Abstract
A fractal interpolation function on a -series local field is defined, and its -type smoothness is shown by virtue of the equivalent relationship between the Hölder type space and the Lipschitz class Lip. The orders of the -type derivatives and the fractal dimensions of the graphs of Weierstrass type function on local fields are given as an example. The -fractal function on is introduced and the conclusion of its smoothness is improved in a more general case; some examples are shown to support the conclusion. Finally, a comparison between the fractal interpolation functions defined on and is given.
1. Introduction
As we know, the traditional method for analyzing given experiment data is by representing data graphically as a subset of ; then the graphical data are analyzed by some geometrical or analytical tools to seek a function with graph , whose values are a good fit to the data over the interval ; this is an interpolation problem. Generally, a function need to satisfy the following: (1) it fits the data at each point ; that is, ; (2) it is some simple function, such as polynomial; (3) it has some order smoothness; for example, “spline interpolant” is smooth in one or two order.
In this paper, we present a new idea: to establish interpolation theory on local fields. We have found that there are not only new concepts but also many new methods in this new branch—interpolation theory on local fields. We may construct a fractal interpolation function that satisfies the above (1), (2), (3), by virtue of the harmonic analysis theory and fractal analysis theory on local fields [1]. However, “some simple function” in (2) and “some order smoothness” in (3) all have new senses which are quite different from those in case.
Barnsley et al. [2–5] defined the fractal interpolation functions on by virtue of the iterated function system.
Definition 1 (see [2]) (IFS, HIFS). Let be a complete metric space with distance for . Let be a given positive integer and let for be continuous mappings; then we call an iterated function system (IFS for short). If, for all , there exists an such that that is, each is contractive with contractive factor ; then the IFS is termed a hyperbolic iterated function system (HIFS for short).
Let be the set of all nonempty compact subsets of the complete metric space . Then is a complete metric space with the Hausdorff metric Define by where . We call the attractor of the IFS if The attractor of a HIFS is the unique set satisfying where is the -fold composition of .
Let be a partition of a closed interval . Set for . Suppose that a set of interpolation points is given, where . Let , , be contractive homeomorphisms such that with some .
Let be continuous mappings satisfying where , , , and .
Set . Define the mappings for by Then constitutes an IFS.
Theorem 2 (see [2, Theorem 1]). The IFS defined above admits a unique attractor , which is the graph of a continuous function passing through the interpolation points .
Definition 3 (see [2]) (FIF). The function described in Theorem 2 is referred to as a fractal interpolation function (FIF for short), associated with the IFS .
Several conclusions on the differentiability of the FIFs defined on have been given out. Barnsley and Harrington [5] introduced the calculus of FIFs. Chen [6] gave some conditions under which the equidistant FIFs, defined by the affine IFS, are nowhere differentiable. Sha and Chen [7] investigated the Hölder smoothness of a class of FIFs and their logical derivatives of order . Chen [8] investigated the smoothness of nonequidistant FIF and obtained the Hölder exponents of such FIFs. Wang [9] investigated the differentiability of the equidistant FIFs generated by the nonlinear IFS. Li et al. [10] obtained the sufficient conditions of Hölder continuity of two kinds of FIFs and proved the sufficient and necessary condition for their differentiability. Navascués [11] introduced -fractal functions and gave some conditions under which the set of their nondifferentiable points is dense in the domain when the scaling factors have the same value. Certainly, differentiability of FIF is important and so that attracts eyes of mathematicians.
By the construction of FIFs, it is reasonable to establish the interpolation theory on local fields since the structures of local fields are suitable to construct some FIFs.
The main purpose of this work is to establish the interpolation theory on the -series local field and to investigate the difference of smoothness between the two FIFs defined on and on . In Section 2 of this paper, some results on the smoothness of a so-called -fractal functions are obtained and some examples supporting corresponding conclusions are given. In Section 3, the -type smoothness on local fields is introduced. In Section 4, a definition of the fractal interpolation functions on is given and the -type smoothness of the fractal interpolation function on is obtained by virtue of the equivalent relationship between the Hölder type space and the Lipschitz class . As a special example of the fractal interpolation functions on local fields, Weierstrass type function on local fields is shown. A linear relationship between the orders of the -type derivatives and the fractal dimensions of the graphs of Weierstrass function on local fields is concluded. Finally, in Section 5, we give a comparison between the FIF on and on .
2. Smoothness of -Fractal Functions
In this section, we focus our research on the smoothness of the -fractal functions on .
