Research Article The Smoothness of Fractal Interpolation Functions on ...

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2014, Article ID 904576, 10 pages http://dx.doi.org/10.1155/2014/904576

Research Article The Smoothness of Fractal Interpolation Functions on R and on 𝑝-Series Local Fields Jing Li1,2 and Weiyi Su1 1 2

Department of Mathematics, Nanjing University, Nanjing 210093, China Nankai University Binhai College, Tianjin 300270, China

Correspondence should be addressed to Jing Li; [email protected] Received 6 November 2013; Revised 5 February 2014; Accepted 10 February 2014; Published 7 April 2014 Academic Editor: Bing Xu Copyright © 2014 J. Li and W. Su. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A fractal interpolation function on a 𝑝-series local field 𝐾𝑝 is defined, and its 𝑝-type smoothness is shown by virtue of the equivalent relationship between the H¨older type space 𝐶𝜎 (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.

1. Introduction As we know, the traditional method for analyzing given experiment data {(𝑥𝑛 , 𝐹𝑛 ) : 𝑥𝑛 ∈ [𝑎, 𝑏] , 𝑛 = 0, 1, . . . , 𝑁, 𝑥0 = 𝑎, 𝑥𝑁 = 𝑏}

(1)

is by representing data graphically as a subset of R2 ; then the graphical data are analyzed by some geometrical or analytical tools to seek a function 𝑓(𝑥) with graph 𝐺 ⊂ R2 , whose values are a good fit to the data over the interval [𝑥0 , 𝑥𝑁]; this is an interpolation problem. Generally, a function 𝑓(𝑥) need to satisfy the following: (1) it fits the data at each point 𝑥𝑛 ; that is, 𝑓(𝑥𝑛 ) = 𝐹𝑛 ; (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 𝑓(𝑥) 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. Barnsley et al. [2–5] defined the fractal interpolation functions on R by virtue of the iterated function system. Definition 1 (see [2]) (IFS, HIFS). Let 𝑀 be a complete metric space with distance 𝑑(𝑥, 𝑦) for 𝑥, 𝑦 ∈ 𝑀. Let 𝑁 > 1 be a given positive integer and let 𝜔𝑛 : 𝑀 → 𝑀 for 𝑛 = 1, 2, . . . , 𝑁 be continuous mappings; then we call {𝑀, 𝜔𝑛 : 𝑛 = 1, 2, . . . , 𝑁} an iterated function system (IFS for short). If, for all 𝑛 = 1, 2, . . . , 𝑁, there exists an 𝑠 ∈ [0, 1) such that 𝑑 (𝜔𝑛 (𝑥) , 𝜔𝑛 (𝑦)) ≤ 𝑠 ⋅ 𝑑 (𝑥, 𝑦) ,

∀𝑥, 𝑦 ∈ 𝑀;

(2)

that is, each 𝜔𝑛 is contractive with contractive factor 𝑠; then the IFS {𝑀, 𝜔𝑛 : 𝑛 = 1, 2, . . . , 𝑁} is termed a hyperbolic iterated function system (HIFS for short). Let 𝐻 be the set of all nonempty compact subsets of the complete metric space 𝑀. Then 𝐻 is a complete metric space with the Hausdorff metric ℎ (𝐴, 𝐵) = max {sup inf 𝑑 (𝑥, 𝑦) , sup inf 𝑑 (𝑥, 𝑦)} , 𝑥∈𝐴 𝑦∈𝐵

𝑥∈𝐵 𝑦∈𝐴

∀𝐴, 𝐵 ∈ 𝐻.

(3)

2

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Define 𝑊 : 𝐻 → 𝐻 by 𝑁

𝑊 (𝐴) = ⋃ 𝜔𝑛 (𝐴) ,

∀𝐴 ∈ 𝐻,

(4)

𝑛=1

where 𝜔𝑛 (𝐴) = {𝜔𝑛 (𝑥) : 𝑥 ∈ 𝐴}. We call 𝐺 ∈ 𝐻 the attractor of the IFS {𝑀, 𝜔𝑛 : 𝑛 = 1, 2, . . . , 𝑁} if 𝑊 (𝐺) = 𝐺.

(5)

The attractor 𝐺 of a HIFS is the unique set satisfying lim ℎ (𝑊𝑚 (𝐵) , 𝐺) = 0,

𝑚→∞

∀𝐵 ∈ 𝐻,

(6)

𝑊 ∘ ⋅ ⋅ ⋅ ∘ 𝑊 is the 𝑚-fold composition of 𝑊. where 𝑊𝑚 = ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑚

Let 𝑥0 < 𝑥1 < ⋅ ⋅ ⋅ < 𝑥𝑁 be a partition of a closed interval 𝐼 = [𝑥0 , 𝑥𝑁]. Set 𝐼𝑛 = [𝑥𝑛−1 , 𝑥𝑛 ] for 𝑛 ∈ {1, 2, . . . , 𝑁}. Suppose that a set of interpolation points 𝑇 = {(𝑥𝑛 , 𝑦𝑛 ) ∈ 𝐼 × [𝑎, 𝑏] : 𝑛 = 0, 1, . . . , 𝑁} is given, where −∞ < 𝑎 < 𝑏 < ∞. Let 𝐿 𝑛 : 𝐼 → 𝐼𝑛 , 𝑛 = 1, 2, . . . , 𝑁, be contractive homeomorphisms such that 𝐿 𝑛 (𝑥0 ) = 𝑥𝑛−1 ,

𝐿 𝑛 (𝑥𝑁) = 𝑥𝑛 ,

󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨𝐿 𝑛 (𝑐1 ) − 𝐿 𝑛 (𝑐2 )󵄨󵄨󵄨 ≤ 𝑟 󵄨󵄨󵄨𝑐1 − 𝑐2 󵄨󵄨󵄨 ,

∀𝑐1 , 𝑐2 ∈ 𝐼,

(7)

with some 0 < 𝑟 < 1. Let 𝐹𝑛 : 𝐼 × [𝑎, 𝑏] → [𝑎, 𝑏] be 𝑁 continuous mappings satisfying 𝐹𝑛 (𝑥0 , 𝑦0 ) = 𝑦𝑛−1 ,

𝐹𝑛 (𝑥𝑁, 𝑦𝑁) = 𝑦𝑛 ,

󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨𝐹𝑛 (𝑥, 𝑡1 ) − 𝐹𝑛 (𝑥, 𝑡2 )󵄨󵄨󵄨 ≤ 𝑞 󵄨󵄨󵄨𝑡1 − 𝑡2 󵄨󵄨󵄨 ,

(8)

where 𝑥 ∈ 𝐼, 𝑡1 , 𝑡2 ∈ [𝑎, 𝑏], 0 < 𝑞 < 1, and 𝑛 = 1, 2, . . . , 𝑁. Set 𝑀 = 𝐼 × R. Define the mappings 𝜔𝑛 : 𝑀 → 𝑀 for 𝑛 ∈ {1, 2, . . . , 𝑁} by 𝜔𝑛 (𝑥, 𝑦) = (𝐿 𝑛 (𝑥) , 𝐹𝑛 (𝑥, 𝑦)) .

