A family of singular functions and its relation to harmonic fractal analysis and fuzzy logic

Abstract We study a parameterized family of singular functions which appears in a paper by H. Okamoto and M. Wunsch (2007). Various properties are revisited from the viewpoint of fractal geometry and probabilistic techniques. Hausdorff dimensions are calculated for several sets related to these functions, and new properties close to fractal analysis and strong negations are explored.


Introduction and preliminaries
Examples of singular functions, that is, monotone increasing continuous functions whose derivatives vanish almost everywhere, have been known since the end of the 19th century (see [1]). Since then, these functions have been studied from a wide variety of fields very distant from one another. Some classes of these functions have been considered in Probability Theory (see [2][3][4]) as well as in Number Theory, where what is known as Minkowski's question mark function is specially relevant (see [5][6][7][8]). It relates to the alternate dyadic and continuous fraction systems of representation. Another example of singular function that relates to representation number systems can be found in [9]. Possibly the best known and most widely studied singular function is Cantor's (see [10] and the references therein) which can be studied with the aid of the 2-or 3-base representation systems, although it is often geometrically built as the limit of a sequence of functions with polygonal graphs. This is also the case of the functions firstly studied by Cahen [11], which Salem [7] introduced using geometric ideas similar to those of Cantor. Other references related to these functions can be found in [7,[12][13][14][15][16].
In recent times, a parameterized family of continuous functions has been considered by Okamoto in [17], and revisited in [18] to see if they are also singular. They contain Bourbaki and Perkins' nowhere differentiable functions as well as Cantor's singular function. In this paper, we study a wide family of two-parametric singular functions f a;b and explore new properties, several of them closely related to fractal analysis and strong negations. We borrow a few ideas from [18,Sec.4], starting with an exact definition of what we mean by f a;b . Definition 1.1. Let a; b 2 0; 1OE and f a;b be defined on the unit interval OE0; 1 by iterations of piecewise affine functions ff n g 1 nD0 , as follows: Let f 0 .x/ D x, and suppose that f n has been properly defined on the whole unit interval. Then, for f nC1 we define This paper is structured in the following way: In Section 2 we show new properties of f a;b ; from the viewpoint of fractal geometry theory. We prove that the graph of f a;b is a compact set (an attractor) that appears as the fixed point of a suitable contraction mapping, which directly implies its monotony and continuity. Moreover, we characterize f a;b as the unique bounded function satisfying a given system of functional equations. We will provide further proof of the singularity of f a;b with the sole aid of probabilistic techniques, using the result that the sequence of Fourier coefficients of its associated measure df a;b does not converge to zero (for a given monotone function S, dS denotes its Stieltjes measure). In the same section, we also establish the Hausdorff dimension of sets related to f a;b , one of them is a measure zero set whose image by f a;b has measure one, and its dimension is obtained as an application of the Besicovitch-Eggleston theorem. To this end, we previously introduced a representation system called a; bsystem. In Section 3 we generalize all the results to a 4-parametric family of functions. Once more, we calculate Hausdorff dimensions for the sets associated with these functions again. The last section is devoted to applications of the family of singular functions studied. With these and the representation system referred to above, we find a bi-parametric family of Katok foliations. In addition, we establish its relation to harmonic analysis on fractals and to strong negations in fuzzy logic.
2 Properties of f a;b