Let and for every . Define an IFS on as follows: where the obey , and they are called the vertical scaling factors, is a continuous function satisfying for every , and is the line passing through and .
In [2, 11], the FIF associated with the IFS (10) is called an -fractal function associated with with respect to the equidistant partition, denoted by .
By [6], it is easy to obtain the explicit representation of . We agree on . Then, for any , there is a sequence such that satisfies and then where the shift map is defined by
Several conclusions on the smoothness of -fractal function have been given like, for instance, the following one.
Theorem 4 (see [9]). If there exists some such that and , then is a.e. differentiable in .
In [11], the FIF (11) and (12) has constant scaling factors . However, in this paper, one assumes that FIF (12) has different scaling factors and then finds conditions for nonsmoothness of . The following lemma is a generalization of Lemma 5.1 of [11].
Lemma 5. If , , , and is differentiable at , then
Proof. Given , we define
Then and as .
For ,
From (12),
By (16), the second term of (17) deduces to
Similarly,
Then
For any ,
and, then, for , there exists , so that
Thus we have
Notice that
and hence we have
Since is differentiable at , it follows that
Note that as , and, by the hypothesis , , then
This completes the proof.
By Lemma 5, we get a conclusion.
Theorem 6. If , , , and does not agree with in a nonempty open subinterval of , then the set of points at which is not differentiable is dense on .
Example 7. Let be the Weierstrass function defined by where . Then is continuous, and the set of its nondifferentiable points is dense on .
Here we set , . The set of data points is given as We see that is generated by the IFS in which the scaling factors are .
In this case, the line passing through and is . Then we get Since , we may say that does not agree with in a nonempty open subinterval of . Moreover, by hypothesis , we get that , . Then we replace with . Using Theorem 6, we get the result (see Figure 1).
(a)
(b)
By Example 7, we conclude that we may choose or some other periodic functions on instead of in (28) to construct a function, the set of whose nondifferentiable points is dense on the domain.
Example 8. Let be the tent map defined by One can extend to a periodic function on the line. For the sake of simplicity we will use the same symbol to represent the extension. Then we have the following: when , the set of points at which the function is not differentiable is dense on (see Figure 2).
(a)
(b)
In Examples 7 and 8, we set the scaling factors to the same value. However, we may give different values to the factors, and these two conclusions are still tenable.
Example 9. Suppose , is generated by the IFS When , is continuous, and the set of its nondifferentiable points is dense on .
Also we set ; the data points are . The scale vector is . Then we have By (11) and (12), we conclude that if then We call this kind of functions Weierstrass-like functions (see Figure 3).
(a)
(b)
3. -Type Smoothness on Local Fields
Let be a local field; that is, it is a locally compact, totally disconnected, nondiscrete, completed topological field [12] with addition and multiplication .
Moreover, if the character of is finite, then must contain a prime field which is isomorphic to the Galois field; otherwise, if the character of is infinite, then must contain a prime field which is isomorphic to the rational number field.
Further, such can be given a valuation by a non-Archimedean norm which satisfies (1) , ; (2) ; (3) . Thus becomes a valued field. So one can prove that there exist a prime and a prime element with , such that every element can be uniquely expressed as with . And we can define a distance on : such that becomes an ultrametric space with ultrametric .
Then, there are 4 cases for a local field .(1)When the character of is finite,(i) is a -series field which is isomorphic to the Galois field ;(ii) is a -degree finite algebraic extension of a -series field with .
At these cases, the operation is mod addition term by term, no carrying, and so is .(2)When the character of is infinite,(i) is a -adic field which is isomorphic to the rational number field ;(ii) is a -degree finite algebraic extension of a -adic field with .
At these cases, the operation is mod addition term by term, carrying from left to right, and so is .
In this section, we concentrate to study the fractal interpolation functions on a -series field, denoted by with a prime .
Let be the character group of ; then is isomorphic to .
Denote by the ring of integers; the ball with center and radius ; the prime ideal in . Moreover, .
There are Haar measures on and , such that and for , respectively.
Let be the base character of which is trivial on but nontrivial on [13]; that is, For and , a character of has the form where .
Definition 10 (see [14]). Let , . If, for a Borel measurable function on , the pseudodifferential operator exists at , it is said to be a -type pointwise derivative of order of at , denoted by . Similarly, if, for a Borel measurable function on , the pseudodifferential operator for exists at , it is said to be a -type pointwise integral of order of at , denoted by .