(9)

Then {𝑀, 𝜔𝑛 : 𝑛 = 1, 2, . . . , 𝑁} constitutes an IFS. Theorem 2 (see [2, Theorem 1]). The IFS {𝑀, 𝜔𝑛 : 𝑛 = 1, 2, . . . , 𝑁} defined above admits a unique attractor 𝐺, which is the graph of a continuous function 𝑓 : 𝐼 → R passing through the interpolation points 𝑇. Definition 3 (see [2]) (FIF). The function 𝑓 described in Theorem 2 is referred to as a fractal interpolation function (FIF for short), associated with the IFS {𝑀, 𝜔𝑛 : 𝑛 = 1, 2, . . . , 𝑁}. 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 H¨older smoothness of a class of FIFs and their logical derivatives of order 𝛼. Chen [8] investigated the smoothness of nonequidistant FIF and obtained the

H¨older 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 H¨older continuity of two kinds of FIFs and proved the sufficient and necessary condition for their differentiability. Navascu´es [11] introduced 𝛼-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 𝑝-series local field 𝐾𝑝 and to investigate the difference of smoothness between the two FIFs defined on R and on 𝐾𝑝 . In Section 2 of this paper, some results on the smoothness of a so-called 𝛼-fractal functions are obtained and some examples supporting corresponding conclusions are given. In Section 3, the 𝑝-type smoothness on local fields is introduced. In Section 4, a definition of the fractal interpolation functions on 𝐾𝑝 is given and the 𝑝-type smoothness of the fractal interpolation function on 𝐾𝑝 is obtained by virtue of the equivalent relationship between the H¨older type space 𝐶𝜎 (𝐾𝑝 ) and the Lipschitz class Lip (𝜎, 𝐾𝑝 ). 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 𝑝type 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 𝐾𝑝 .

2. Smoothness of 𝛼-Fractal Functions In this section, we focus our research on the smoothness of the 𝛼-fractal functions on R. Let 𝐼 = [0, 1] and 𝑥𝑛 = 𝑛/𝑁 for every 𝑛 ∈ {0, 1, . . . , 𝑁}. Define an IFS on R as follows: 𝑥 (𝑛 − 1) + 𝐿 (𝑥) 𝑥 𝑁 𝑁 )=( 𝜔𝑛 ( ) = ( 𝑛 ), 𝑦 𝐹𝑛 (𝑥, 𝑦) 𝛼𝑛 𝑦 + 𝑔 ∘ 𝐿 𝑛 (𝑥) − 𝛼𝑛 𝑏 (𝑥) 𝑛 = 1, 2, . . . , 𝑁, (10) where the 𝛼𝑛 obey −1 < 𝛼𝑛 < 1, and they are called the vertical scaling factors, 𝑔 ∈ C(𝐼) is a continuous function satisfying 𝑔(𝑥𝑛 ) = 𝑦𝑛 for every 𝑛 ∈ {0, 1, . . . , 𝑁}, and 𝑏(𝑥) = 𝑦0 + (𝑦𝑁 − 𝑦0 )𝑥 is the line passing through (𝑥0 , 𝑦0 ) and (𝑥𝑁, 𝑦𝑁). In [2, 11], the FIF associated with the IFS (10) is called an 𝛼-fractal function associated with 𝑔 with respect to the equidistant partition, denoted by 𝑔𝛼 (𝑥).

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3

By [6], it is easy to obtain the explicit representation of 𝑔𝛼 . We agree on ∏1𝑗=0 𝛼𝑛𝑗 = 1. Then, for any 𝑥 ∈ 𝐼, there is a sequence {𝑛𝑗 }+∞ 𝑗=1 , 𝑛𝑗 ∈ {1, 2, . . . , 𝑁} such that 𝑥 satisfies +∞ 𝑛 𝑗

𝑥 = (𝑛1 𝑛2 𝑛3 ⋅ ⋅ ⋅ ) = ∑

−1

𝑁𝑗

𝑗=1

,

(11)

By (16), the second term of (17) deduces to +∞

𝑘

𝑘=𝑟

𝑗=1

∑ (∏𝛼𝑖𝑗 ) (𝑔 (0) − 𝛼1 𝑏 (0)) 𝑟

= (∏𝛼𝑖𝑗 ) (𝑦0 − 𝛼1 𝑦0 ) (1 + 𝛼1 + 𝛼12 + ⋅ ⋅ ⋅ )

and then +∞

𝑘

𝑘=0

𝑗=1

(18)

𝑗=1

𝑔𝛼 (𝑥) = ∑ (∏𝛼𝑛𝑗 ) (𝑔 (𝜎𝑘 𝑥) − 𝛼𝑛𝑘+1 𝑏 (𝜎𝑘+1 𝑥)) ,

(12)

𝑟

= 𝑦0 (∏𝛼𝑖𝑗 ) . 𝑗=1

where the shift map 𝜎𝑘 : [0, 1] → [0, 1] is defined by +∞

𝑛𝑘+𝑙 − 1 , 𝑁𝑙 𝑙=1

𝜎𝑘 (𝑛1 𝑛2 ⋅ ⋅ ⋅ 𝑛𝑘 ⋅ ⋅ ⋅ ) = (𝑛𝑘+1 𝑛𝑘+2 ⋅ ⋅ ⋅ ) = ∑

Similarly, (13)

𝑘 ∈ N. Several conclusions on the smoothness of 𝛼-fractal function have been given like, for instance, the following one.

𝑟−1

𝑘

𝑘=0

𝑗=1

𝑔𝛼 (𝑥𝑟󸀠 ) = ∑ (∏𝛼𝑖𝑗 ) (𝑔 (𝜎𝑘 𝑥𝑟󸀠 ) − 𝛼𝑖𝑘+1 𝑏 (𝜎𝑘+1 𝑥𝑟󸀠 )) (19) 𝑟

Theorem 4 (see [9]). If there exists some 𝑛 ∈ {1, 2, . . . , 𝑁} such that 𝛼𝑛 = 0 and 𝑔 ∈ C1 [0, 1], then 𝑔𝛼 is a.e. differentiable in (0, 1). In [11], the FIF (11) and (12) has constant scaling factors 𝛼1 = 𝛼2 = ⋅ ⋅ ⋅ = 𝛼𝑁 = 𝑎. However, in this paper, one assumes that FIF (12) has different scaling factors and then finds conditions for nonsmoothness of 𝑔𝛼 . The following lemma is a generalization of Lemma 5.1 of [11]. Lemma 5. If 𝑔 ∈ C1 [0, 1], |𝛼𝑛 | ≥ 1/𝑁, 𝑛 = 1, 2, . . . , 𝑁, and 𝑔𝛼 is differentiable at 𝑥 ∈ [0, 1], then lim 𝑔󸀠 (𝜎𝑘 𝑥) = 𝑦𝑁 − 𝑦0 .

