The Smoothness of Fractal Interpolation Functions on R and on p-Series Local Fields

A fractal interpolation function on ap-series local fieldKp is defined, and itsp-type smoothness is shown by virtue of the equivalent relationship between the Hölder type space C (Kp) and the Lipschitz class Lip(σ,Kp). The orders of the p-type derivatives and the fractal dimensions of the graphs of Weierstrass type function on local fields are given as an example. The α-fractal function on R 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 R and Kp is given.


Introduction
As we know, the traditional method for analyzing given experiment data {(  ,   ) :   ∈ [, ] ,  = 0, 1, . . ., ,  0 = ,   = } (1) is by representing data graphically as a subset of R 2 ; then the graphical data are analyzed by some geometrical or analytical tools to seek a function () with graph  ⊂ R 2 , whose values are a good fit to the data over the interval [ 0 ,   ]; 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 R case.
Let  be the set of all nonempty compact subsets of the complete metric space .Then  is a complete metric space with the Hausdorff metric ℎ (, ) = max {sup where   () = {  () :  ∈ }.We call  ∈  the attractor of the IFS {,   :  = 1, 2, . . ., } if The attractor  of a HIFS is the unique set satisfying lim where with some 0 <  < 1.
Several conclusions on the differentiability of the FIFs defined on R 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 R 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 Lip (,   ).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 R and on   .

Smoothness of 𝛼-Fractal Functions
In this section, we focus our research on the smoothness of the -fractal functions on R.
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 ∏ 1 =0    = 1.Then, for any  ∈ , there is a sequence {  } +∞ =1 ,   ∈ {1, 2, . . ., } such that  satisfies and then where the shift map Several conclusions on the smoothness of -fractal function have been given like, for instance, the following one.

, 𝑁, and
Proof. Then From (12), By ( 16), the second term of (17) deduces to Similarly, Then For any 0 ≤  < , and, then, for  ∈ C 1 [0, 1], 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 This completes the proof.
By Lemma 5, we get a conclusion.
Here we set  = 2,  = [0, 2].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 ( 0 ,  0 ) and (  ,   ) is () = 2  /(2  − 1).Then we get Since   () = − sin , we may say that   () does not agree with 0(=   −  0 ) in a nonempty open subinterval of [0, 2].Moreover, by hypothesis 0 <  ≤ 1, we get that |  | ≥ 1/2,  = 1, 2. Then we replace   () with   (2).Using Theorem 6, we get the result (see Figure 1).By Example 7, we conclude that we may choose sin  or some other periodic functions on [0, 1] instead of cos  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 0 <  ≤ 1, the set of points at which the function is not differentiable is dense on [0, 1] (see Figure 2).) , ) . (34) When  1 ,  2 ∈ (0, 1], () is continuous, and the set of its nondifferentiable points is dense on .

𝑝-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.(1) When the character of  is finite, (i)  is a -series field which is isomorphic to the Galois field GF(); (ii)  is a -degree finite algebraic extension of a series field with  ∈ N.
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 Q; (ii)  is a -degree finite algebraic extension of a adic field with  ∈ N.
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  ≥ 2.
Let Γ  be the character group of   ; then   is isomorphic to Γ  .
Definition 10 (see [14]).Let ⟨⟩ = max{1, ||},  ≥ 0. If, for a Borel measurable function  :   → C 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  :   → C on   , the pseudodifferential operator for  ≥ 0 exists at  ∈   , it is said to be a -type pointwise integral of order  of  at , denoted by  ⟨⟩ ().
We construct IFS on   as follows.
where dim  , dim  , and dim   are the Hausdorff, Box, and Packing dimension of .

A Comparison between the FIF on R and on 𝐾 𝑝
The structure of local fields is quite different from that of Euclidean spaces [1], so that there are essential different properties between FIF on R and that on   .Firstly, we note that, in R case, the differentiability of FIF   () could not be guaranteed by that of (), even () has more higher order smoothness.See Example 7; the function () = cos  + 1/(2  − 1) is infinitely often differentiable; however, the set of nondifferential points of   () is dense in [0, 2].
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 ∈ inf ∈  (, ) , sup ∈ inf ∈  (, )} , ∀,  ∈ .

Figure 1 :Figure 2 :
Figure 1: The graph of () in Example 7 with  = 0.5 and the graph of the corresponding Weierstrass function   () with  = 0.5.