Analytic properties
First, as we pointed out above, we examine new properties of f a;b from a geometric point of view. But the tool we shall use is the Hausdorff-Pompeiu metric (see [20,21] Thus, the graph of f a;b is a self-affine subset of the unit square OE0; 1 2 ; obtained as the fixed point of a suitable contraction C in the space K OE0; 1 2 Á , via the Contaction Mapping Theorem. The underlying idea is graphically expressed in Fig. 1. The graph of the polygonal f n is defined by iteration as the image, by a contraction C that is defined below, of the graph of f n 1 with the start f 0 D identity.
For convenience, let us put a 1 D aI a 2 D b aI a 3 D 1 b: where C 1 ; C 2 and C 3 are contractions given by (1) (a; b; a 2 ; a 3 2 0; 1OE) is a contraction for the Hausdorff metric in K OE0; 1 2 Á .
The graph of f a;b is the unique invariant set for the iterated function system defined by (1), that is the fixed point of C . The next result follows as a consequence of the definition of f a;b and its graph is a compact set.  In [22], for Okamoto's one-parameterized function it is proved that f a D f a;1 a is non-differentiable almost everywhere at the critical parameter value. In [19], the performance of f 0 a is explored almost everywhere for different values of a. For f a;b ; we have the following general result. Proof. If f 0 a;b .0 C / exists, then it must vanish. To prove it, we take into account that f a;b 1 3 n D a n and f a;b 2 3 n D ba n 1 . This implies that the slope of the segments joining the point .0; 0/ with 1 3 n ; a n , 2 3 n ; ba n 1 and 1 3 n 1 ; a n 1 Á are, repectively, .3a/ n , 3b.3a/ n 1 2 and .3a/ n 1 . If f 0 a;b .0 C / exists and does not vanish, then the limits of the quotients .3a/ n .3a/ n 1 D 3a and 3b.3a/ n 1 2.3a/ n 1 D 3b 2 exist and are equal to 1. That is, .a; b/ D 1 3 ; 2 3 which is a contradiction.
A similar argument allows us to obtain f 0 a;b .1 / D 0, if it exists. By repeting this reasoning or using the system of functional equations (2), it follows that if exist, the numbers Let us suppose that x cannot be expanded in the form m 3 k : If f 0 a;b .x/ exists and does not vanish, for sequences .x k / and .y k / that converge to x such that x k Ä x < y k : Let us take x k WD max j 3 k W j D 0; : : : ; 3 k and and y k WD min j 3 k W j D 0; : : : ; 3 k and x < j 3 k ; then we have Thus, if 3a i ¤ 1 for the three values, this is a contradiction. If some of them equal 1, then the other two differ from 1, because we have avoided the case a D 1=3 and b D 2=3. Therefore, if we obtain 3a i D 1, we shall change that quotient by This quotient is 3.a i Ca j / 2 ; with i ¤ j , and differs from 1. For each k we have described the possibility of taking the quotient on a finite set (where 1 is not included). Consequently, the limit f 0 a;b .x/ must be zero.
Note that the theorem above provides a new proof for the singularity of f a;b :

Singular functions as the convolution of distribution functions
We shall prove that f a;b can be obtained as an infinite convolution S of atomic probabilities, and that the sequence of Fourier coefficients of dS does not converge to zero. A number of definitions and results to be used later are recorded bellow. If F 1 and F 2 are distribution functions, then the function F .x/ D R R F 1 .y x/dF 2 .y/ is called the convolution of the distribution functions F 1 and F 2 . This is a new distribution function denoted as F D F 1 F 2 . The convolution F 1 F 2 provides the distribution function of the sum of two independent random variables with distribution functions F 1 and F 2 .