𝑗 Proof. Given 𝑥 = (𝑖1 𝑖2 ⋅ ⋅ ⋅ 𝑖𝑟 ⋅ ⋅ ⋅ ) = ∑+∞ 𝑗=1 (𝑖𝑗 − 1)/𝑁 , we define 𝑟

𝑗=1

𝑥𝑟󸀠

𝑟

= (𝑖1 𝑖2 ⋅ ⋅ ⋅ 𝑖𝑟 𝑁𝑁 ⋅ ⋅ ⋅ ) = ∑

𝑗=1

𝑖𝑗 − 1 𝑁𝑗

𝑖𝑗 − 1 𝑁𝑗

𝑘

𝑘=0

𝑗=1

𝑘

𝑘=𝑟

𝑗=1

𝑟−1

𝑘

𝑘=0

𝑗=1

= ∑ (∏𝛼𝑖𝑗 ) {[𝑔 (𝜎𝑘 𝑥𝑟󸀠 ) − 𝑔 (𝜎𝑘 𝑥𝑟 )] −𝛼𝑖𝑘+1 [𝑏 (𝜎𝑘+1 𝑥𝑟󸀠 ) − 𝑏 (𝜎𝑘+1 𝑥𝑟 )]} 𝑟

+ (𝑦𝑁 − 𝑦0 ) (∏𝛼𝑖𝑗 ) .

1 + 𝑟. 𝑁

(20) (15) For any 0 ≤ 𝑘 < 𝑟,

𝜎𝑘 𝑥𝑟󸀠 = (𝑁𝑁 ⋅ ⋅ ⋅ ) = 1.

𝜎𝑘 𝑥𝑟󸀠 − 𝜎𝑘 𝑥𝑟 =

+∞

1 (𝑁 − 1) = 𝑟−𝑘 , 𝑗 𝑁 𝑁 𝑗=𝑟−𝑘+1 ∑

(21)

(16) and, then, for 𝑔 ∈ C1 [0, 1], there exists 𝜉𝑘𝑟 ∈ (𝜎𝑘 𝑥𝑟 , 𝜎𝑘 𝑥𝑟󸀠 ), so that

𝑔𝛼 (𝑥𝑟 ) = ∑ (∏𝛼𝑖𝑗 ) (𝑔 (𝜎𝑘 𝑥𝑟 ) − 𝛼𝑖𝑘+1 𝑏 (𝜎𝑘+1 𝑥𝑟 )) +∞

𝑔𝛼 (𝑥𝑟󸀠 ) − 𝑔𝛼 (𝑥𝑟 )

𝑗=1

From (12), 𝑟−1

Then

,

Then 𝑥𝑟 ≤ 𝑥 ≤ 𝑥𝑟󸀠 and 𝑥𝑟󸀠 − 𝑥𝑟 → 0 as 𝑟 → ∞. For ∀𝑘 ≥ 𝑟, 𝜎𝑘 𝑥𝑟 = (11 ⋅ ⋅ ⋅ ) = 0,

𝑗=1

(14)

𝑘 → +∞

𝑥𝑟 = (𝑖1 𝑖2 ⋅ ⋅ ⋅ 𝑖𝑟 11 ⋅ ⋅ ⋅ ) = ∑

+ 𝑦𝑁 (∏𝛼𝑖𝑗 ) .

+ ∑ (∏𝛼𝑖𝑗 ) (𝑔 (𝜎𝑘 𝑥𝑟 ) − 𝛼𝑖𝑘+1 𝑏 (𝜎𝑘+1 𝑥𝑟 )) . (17)

𝑔 (𝜎𝑘 𝑥𝑟󸀠 ) − 𝑔 (𝜎𝑘 𝑥𝑟 ) =

𝑔󸀠 (𝜉𝑘𝑟 ) , 𝑁𝑟−𝑘

(𝑦 − 𝑦 ) 𝑏 (𝜎𝑘+1 𝑥𝑟󸀠 ) − 𝑏 (𝜎𝑘+1 𝑥𝑟 ) = 𝑁𝑟−𝑘−10 . 𝑁

(22)

4

Discrete Dynamics in Nature and Society Theorem 6. If 𝑔 ∈ C1 [0, 1], |𝛼𝑛 | ≥ 1/𝑁, 𝑛 = 1, 2, . . . , 𝑁, and 𝑔󸀠 (𝑥) does not agree with 𝑦𝑁 − 𝑦0 in a nonempty open subinterval of 𝐼 = [0, 1], then the set of points at which 𝑔𝛼 is not differentiable is dense on 𝐼.

Thus we have 𝑔𝛼 (𝑥𝑟󸀠 ) − 𝑔𝛼 (𝑥𝑟 ) 𝑥𝑟󸀠 − 𝑥𝑟 𝑟−1

𝑘

𝑘=0

𝑗=1

Example 7. Let 𝑊𝜆 be the Weierstrass function defined by

= ∑ (∏𝛼𝑖𝑗 ) [𝑔󸀠 (𝜉𝑘𝑟 ) − (𝑦𝑁 − 𝑦0 ) 𝑁𝛼𝑖𝑘+1 ]

+∞

𝑊𝜆 (𝑥) = ∑ 2−𝜆𝑗 cos (2𝑗 𝑥) ,

⋅ 𝑁𝑘 + (𝑦𝑁 − 𝑦0 ) (∏𝛼𝑖𝑗 ) ⋅ 𝑁𝑟 𝑘

𝑘=0

𝑗=1

= ∑ (∏ (𝑁𝛼𝑖𝑗 )) (𝑔󸀠 (𝜉𝑘𝑟 ) − (𝑦𝑁 − 𝑦0 ) 𝑁𝛼𝑖𝑘+1 ) 𝑟

𝑗=1

(23) Notice that 𝑟

𝑟−1

𝑘+1

𝑘

𝑗=1

𝑘=0

𝑗=1

𝑗=1

∏ (𝑁𝛼𝑖𝑗 ) = 1 + ∑ (∏ (𝑁𝛼𝑖𝑗 ) − ∏ (𝑁𝛼𝑖𝑗 )) ,

(24)

and hence we have 𝑔𝛼 (𝑥𝑟󸀠 ) − 𝑔𝛼 (𝑥𝑟 )

