Cinfinity 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 Bairetypical Cinfinity 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 Bairetypical Cinfinity 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 Bairetypical Cinfinity function has a Bairetypical 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 Bairetypical Cinfinity function. Thus, the Bairetypical Cinfinity 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 Bairetypical Cinfinity 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 Cinfinity 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 porositytypical Cinfinity function or for the prevalent Cinfinity function (i.e. the complement of a Christensen null set; see ).
An investigation for the porositytypical Cinfinity 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 Cinfinity 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/KuratowskiUlam 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 Cinfinity functions are Fabius [14], Kabaya/Iri [17], and Fournier/Gauthier [15]. In the first two papers certain distribution functions are shown to be Cinfinity and nowhere analytic for any sequence of independent and uniformly distributed random variables. The last paper proves that the Bairetypical complexvalued sequence gives rise to a Maclaurin series whose radius of convergence is 0. Results related to this last paper can also be found in SalÃÂÃÂ¡t/TÃÂÃÂ³th [24] and SÃÂÃÂ¡lat/Taylor/TÃÂÃÂ³th [25].
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VI. REFERENCES
[1] Teresa Bartczak, "The Cauchy singular points of a function of several variables" (Polish), Zeszyty Nauk. Uniw. LÃÂÃÂ³dzk. Nauki Mat. Przyrodn. Ser. II Zeszyt 52 Mat. (1973), 85108. [MR 48 #11428; Zbl 264.26016]
[2] Luis G. Bernal, "Functions with successive derivatives everywhere large or small" (Spanish), Collect. Math. 38 (1987), 117122. [MR 90c:26013]
[3] Gerald G. Bilodeau, The origin and early development of nonanalytic infinitely differentiable functions, Arch. Hist. Exact Sci. 27 (1982), 115135. [MR 84g:26017; Zbl 503.01005]
[4] Ralph P. Boas, "When is a Cinfinity function analytic?", Mathematical Intelligencer 11(4) (1989), 3437. [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 dÃÂÃÂ©finies par des inÃÂÃÂ©galitÃÂÃÂ©s portant sur leurs derivÃÂÃÂ©es successives" [On classes of functions defined by inequalities holding on their successive derivatives], ActualitÃÂÃÂ©s 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), 618624. [MR 86b:26034; Zbl 598.26035]
[8] Frank S. Cater, "Most Cinfinity functions are nowhere Gevrey differentiable of any order", Real Analysis Exchange 27 (200102), 7779.
[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", JulyAugust 1971, Aarhus University, Denmark], 2631. [MR 52 #6685; Zbl 241.28008]
[10] Richard B. Darst, "Most infinitely differentiable functions are nowhere analytic", Canad. Math. Bull. 16 (1973), 597598. [MR 49 #10838; Zbl 285.26015]
[11] Richard B. Darst, "Localization properties of basic classes of Cinfinity functions", Proc. Amer. Math. Soc. 46 (1974), 2428. [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), 15. [MR 85h:26028]
[14] J. Fabius, "A probabilistic example of a nowhere analytic Cinfinityfunction", Z. Wahrscheinlichkeitstheorie Verw. Geb. 5 (1966), 173174. [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), 3940. [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. 378B]
[17] Keiko Kabaya and Masao Iri, "Sum of uniformly distributed random variables and a family of nonanalytic Cinfinityfunctions", Japan J. Appl. Math. 4(1) (1987), 122. [MR 89d:26023; Zbl 625.60054]
[18] Sung S. Kim and Kil H. Kwon, "Smooth (Cinfinity) but nowhere analytic functions", Amer. Math. Monthly 107 (2000), 264266. [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), 155158. [Errata in Annals 68 (1958), 202] [MR 19, 551d; Zbl 78.08104]
[20] Lawrence E. May, "On Cinfinity functions analytic on sets of small measure", Canad. Math. Bull 12 (1969), 2530. [MR 39 #2933; Zbl 176.01201]
[21] Kent G. Merryfield, "A nowhere analytic Cinfinity function", Missouri J. Math. Sci. 4(3) (1992), 132138.
[22] Dietrich Morgenstern, "Unendlich oft differenzierbare nichtanalytische funktionen" [Infinitely differentiable nonanalytic functions], Math. Nachrichten 12 (1954), page 74. [MR 16, 342b; Zbl 58.28902]
[23] Taje I. Ramsamujh, "Nowhere analytic Cinfinity functions", J. Math. Analysis Appl. 160 (1991), 263266. [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 Cinfinity 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 SalÃÂÃÂ¡t and JÃÂÃÂ¡nos T. TÃÂÃÂ³th, "On radii of convergence of power series", Bull. Math. Soc. Sci. Math. Roum. 38 (199495), 183198. [Zbl 886.40002]
[25] Tibor SalÃÂÃÂ¡t, S. James Taylor, and JÃÂÃÂ¡nos T. TÃÂÃÂ³th, "Radii of convergence of power series", Real Analysis Exchange 24 (199899), 263273. [MR 2000h:40004]
[26] Helmut Salzmann and Karl Zeller, "SingularitÃÂÃÂ¤ten unendlich oft differenzierbarer funktionen" [Singularities of infinitely differentiable functions], Math. Zeitschrift 62 (1955), 354367. [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 MathÃÂÃÂ©matique UniversitÃÂÃÂ© de LiÃÂÃÂ¨ge, 1995, 8 pages.
[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] JÃÂÃÂ³zef Siciak, "Regular and singular points of Cinfinity functions" (Polish), Zesz. Nauk. Politech. Sl., Mat.Fiz. 48(853) (1986), 127146. [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 Cinfinity function is given in Chapter 5.] [MR 95h:60001; Zbl 803.60001]
[31] Rudolf VÃÂÃÂ½bornÃÂÃÂ½, "A Cinfinity nowhere analytic function. A forgotten example from 1888", Austral. Math. Soc. Gaz. 15(1) (1988), 78. [MR 89d:26025]
[32] Yohei Yamazaki, "A simple example of a Cinfinity function that is nowhere analytic" (Japanese), Sugaku 27 (1975), 366367. [MR 58 #31185]
[33] Helene Zahorska, "Über die singulÃÂÃÂ¤ren 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]
[34] Zygmunt Zahorski, "ProblÃÂÃÂ¨mes de la thÃÂÃÂ©orie des ensembles et des fonctions" [Problems in the theory of sets and of functions], C. R. Acad. Sci. Paris 223 (1946), 449451. [MR 8, 141c; Zbl 60.13705]
[35] Zygmunt Zahorski, "Sur l'ensemble des points singuliers d'une fonction d'une variable rÃÂÃÂ©elle admettant les dÃÂÃÂ©rivÃÂÃÂ©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), 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]
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