Let F be a distribution function, then its characteristic function e F is the expected value of e ixt , that is, By imposing conditions on the distribution functions, Jessen and Wintner in their paper [3], obtain convergence criteria for the convolution of distribution functions. To be more specific, in [3] they obtain the following results.
Lemma 2.6 ([3, Th. 4]). If M 2 .F n / is finite for every n; then the convergence of two series is a convergent infinite convolution of distribution functions F n each of which is purely discontinuous, then F is either purely discontinuous, or singular, or absolutely continuous.
The next result can be found in [23, pg. 46].
Finally, in [26] the following useful result is proven.
Lemma 2.9. If F n and F have characteristic functions e F n and e F , respectively, and .F n / is weakly convergent to F , then e F n .x/ ! e F .x/ for each x.
Definition 2.10. Let be the infinite convolution function S WD F 1 F 2 F 3 , where F 1 ; F 2 ; F 3 ; ::: are distribution functions given, for each positive integeer n, by This function S is well defined. In fact, let us note that, for every n: .
The convergence of the corresponding series ensures, by Lemma 2.6, the weak convergence of the convolution, that is, S.x/ exists for every x.
Theorem 2.11. With the notation as in the above definition, we have: b) The sequence of Fourier coefficients of e S, i.e. S p D R 1 0 e 2 pix dS.x/, does not converge to zero. c) S is a singular function.
Proof. a) The characteristic function of F n is a 1 C a 2 e i t 3 n C a 3 e i 2t 3 n . If H D F G; then their respective characteristic functions satisfy the relation e H D e F e G; and as a consequence, by Lemma 2.6, we obtain that the b) Let us denote by S p WD e S .2 p/ the Fourier coefficients of dS . The expression of e S allows us to obtain that S 3 q p D S p for all q 0: Since the measure is not equal to the Lebesgue measure, then there exists n > 0 such that S p ¤ 0; and, as a consequence, the Fourier coefficients do not converge to zero.
c) The functions in the convolution are purely discontinuous, thus S is also pure. But it is neither discontinuous, by Theorem 2.8, nor absolutely continuous, because by b) it does not satisfy the Riemann-Lebesgue theorem. Therefore, it must be singular.
For a nonnegative integer m < 3 n ; let us consider its expansion in the base 3 m D r 0 3 0 C r 1 3 1 C r 2 3 2 C C r n 1 3 n 1 ; with r i 2 f0; 1; 2g : Set c.m/ (resp., u.m/ and d.m/) the number of times that the digit 0 appears (resp., 1s and 2s) in the above expansion of m.
Lemma 2.12. The distribution function F 1 F 2 F n has the following associated probability Proof. The proof is reached by induction. The statement is true for the first values. Let us suppose that it is true for n 1, and we will prove it for n. (Let us denote by P n the associated probability with F n .) If m is congruent with zero module 3, i.e. m D 3k; then Taking into account that the expansion of m has one less 0 than the expansion of k; and the same number of 1s and 2s, by applying the induction hypothesis we obtain the desired result. For the cases m Á 1 (mod 3) and m Á 2 (mod 3) we proceed in a similar way.
Applying the above lemma and making an inductive hypothesis we obtain the following result.
Now, we show the relationship between f a;b and S: Theorem 2.14. f a;b is the infinite convolution function S .
Proof. We have to prove that the graph of G n D F 1 F 2 F n and C n .d / are the same set, where d denotes the segment .0; 1/ .1; 1/ and C is given by (2.1), but for a finite set of points. Namely, the intersection of C n .d / and the line x D k 3 n for 1 Ä k Ä 3 n 1 has two points: k 3 n ; G n k 3 n ÁÁ and k 3 n ; G n k 3 n