𝑘=0

𝑗=1

= ∑ (∏ (𝑁𝛼𝑖𝑗 )) (𝑔󸀠 (𝜉𝑘𝑟 ) − (𝑦𝑁 − 𝑦0 ) 𝑁𝛼𝑖𝑘+1 ) + (𝑦𝑁 − 𝑦0 ) + (𝑦𝑁 − 𝑦0 ) 𝑟−1

𝑘+1

𝑘

𝑘=0

𝑗=1

𝑗=1

𝑘

𝑘=0

𝑗=1

(25)

= 𝑦𝑁 − 𝑦0 + ∑ (∏ (𝑁𝛼𝑖𝑗 )) (𝑔󸀠 (𝜉𝑘𝑟 ) − (𝑦𝑁 − 𝑦0 )) . Since 𝑔𝛼 is differentiable at 𝑥, it follows that

(30)

1 . (2𝜆 − 1)

(31)

Since 𝑔󸀠 (𝑥) = − sin 𝑥, we may say that 𝑔󸀠 (𝑥) does not agree with 0(= 𝑦𝑁 − 𝑦0 ) in a nonempty open subinterval of [0, 2𝜋]. Moreover, by hypothesis 0 < 𝜆 ≤ 1, we get that |𝛼𝑛 | ≥ 1/2, 𝑛 = 1, 2. Then we replace 𝑊𝜆 (𝑥) with 𝑊𝜆 (2𝜋𝑥). Using Theorem 6, we get the result (see Figure 1). By Example 7, we conclude that we may choose sin 𝑥 or some other periodic functions on [0, 1] instead of cos 𝑥 in (28) to construct a function, the set of whose nondifferentiable points is dense on the domain. Example 8. Let 𝜑 be the tent map defined by

󸀠

(𝑔𝛼 ) (𝑥) = 𝑦𝑁 − 𝑦0 𝑟−1

𝑘

𝑘=0

𝑗=1

+ lim ∑ (∏ (𝑁𝛼𝑖𝑗 )) (𝑔󸀠 (𝜉𝑘𝑟 ) − (𝑦𝑁 − 𝑦0 )) . 𝑟 → +∞

We see that 𝑊𝜆 (𝑥) is generated by the IFS 𝑥 𝑥 2 𝑤1 ( ) = ( −𝜆 𝑥 ), 𝑦 2 𝑦 + cos ( ) 2 𝑥 +𝜋 𝑥 𝑤2 ( ) = ( −𝜆 2 𝑥 ), 𝑦 2 𝑦 − cos ( ) 2

𝑔 (𝑥) = cos 𝑥 +

× ∑ (∏ (𝑁𝛼𝑖𝑗 ) − ∏ (𝑁𝛼𝑖𝑗 )) 𝑟−1

(2 − 2𝜆 ) 2𝜆 2𝜆 ) , (𝜋, ) , (2𝜋, )} . (29) (2𝜆 − 1) (2𝜆 − 1) (2𝜆 − 1)

in which the scaling factors are 𝛼1 = 𝛼2 = 2−𝜆 . In this case, the line passing through (𝑥0 , 𝑦0 ) and (𝑥𝑁, 𝑦𝑁) is 𝑏(𝑥) = 2𝜆 /(2𝜆 − 1). Then we get

𝑥𝑟󸀠 − 𝑥𝑟 𝑘

Here we set 𝑁 = 2, 𝐼 = [0, 2𝜋]. The set of data points is given as {(0,

+ (𝑦𝑁 − 𝑦0 ) ∏ (𝑁𝛼𝑖𝑗 ) .

𝑟−1

(28)

where 0 < 𝜆 ≤ 1. Then 𝑊𝜆 (𝑥) is continuous, and the set of its nondifferentiable points is dense on [0, 2𝜋].

𝑗=1

𝑟−1

𝑥 ∈ [0, 2𝜋] ,

𝑗=0

𝑟

(26) Note that 𝜉𝑘𝑟 → 𝜎𝑘 𝑥 as 𝑟 → +∞, and, by the hypothesis |𝛼𝑛 | ≥ 1/𝑁, 𝑛 = 1, 2, . . . , 𝑁, then lim 𝑔󸀠 (𝜎𝑘 𝑥) = 𝑦𝑁 − 𝑦0 .

𝑘 → +∞

This completes the proof. By Lemma 5, we get a conclusion.

(27)

1 { 𝑥, 𝑥 ∈ [0, ] , { { 2 (32) 𝜑 (𝑥) = { { 1 { 1 − 𝑥 𝑥 ∈ ( , 1] . { 2 One can extend 𝜑 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 < 𝜆 ≤ 1, the set of points at which the function +∞

𝜑𝛼 (𝑥) = ∑ 2−𝜆𝑗 𝜑 (2𝑗 𝑥) ,

𝑥 ∈ [0, 1]

𝑗=0

is not differentiable is dense on [0, 1] (see Figure 2).

(33)

Discrete Dynamics in Nature and Society

5

3.5

4 3

3

2

2.5

1 2 0 1.5

1

−1

0

1

2

3

4

5

6

7

−2

0

1

2

3

(a)

4

6

5

7

(b)

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

1.4

0.45

1.2

0.4

1

0.35 0.3

0.8

0.25 0.6

0.2 0.15

0.4

0.1

0.2

0.05 0

0

0.2

0.4

0.6

0.8

1

0

0

(a)

0.2

0.4

0.6

0.8

1

(b)

Figure 2: The graph of 𝜑(𝑥) in Example 8 and the graph of 𝜑𝛼 (𝑥) with 𝜆 = 0.55.

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 𝑊(𝑥), 𝑥 ∈ 𝐼 = [0, 2𝜋] is generated by the IFS 𝑥 2

𝑥 𝑤1 ( ) = ( 𝑥 ), 𝑦 2−𝜆 1 𝑦 + sin ( ) 2 𝑥 +𝜋 2

Also we set 𝑁 = 2; the data points are {(0, 0), (𝜋, 0), (2𝜋, 0)}. The scale vector is 𝛼 = (2−𝜆 1 , 2−𝜆 2 ). Then we have 𝑏 (𝑥) = 0,

𝑔 (𝑥) = sin 𝑥.

By (11) and (12), we conclude that if +∞ 𝑛 − 1 𝑥 𝑗 =∑ 𝑗 , 2𝜋 𝑗=1 2

(34)

𝑥 𝑤2 ( ) = ( 𝑥 ). 𝑦 2−𝜆 2 𝑦 − sin ( ) 2 When 𝜆 1 , 𝜆 2 ∈ (0, 1], 𝑊(𝑥) is continuous, and the set of its nondifferentiable points is dense on 𝐼.