4. IFS and FIF on Local Fields
We now turn to consider the interpolation theory on a -series field .
Suppose the set of interpolating points is given, and for every , where . Let .
Definition 11. Let be the -series field, let be the metric space with in (39), and let be contractive mappings with contractive factors , , respectively. Then is called a hyperbolic iterated function system (HIFS, for short), with contractive factor , where .
We construct IFS on as follows.
For , take mappings by and mappings by , with and in Lipschitz class [15], where . Then determine mappings by Thus, forms an IFS on with a multi-index , is called a vertical scaling factor of , and let .
Lemma 12. Suppose , ; then .
Proof. Let , . Then . When , we have ; hence, for any ; when , for any .
Similar to that in [3], we have the following theorem.
Theorem 13. Let be the IFS defined in (45) with and with , . Then there exists an ultrametric on , such that the IFS in (45) is hyperbolic with respect to . Moreover, there exists a unique nonempty compact set such that
Proof. We denote defined by
where is defined in (39), and factor will be determined later.
For every ,
Since , , by Lemma 12, we have
By , we set , , and , then
The proof completes.
Definition 14. One calls in Theorem 13 an attractor of the HIFS in (45). The set is the graph of a function which obeys for . One refers to such function as a fractal interpolation function on (FIF, for short), associated with the HIFS .
Proposition 15. Let be the FIF associated with the HIFS in (45). For any , , , one has where one agrees on and the shift operator is defined by ; that is,
In a simple case, we suppose , and, for any , , . Then we have the following.
Lemma 16. Let be the FIF associated with the HIFS where , , and with ; then is continuous in ; moreover,
Proof. For any given , let be a nonnegative integer such that . By (51), we have
When , then ; we have
it follows that .
When , then ; we have
it follows that .
We complete the proof.
Lemma 17 (see [16, Theorem 1]). If , then, for any , the -type derivative of order of exists, and .
Lemma 18 (see [15, Theorem 1]). For a local fields , one has
From the above three Lemmas, we can conclude the following.
Theorem 19. Let be the FIF associated with the HIFS in (53), where , , and with ; then is -order differentiable in the sense of (42) with any .
Moreover, we get a similar conclusion in a more general case.
Theorem 20. Let be the FIF associated with the HIFS in (45), where , and for each . Let , and with . Then is -order differentiable in the sense of (42) with any .
Proof. For and each that has a compact support , there exists independent of , such that for any .
Suppose and ; then
The proof is complete.
Example 21. The Weierstrass type function on the dyadic series field is defined by [17] Then is -order differentiable in the sense of (42) with .
Since every can be expressed as , , , it follows that Thus is the FIF associated with the HIFS where , , and .
For every , , by Theorem 20, we can get the result (see Figure 4).
(a)
(b)
Denote by the graph of the Weierstrass type function on the dyadic series field defined in Example 21 and by the graph of -order -type derivative of ; then [17] gives the following elegant results:(1)the fractal dimensions of the graph satisfy for a.e. (2)the fractal dimensions of the graph satisfy for a.e. where , , and are the Hausdorff, Box, and Packing dimension of .
5. A Comparison between the FIF on and on
The structure of local fields is quite different from that of Euclidean spaces [1], so that there are essential different properties between FIF on and that on .
Firstly, we note that, in case, the differentiability of FIF could not be guaranteed by that of , even has more higher order smoothness. See Example 7; the function is infinitely often differentiable; however, the set of nondifferential points of is dense in .
Compare with the case in ; the differentiability of FIF has -order smoothness depending on the and of IFS. Since we use the results in Lemmas 17 and 18 which only hold for but not for . The relationship , on helps us to get the smoothness of the Weierstrass type function in Example 21. For more examples see [1, 18].
Secondly, in the case, the fractal dimensions of FIF have few research results. In the case, not only may we have the -type smoothness of FIF but also may determine the space which FIF lives. Moreover, for some special IFS, not only may we determine the fractal dimensions of FIF but also may determine the fractal dimensions of the -type derivatives of FIF, for instance, for the Weierstrass type functions in Example 21, and for more examples see [1]. However, for a general FIF, the relations between the orders of smoothness of FIF and the fractal dimensions of FIF are still open problems.
We also note that, in the case, the interpolation points are , and , , where is a natural number; one may choose it as one wants. However, in the case, , with a prime .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to express their gratitude to referees for their valuable comments and suggestions. This work is supported by NSFC (10571084).