ÁÁ
; where the minus sign means left-side limit: For the other values of x; the graph of G n and the set C n .d / coincide. Set C n .x/ WD max fy W .x; y/ 2 C n .d /g : It is immediate that Thus we will show that for n; it follows that C n . k 3 n / D G n . k 3 n / for 0 Ä k < 3 n : Let us suppose that it is also true for n 1: ; the proof is similar to the above. c) If G n mC2:3 n 1 3 n ; the proof also follows similarly.
This ensures that the statement is also true for n. If G n are step functions that coincide with f a;b at points in the form m 3 n with 0 Ä m < 3 n ; then lim In fact, f a;b is a singular function and additionally we know its characteristic function, as well as the fact that the sequence of its Fourier coefficients does not converge to zero.

Fractal sets associated with f a;b
This section is devoted to describing several sets related to the function f a;b and to computing their Hausdorff dimensions. We need some additional results.
We are going to introduce a representation system of real numbers in OE0; 1 :The method employed is based on the imitation of the action of f a;b on the Y axis. With notations already used above, let us divide the unit interval into subintervals OE0; a ; OEa; b and OEb; 1. Thus, if y 2 OE0; 1 ; then y can be written as one of the following expressions: , and z 2 OE0; 1. We can now apply the above algorithm to z 2 OE0; 1, and by iteration we have the formal relation Observe that if y WD b 2 C a 2 z, with z 2 f0; 1g, then y has two formal equalities. But this fact is of no relevance to our task because it is true on a denumerable set, hence on a set of measure zero. The nature of the construction also ensures that for different points their corresponding formal equalities are different, as well.
converges to y, that is, the formal equality above is in fact an equality.
Proof. Note that s 1 : : : s n d nC1 Ä max fa 1 ; a 2 ; a 3 ; bg nC1 . Thus, by construction which implies that the series converges to y: Next, the family of parameterized singular functions is described with the help of the a; b-representation system. Applying the Law of Large Numbers we obtain the following result.
Corollary 3.4. With the above notation the set of points whose proportion of times that d n D 0 is a 1 , for d n D a is a 2 times or d n D b is a 3 ; has measure 1.
We are now going to exhibit a set of measure zero whose image under f a;b is of measure one and vice versa. x/ entails that if c n D 0; 1; 2; then d n D 0; a; b, respectively. Therefore, the set of points whose proportions of 0s, 1s and 2s are a 1 ; a 2 and a 3 ; respectively, has measure zero and is mapped on a set of measure 1.
In addition, we shall give another analogue property using the measure of the normal numbers in OE0; 1 : We recall that a number is normal in the 3-base if the proportions of 0s, 1s and 2s in this base is 1/3 for each case, and that the set of normal numbers in the 3-base is of measure 1 (see, for instance, [24] or [25]). Theorem 3.6. There exists a set of measure one that is mapped by f a;b on a set of measure zero and Hausdorff dimension ln 27 ln .a 1 a 2 a 3 / : Proof. A normal number in the base 3 has the same proportion for each of the three digits in its expansion. Their images will have the same proportion in the random variables d n : Corollary 3.4 shows that the proportions must be a 1 ; a 2 and a 3 , on a set of measure 1. Thus, the image of normal numbers is a measure zero set.
To calculate the Hausdorff dimension for this zero-measure set, we use that the measure df a;b concentrates its mass on it. Thus, applying [20,Lemma 4.9], by substitution of balls by neigbourhoods in the following way: If then we use the representation method and, in the n-th iteration, we do z D 0 and z D 1; obtaining the extremes of an interval containing x, namely: OEd 1 C C s 1 s 2 : : : s n 1 d n ; d 1 C C s 1 s 2 : : : s n 1 d n C s 1 s 2 : : : s n : Clearly, its Lebesgue measure is s 1 s 2 : : : s n , and, by construction, its df 1 ba -measure is 1 3 n . Therefore, the Hausdorff dimension is given by the number

Generalization
Although their description is complex, a geometric generalization of the preceding functions can be easily carried out. We directly proceed in the following way, avoiding proofs in the results that follow. They can be derived from the techniques and ideas already used in the proofs of the preceding results.  :

Applications
In this section we apply the main results obtained in this paper to provide nontrivial examples in Measure Theory and Fractal Analysis. On the one hand, we will construct an example of Katok foliation, and on the other hand, we study singular functions and their interaction with harmonic functions defined on a self-similar set.

Katok Foliation
The first example of a pathological foliation was constructed by Katok. A different version of this construction on the square appeared in Milnor's work [27], which showed examples of foliations of the unit square such that a full measure set intersects each leaf of the foliation at exactly one point.
Here we introduce all the necessary notions required for the precise formulation of our results.
; then g x .t / is obtained by the substitution of d i ; s i with one of the values 0; t; t 2 ; 1 t; t .1 t /, depending on d i . For each substitution, we obtain an expansion series in powers of t . It is clear that g x .0/ D 0 and g x .1/ D 1 W the series expansion begins with a term in the t n power, thus g x .0/ D 0. For the latter, we can write in the form t n C a series of positive terms, with 0 Ä g x .t / Ä 1: Hence, by continuity, the equality follows. Proof. If .t; g x .t// and .t; g x 0 .t // are equal, the series expansion in the t 2 ; t-representation system of x and x 0 is the same. But this would imply x D x 0 , a contradiction. Proof. Singularity follows from the above results. Let x 2 0; 1OE : The function g x is continuous with g x .0/ D 0 and g x .1/ D 1, and as a consequence, it takes all the values 0 < y < 1. and g x defined as above, is a Katok foliation.
Proof. E is a set of measure 1 by Fubini's theorem. The analyticity was already obtained above. To deduce that the graph of each function in the foliation intersects at most at one point is a consequence of the fact that the variables d i (that only depend on x) appear in the same proportion in the image of g x : Because this proportion is a limit, it is possible that it does not exist. If it exists, then we find a value t with the desired proportions.
Remark 5.6. In exchange of t 2 ; t for t u ; t v with u > v; one can obtain a bi-parametric family of foliations.