(35)

𝑛𝑗 ∈ {1, 2} , 𝑗 ∈ N,

(36)

then +∞

𝑘

𝑘=0

𝑗=1

−𝜆 𝑛𝑗

𝑊 (𝑥) = 𝑔𝛼 (𝑥) = ∑ (∏2

) sin (2𝑘 𝑥) .

(37)

We call this kind of functions Weierstrass-like functions (see Figure 3).

6

Discrete Dynamics in Nature and Society 1

2.5

0.8

2

0.6

1.5

0.4

1

0.2

0.5

0

0

−0.2

−0.5

−0.4 −0.6

−1

−0.8

−1.5

−1

0

1

2

3

4

5

6

7

(a)

−2

0

1

2

3

4

5

6

7

(b)

Figure 3: The graph of 𝑔(𝑥) in Example 9 and the graph of 𝑔𝛼 (𝑥) with 𝜆 1 = 0.4 and 𝜆 2 = 0.6.

3. 𝑝-Type Smoothness on Local Fields Let 𝐾 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 𝐾 is finite, then 𝐾 must contain a prime field which is isomorphic to the Galois field; otherwise, if the character of 𝐾 is infinite, then 𝐾 must contain a prime field which is isomorphic to the rational number field. Further, such 𝐾 can be given a valuation by a nonArchimedean norm | ⋅ | : 𝐾 → [0, +∞) which satisfies (1) |𝑥| ≥ 0, |𝑥| = 0 ⇔ 𝑥 = 0; (2) |𝑥𝑦| = |𝑥||𝑦|; (3) |𝑥 + 𝑦| ≤ max{|𝑥|, |𝑦|}. Thus 𝐾 ≡ (𝐾, ⊕, ⊗, | ⋅ |) becomes a valued field. So one can prove that there exist a prime 𝑝 ≥ 2 and a prime element 𝛽 ∈ 𝐾 with |𝛽| = 𝑝−1 , such that every element 𝑥 ∈ 𝐾 can be uniquely expressed as +∞

𝑥 = (𝑥−𝑠 𝑥−𝑠+1 ⋅ ⋅ ⋅ 𝑥−1 𝑥0 𝑥1 ⋅ ⋅ ⋅ ) = ∑ 𝑥𝑗 𝛽𝑗 , 𝑗=−𝑠

𝑥𝑗 ∈ {0, 1, . . . , 𝑝 − 1} ,

(38)

𝑗 = −𝑠, −𝑠 + 1, . . . , 𝑠 ∈ N,

with |𝑥| = 𝑝𝑠 . And we can define a distance on 𝐾: 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 |𝑥| − 󵄨󵄨󵄨𝑦󵄨󵄨󵄨󵄨󵄨󵄨 , |𝑥| ≠ 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 , { {󵄨󵄨󵄨 󵄨 󵄨 𝑑𝐾 (𝑥, 𝑦) = {󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨 , |𝑥| = 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 , { 𝑥 = 𝑦, {0,

∀𝑥, 𝑦 ∈ 𝐾,

(39)

such that 𝐾 becomes an ultrametric space with ultrametric 𝑑𝐾 (𝑥, 𝑦). Then, there are 4 cases for a local field 𝐾 ≡ (𝐾, ⊕, ⊗, | ⋅ |). (1) When the character of 𝐾 is finite, (i) 𝐾 is a 𝑝-series field which is isomorphic to the Galois field GF(𝑝); (ii) 𝐾 is a 𝑐-degree finite algebraic extension of a 𝑝series field with 𝑐 ∈ N.

At these cases, the operation ⊕ is mod 𝑝 addition term by term, no carrying, and so is ⊗. (2) When the character of 𝐾 is infinite, (i) 𝐾 is a 𝑝-adic field which is isomorphic to the rational number field Q; (ii) 𝐾 is a 𝑐-degree finite algebraic extension of a 𝑝adic field with 𝑐 ∈ N. At these cases, the operation ⊕ is mod 𝑝 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 𝑝-series field, denoted by 𝐾𝑝 with a prime 𝑝 ≥ 2. Let Γ𝑝 be the character group of 𝐾𝑝 ; then 𝐾𝑝 is isomorphic to Γ𝑝 . Denote by 𝐷 = {𝑥 ∈ 𝐾𝑝 : |𝑥| ≤ 1} the ring of integers; 𝐵𝑘 = {𝑥 ∈ 𝐾𝑝 : |𝑥| ≤ 𝑝−𝑘 } the ball with center 0 ∈ 𝐾𝑝 and radius 𝑝−𝑘 , 𝑘 ∈ Z; 𝐵1 = 𝛽𝐷 = {𝑥 ∈ 𝐾𝑝 : |𝑥| ≤ 𝑝−1 } the prime ideal in 𝐾𝑝 . Moreover, Γ𝑘 = {𝜉 ∈ Γ𝑝 : |𝜉| ≤ 𝑝𝑘 }, 𝑘 ∈ Z. There are Haar measures on 𝐾𝑝 and Γ𝑝 , such that 𝜇(𝐵𝑘 ) ≡ |𝐵𝑘 | = 𝑝−𝑘 and ](Γ𝑘 ) ≡ |Γ𝑘 | = 𝑝𝑘 for 𝑘 ∈ Z, respectively. Let 𝜒(𝑥) be the base character of 𝐾𝑝 which is trivial on 𝐷 but nontrivial on 𝐵−1 [13]; that is, 𝑒2𝜋𝑖/𝑝 , 𝑗 = 1, 𝜒 (𝛽−𝑗 ) = { 1, otherwise.

(40)

+∞ 𝑗 𝑗 For 𝑥 = ∑+∞ 𝑗=−𝑠 𝑥𝑗 𝛽 and 𝜆 = ∑𝑗=−𝑡 𝜆 𝑗 𝛽 ∈ 𝐾𝑝 , a character 𝜒𝜆 ∈ Γ𝑝 of 𝐾𝑝 has the form

𝜒𝜆 (𝑥) = 𝜒 (𝜆𝑥) = 𝑒(2𝜋𝑖/𝑝) ∑𝑗 𝑥𝑗 𝜆 −𝑗−1 ,

(41)

where 𝑥𝑗 , 𝜆 𝑘 ∈ {0, 1, . . . , 𝑝 − 1}, 𝑗 = −𝑠, −𝑠 + 1, . . . ; 𝑘 = −𝑡, −𝑡 + 1, . . . ; 𝑠, 𝑡 ∈ N.

Discrete Dynamics in Nature and Society

7

Definition 10 (see [14]). Let ⟨𝜉⟩ = max{1, |𝜉|}, 𝑚 ≥ 0. If, for a Borel measurable function 𝑓 : 𝐾𝑝 → C on 𝐾𝑝 , the pseudodifferential operator 𝑇⟨⋅⟩𝑚 𝑓 (𝑥) = ∫ {∫ ⟨𝜉⟩𝑚 𝑓 (𝑡) 𝜒𝜉 (𝑡 − 𝑥) d𝑡} d𝜉 Γ𝑝

𝐾𝑝

(42)

exists at 𝑥 ∈ 𝐾𝑝 , it is said to be a 𝑝-type pointwise derivative of order 𝑚 of 𝑓 at 𝑥, denoted by 𝑓⟨𝑚⟩ (𝑥). Similarly, if, for a Borel measurable function 𝑓 : 𝐾𝑝 → C on 𝐾𝑝 , the pseudodifferential operator for 𝑚 ≥ 0 𝑇⟨⋅⟩−𝑚 𝑓 (𝑥) = ∫ {∫ ⟨𝜉⟩−𝑚 𝑓 (𝑡) 𝜒𝜉 (𝑡 − 𝑥) d𝑡} d𝜉 Γ𝑝

𝐾𝑝

(43)

Theorem 13. Let {𝐿, 𝑤𝑛 : 𝑛 = 0, 1, . . . , 𝑝−1} be the IFS defined in (45) with |𝛼𝑛 | < 1 and 𝜑𝑛 ∈ Lip (𝛾𝑛 , 𝐾𝑝 ) with 𝛾𝑛 ≥ 1, 𝑛 ∈ {0, 1, . . . , 𝑝 − 1}. Then there exists an ultrametric 𝑑𝐿∗ on 𝐿, such that the IFS in (45) is hyperbolic with respect to 𝑑𝐿∗ . Moreover, there exists a unique nonempty compact set 𝐺 ⊂ 𝐿 such that 𝑝−1

𝐺 = ⋃ 𝑤𝑛 (𝐺) . Proof. We denote 𝑑𝐿∗ : 𝐿 × 𝐿 → [0, +∞) defined by 𝑑𝐿∗ ((𝑥1 , 𝑦1 ) , (𝑥2 , 𝑦2 )) = 𝑑𝐾𝑝 (𝑥1 , 𝑥2 ) + 𝜃𝑑R (𝑦1 , 𝑦1 ) , 𝑥1 , 𝑥2 ∈ 𝐷,

exists at 𝑥 ∈ 𝐾𝑝 , it is said to be a 𝑝-type pointwise integral of order 𝑚 of 𝑓 at 𝑥, denoted by 𝑓⟨𝑚⟩ (𝑥).

4. IFS and FIF on Local Fields We now turn to consider the interpolation theory on a 𝑝series field 𝐾𝑝 . Suppose the set of interpolating points {(𝑥(𝑛) , 𝑦(𝑛) ) ∈ 𝐷 × R : 𝑛 = 0, 1, . . . , 𝑝 − 1} is given, and +∞

𝑥(𝑛) = ∑ 𝑛 ⋅ 𝛽𝑗 ∈ 𝑛 ⋅ 𝛽0 + 𝐵1

(44)

𝑗=0

for every 𝑛 ∈ {0, 1, . . . , 𝑝 − 1}, where 𝐷 = {𝑥 ∈ 𝐾𝑝 : |𝑥| ≤ 1}. Let 𝐿 = 𝐷 × R. Definition 11. Let 𝐾𝑝 be the 𝑝-series field, let (𝐾𝑝 , 𝑑) be the metric space with 𝑑 in (39), and let 𝑤𝑛 : 𝐿 → 𝐿 be contractive mappings with contractive factors 𝑠𝑛 , 𝑛 = 0, 1, . . . , 𝑝 − 1, respectively. Then {𝐿, 𝑤𝑛 : 𝑛 = 0, 1, . . . , 𝑝 − 1} is called a hyperbolic iterated function system (HIFS, for short), with contractive factor 𝑠 ∈ [0, 1), where 𝑠 = max0≤𝑛≤𝑝−1 {𝑠𝑛 }. We construct IFS on 𝐾𝑝 as follows. For 𝑛 = 0, 1, . . . , 𝑝 − 1, take 𝑝 mappings 𝐿 𝑛 : 𝐷 → 𝐷 by 𝐿 𝑛 (𝑥) = 𝑛𝛽0 +𝑥𝛽1 and 𝑝 mappings 𝐹𝑛 : 𝐿 → R by 𝐹𝑛 (𝑥, 𝑦) = 𝛼𝑛 𝑦 + 𝜑𝑛 (𝑥), with |𝛼𝑛 | < 1 and 𝜑𝑛 : 𝐷 → R in Lipschitz class Lip (𝛾𝑛 , 𝐾𝑝 ) [15], where 𝛾𝑛 ≥ 1. Then determine 𝑝 mappings 𝑤𝑛 : 𝐿 → 𝐿 by 𝐿 𝑛 (𝑥) 𝑛𝛽0 + 𝑥𝛽1 𝑥 ). 𝑤𝑛 ( ) = ( )=( 𝑦 𝐹 (𝑥, 𝑦) 𝛼 𝑦 + 𝜑 (𝑥) 𝑛

𝑛

(45)

𝑛

Thus, {𝐿, 𝑤𝑛 : 𝑛 = 0, 1, . . . , 𝑝 − 1} forms an IFS on 𝐾𝑝 with a multi-index 𝛼 = (𝛼0 , 𝛼1 , . . . , 𝛼𝑝−1 ), 𝛼𝑛 is called a vertical scaling factor of 𝑤𝑛 , and let 𝛾 = max0≤𝑛≤𝑝−1 𝛾𝑛 . 𝜌

Lemma 12. Suppose 𝜌 ≥ 1, 𝑥 ∈ [0, 1]; then 𝑥 ≤ 𝜌𝑥. 𝜌

󸀠

𝜌−1

Proof. Let 𝑓(𝑥) = 𝑥 − 𝜌𝑥, 𝑥 ∈ [0, 1]. Then 𝑓 (𝑥) = 𝜌(𝑥 − 1). When 𝜌 > 1, we have 𝑓󸀠 (𝑥) < 0; hence, 𝑓(𝑥) ≤ 𝑓(0) = 0 for any 𝑥 ∈ [0, 1]; when 𝜌 = 1, 𝑓(𝑥) ≡ 0 for any 𝑥 ∈ [0, 1]. Similar to that in [3], we have the following theorem.