Fractal Analysis
Applications of fractal sets in physics phenomena have brought about an important development of fractal techniques in the last twenty years, specifically in harmonic functions in fractals. A good introduction to this topic can be found in [28]. A study of the harmonic functions on the Sierpinski triangle can be found in [29]. In this subsection we study the harmonic functions on these sets and their relationship with singular functions which are the object of our research in this paper. We focus our study on a self-similar subset in the unit square OE0; 1 2 ; which is determined as the fixed point of a suitable contraction in the metric space K OE0; 1 2 Á ; endowed with the Hausdorff metric.
Definition 5.7. For r such that 0 < r < 1=2; let us set functions: and let The only fixed point of G will be denoted by N r .
N r is a fractal set, and we will study harmonic functions on it. This type of sets has already been used to show the first example of copulas with fractal support (see for instance [30] or [31]). Let us recall that the fixed point N r in the Banach contraction mapping is obtained as the limit by iterations of G starting at any choosen point. Taking as the starting point the compact unit square, for r D 1=3, the first two are as Figure 2 shows. We name the vertices of OE0; 1 2 : p 1 D .0; 1/ ; p 2 D .1; 1/ ; p 3 D .0; 0/ and p 4 D .1; 0/ ; and for the vertices in the first iteration: q 1 D .r; 1/ q 2 D .1 r; 1/ q 3 D .0; 1 r/ q 4 D .r; 1 r/ q 5 D .1 r; 1 r/ q 6 D .1; 1 r/ q 7 D .0; r/ q 8 D .r; r/ q 9 D .1 r; r/ q 10 D .1; r/ q 11 D .r; 0/ q 12 D .1 r; 0/ (see Figure 3).
For the sake of our paper, we state that a function h defined in C is harmonic if : And the same occurs on each of the squares that appear at each stage. G can be extended to C by continuity, because the set of vertices of the squares form a dense set. These functions will be denoted by h s 1 s 2 s 3 s 4 . These functions and the rest that follows in this section, depend on the parameter r, but we omit it for simplicity. We will study their performance on the diagonal of the set, and we will do it for the particular case of h˛1 0˛; which we denote by hf or short. Actually, the only condition we impose is that s 1 D s 4 D˛; because it is necessary to take it into account: When we are restricted to the diagonal, it is necessary to study the subsets G 1 .OE0; 1 2 /, G 3 .OE0; 1 2 / and G 5 .OE0; 1 2 / W they are the new squares having a non-empty intersection with it. The self-similarity of the set and the way the function takes the values at the vertices of the new squares, allow us to write the following equations: and if˛D 1=2, then they become the functional equations for g 1 2 : 8 < : x/ : Now attending to the functional equations of f r;1 r (with 0 < r < 1=2) we have the following result. Proof. The equalities already established for the family of functions g˛ensure the way to divide the unit interval OE0; 1 into subintervals where the corresponding function is the scale copy of g 1 2 : Thus, we complete the proof, because the inverse of a singular function is also singular.
Corollary 5.11. There exists a set of measure zero and Hausdorff dimension ln 27 ln r 2 .1 2r/ that is mapped on a set of measure one by g˛.

Strong Negations
In the framework of Fuzzy Logic, a strictly decreasing mapping n W OE0; 1 ! OE0; 1 is a strong negation if n 2 .x/ WD n ı n .x/ D x for all x in OE0; 1 (see [32]). Next, we are going to introduce a family of strong negations related to the functions we have studied.
Proof. Continuity and monotony are evident. Through functional equations (3), we deduce that n a;b is the only function defined in OE0; 1 that is bounded and satisfies the relations: The others follow is an analogous way. Finally, its uniqueness follows from that of f ab.1 b/.1 a/ in (3) or from the fixed point Banach theorem.
To show that n 2 a;b .x/ D x it is sufficient to use equations (4). From them, we deduce that n 2 a;b satisfies the functional equations However, the solution of this system is unique in the unit interval, and this is the identity function. Thus, n 2 a;b equals identity in OE0; 1.

Conclusions
We have studied a wide family of two-parametric singular functions. To this end, we introduce a biparametric representation system that allows us to give an explicit expression of these functions and to calculate the Hausdorff dimension of several sets with distinguished properties for f a;b .
In addition, we have examined some applications of the study of these families of singular functions obtaining Katok foliations, results concerning its relations with fractal harmonic analysis and with strong negations.
For further investigations we are interested in the study of these functions in the context of homeomorphisms between the supports of copulas with fractal support [31], as they appear in the disintegration of the measure associated to some self-similar copulas, and in the study of functions generated as f a;b ; but with random a and b.