(46)

𝑛=0

𝑦1 , 𝑦2 ∈ R, (47)

where 𝑑𝐾𝑝 : 𝐾𝑝 ×𝐾𝑝 → [0, +∞) is defined in (39), and factor 𝜃 > 0 will be determined later. For every 𝑛 ∈ {0, 1, . . . , 𝑝 − 1}, 𝑑𝐿∗ (𝑤𝑛 (𝑥1 , 𝑦1 ) , 𝑤𝑛 (𝑥2 , 𝑦2 )) = 𝑑𝐾𝑝 (𝐿 𝑛 (𝑥1 ) , 𝐿 𝑛 (𝑥2 )) + 𝜃𝑑R (𝐹𝑛 (𝑥1 , 𝑦1 ) , 𝐹𝑛 (𝑥2 , 𝑦2 ))

(48)

−1

≤ 𝑝 𝑑𝐾𝑝 (𝑥1 , 𝑥2 ) 𝛾𝑛 󵄨 󵄨 󵄨󵄨 + 𝜃 {󵄨󵄨󵄨𝛼𝑛 󵄨󵄨󵄨 󵄨󵄨󵄨𝑦1 − 𝑦2 󵄨󵄨󵄨 + [𝑑𝐾𝑝 (𝑥1 , 𝑥2 )] } .

Since 𝛾𝑛 ≥ 1, |𝛼𝑛 | < 1, by Lemma 12, we have 𝑑𝐿∗ (𝑤𝑛 (𝑥1 , 𝑦1 ) , 𝑤𝑛 (𝑥2 , 𝑦2 )) 󵄨 󵄨 󵄨󵄨 ≤ 𝑝−1 𝑑𝐾𝑝 (𝑥1 , 𝑥2 ) + 𝜃 {󵄨󵄨󵄨𝛼𝑛 󵄨󵄨󵄨 󵄨󵄨󵄨𝑦1 − 𝑦2 󵄨󵄨󵄨 + 𝛾𝑛 𝑑𝐾𝑝 (𝑥1 , 𝑥2 )} 󵄨 󵄨 = (𝑝−1 + 𝜃𝛾𝑛 ) 𝑑𝐾𝑝 (𝑥1 , 𝑥2 ) + 𝜃 󵄨󵄨󵄨𝛼𝑛 󵄨󵄨󵄨 𝑑R (𝑦1 , 𝑦1 ) . (49) By 𝛾 = max0≤𝑛≤𝑝−1 𝛾𝑛 , we set 𝜃 = (𝑝 − 1)/𝑝𝛾, 𝑎 = 1/𝑝 + (𝑝 − 1)𝛾𝑛 /𝑝𝛾 < 1, and 𝑏 = max{|𝛼𝑛 |} < 1, then 𝑑𝐿∗ (𝑤𝑛 (𝑥1 , 𝑦1 ) , 𝑤𝑛 (𝑥2 , 𝑦2 )) ≤ 𝑎𝑑𝐾𝑝 (𝑥1 , 𝑥2 ) + 𝜃𝑏𝑑R (𝑦1 , 𝑦1 ) ≤ max {𝑎, 𝑏} 𝑑𝐿∗ ((𝑥1 , 𝑦1 ) , (𝑥2 , 𝑦2 )) ,

(50)

𝑛 = 0, 1, . . . , 𝑝 − 1. The proof completes. Definition 14. One calls 𝐺 in Theorem 13 an attractor of the HIFS {𝐿, 𝑤𝑛 : 𝑛 = 0, 1, . . . , 𝑝 − 1} in (45). The set 𝐺 ⊂ 𝐿 is the graph of a function 𝑓 : 𝐷 → R which obeys 𝑓(𝑥(𝑛) ) = 𝑦(𝑛) for 𝑛 = 0, 1, . . . , 𝑝−1. One refers to such function 𝑓 as a fractal interpolation function on 𝐾𝑝 (FIF, for short), associated with the HIFS {𝐿, 𝑤𝑛 : 𝑛 = 0, 1, . . . , 𝑝 − 1}.

8

Discrete Dynamics in Nature and Society

Proposition 15. Let 𝑓 : 𝐷 → R be the FIF associated with 𝑘 the HIFS in (45). For any 𝑥 = ∑+∞ 𝑘=0 𝑥𝑘 𝛽 ∈ 𝐷 ⊂ 𝐾𝑝 , 𝑥𝑘 ∈ {0, 1, . . . , 𝑝 − 1}, 𝑘 = 0, 1, 2, . . ., one has 𝑗−1

+∞ { { { ∑ (∏𝛼𝑥 ) 𝜑𝑥 (𝜎𝑗+1 𝑥) , 𝑥 ∈ 𝐷, 𝑗 𝑓 (𝑥) = { 𝑗=0 𝑘=0 𝑘 { { otherwise, {0,

(52)

In a simple case, we suppose 𝛼0 = 𝛼1 = ⋅ ⋅ ⋅ = 𝛼𝑝−1 = 𝛼, and, for any 𝑥 ∈ 𝐷, 𝜑0 (𝑥) = 𝜑1 (𝑥) = ⋅ ⋅ ⋅ = 𝜑𝑝−1 (𝑥) = 𝜑(𝑥) ∈ Lip (𝛾, 𝐾𝑝 ), 𝛾 ≥ 1. Then we have the following. Lemma 16. Let 𝑓 : 𝐷 → R be the FIF associated with the HIFS 𝐿 𝑛 (𝑥) 𝑛𝛽0 + 𝑥𝛽1 𝑥 )=( 𝑤𝑛 ( ) = ( ), 𝑦 𝐹 (𝑥, 𝑦) 𝛼𝑦 + 𝜑 (𝑥) 𝑛

|𝛼|𝑟 (𝑝𝛾 − 1) 󵄨 󵄨󵄨 󵄨󵄨𝑓 (𝑥) − 𝑓 (𝑦)󵄨󵄨󵄨 ≤ (|𝛼| 𝑝𝛾 − 1) (1 − |𝛼|)

(51)

where one agrees on ∏0𝑘=−1 𝛼𝑥𝑘 = 1 and the shift operator 𝜎 : 𝐷 → 𝐷 is defined by 𝜎𝑥 = (𝛽−1 𝑥)|𝐷; that is, 𝜎 (𝑥0 𝛽0 + 𝑥1 𝛽1 + 𝑥2 𝛽2 + ⋅ ⋅ ⋅ ) = 𝑥1 𝛽0 + 𝑥2 𝛽1 + ⋅ ⋅ ⋅ .

When 0 < log𝑝 (1/|𝛼|) < 𝛾, then |𝛼|𝑝𝛾 > 1; we have

𝑝𝛾 − 1 󵄨log (1/|𝛼|) 󵄨󵄨 ; = 󵄨𝑥 − 𝑦󵄨󵄨󵄨 𝑝 (|𝛼| 𝑝𝛾 − 1) (1 − |𝛼|) 󵄨

(57)

it follows that 𝑓 ∈ Lip (log𝑝 (1/|𝛼|), 𝐾𝑝 ). We complete the proof. Lemma 17 (see [16, Theorem 1]). If 𝑓 ∈ 𝐶𝜆 (𝐾) (𝜆 ≥ 0), then, for any 𝜏 ∈ [0, 𝜆], the 𝑝-type derivative of order 𝜏 of 𝑓 exists, and 𝑇⟨⋅⟩𝜏 𝑓 ∈ 𝐶𝜆−𝜏 (𝐾). Lemma 18 (see [15, Theorem 1]). For a local fields 𝐾, one has Lip (𝜆, 𝐾) = 𝐶𝜆 (𝐾) ,

𝜆 ∈ (0, +∞) .

(58)

(53) From the above three Lemmas, we can conclude the following.

𝑛 = 0, 1, . . . , 𝑝 − 1, where |𝛼| < 1, 𝛾 ≥ 1, and 𝜑 ∈ Lip (𝛾, 𝐾𝑝 ) with 𝛾 ≠ log𝑝 (1/|𝛼|); then 𝑓 is continuous in 𝐷; moreover, (54)

Theorem 19. Let 𝑓 : 𝐷 → R be the FIF associated with the HIFS in (53), where |𝛼| < 1, 𝛾 ≥ 1, and 𝜑 ∈ Lip (𝛾, 𝐾𝑝 ) with 𝛾 ≠ log𝑝 (1/|𝛼|); then 𝑓 is 𝑚-order differentiable in the sense of (42) with any 𝑚 ∈ [0, min{𝛾, log𝑝 (1/|𝛼|)}).

Proof. For any given 𝑥, 𝑦 ∈ 𝐷, let 𝑟 be a nonnegative integer such that |𝑥 − 𝑦| = 𝑝−𝑟 . By (51), we have

Moreover, we get a similar conclusion in a more general case.

𝑓 ∈ Lip (min {𝛾, log𝑝

1 } , 𝐾𝑝 ) . |𝛼|

󵄨 󵄨󵄨 󵄨󵄨𝑓 (𝑥) − 𝑓 (𝑦)󵄨󵄨󵄨 󵄨󵄨+∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑗 𝑗+1 𝑗+1 󵄨 󵄨 = 󵄨󵄨 ∑ 𝛼 [𝜑 (𝜎 𝑥) − 𝜑 (𝜎 𝑦)]󵄨󵄨󵄨󵄨 󵄨󵄨 𝑗=0 󵄨󵄨 󵄨 󵄨

Theorem 20. Let 𝑓 : 𝐷 → R be the FIF associated with the HIFS in (45), where |𝛼𝑛 | < 1, 𝛾𝑛 ≥ 1 and 𝜑𝑛 ∈ Lip (𝛾𝑛 , 𝐾𝑝 ) for each 𝑛 ∈ {0, 1, . . . , 𝑝 − 1}. Let 𝛼 = max0≤𝑛≤𝑝−1 |𝛼𝑛 |, and 𝛾 = min0≤𝑛≤𝑝−1 𝛾𝑛 with 𝛾 ≠ log𝑝 (1/𝛼). Then 𝑓 is 𝑚order differentiable in the sense of (42) with any 𝑚 ∈ [0, min{𝛾, log𝑝 (1/𝛼)}).

+∞ 󵄨 󵄨𝛾 ≤ ∑ |𝛼|𝑗 ⋅ 󵄨󵄨󵄨󵄨𝜎𝑗+1 𝑥 − 𝜎𝑗+1 𝑦󵄨󵄨󵄨󵄨 𝑗=0

𝑟−2

(𝑟−𝑗−1)𝛾

1 ≤ ∑|𝛼|𝑗 ( ) 𝑝 𝑗=0

+∞

󵄨 󵄨𝛾 + ∑ |𝛼|𝑗 ⋅ 󵄨󵄨󵄨󵄨𝜎𝑗+1 𝑥 − 𝜎𝑗+1 𝑦󵄨󵄨󵄨󵄨 𝑗=𝑟−1

≤

𝑝(1−𝑟)𝛾 − |𝛼|𝑟−1 |𝛼|𝑟−1 + 𝛾 1 − |𝛼| 𝑝 1 − |𝛼|

=

𝑝(1−𝑟)𝛾 (1 − |𝛼|) − |𝛼|𝑟 (𝑝𝛾 − 1) . (1 − |𝛼| 𝑝𝛾 ) (1 − |𝛼|) (55)

When log𝑝 (1/|𝛼|) > 𝛾 ≥ 1, then 0 < |𝛼|𝑝𝛾 < 1; we have 𝑝(1−𝑟)𝛾 𝑝𝛾 󵄨 󵄨𝛾 󵄨󵄨 󵄨󵄨 = 󵄨󵄨𝑓 (𝑥) − 𝑓 (𝑦)󵄨󵄨󵄨 ≤ 󵄨𝑥 − 𝑦󵄨󵄨󵄨 ; 1 − |𝛼| 𝑝𝛾 1 − |𝛼| 𝑝𝛾 󵄨 it follows that 𝑓 ∈ Lip (𝛾, 𝐾𝑝 ).

Proof. For 𝜑𝑛 ∈ Lip (𝛾𝑛 , 𝐾𝑝 ) and each 𝜑𝑛 that has a compact support 𝐷, there exists 𝑀 > 0 independent of 𝑥, such that 𝑝−1 ∑𝑛=0 |𝜑𝑛 (𝑥)| ≤ 𝑀 for any 𝑥 ∈ 𝐷. Suppose 𝑥, 𝑦 ∈ 𝐷 and |𝑥 − 𝑦| = 𝑝−𝑟 ; then 󵄨 󵄨󵄨 󵄨󵄨𝑓 (𝑥) − 𝑓 (𝑦)󵄨󵄨󵄨 󵄨󵄨+∞ 󵄨󵄨 𝑗−1 𝑗−1 󵄨󵄨 󵄨󵄨 𝑗+1 𝑗+1 󵄨 󵄨 = 󵄨󵄨 ∑ [(∏𝛼𝑥𝑘 ) 𝜑𝑥𝑗 (𝜎 𝑥) − (∏𝛼𝑦𝑘 ) 𝜑𝑦𝑗 (𝜎 𝑦)]󵄨󵄨󵄨󵄨 󵄨󵄨 𝑗=0 󵄨󵄨 𝑘=0 𝑘=0 󵄨 󵄨 𝑟−1 𝑗−1 +∞ 󵄨󵄨 󵄨 󵄨 󵄨󵄨 ≤ ∑ (∏ 󵄨󵄨󵄨󵄨𝛼𝑥𝑘 󵄨󵄨󵄨󵄨) 󵄨󵄨󵄨𝜑𝑥𝑗 (𝜎𝑗+1 𝑥) − 𝜑𝑥𝑗 (𝜎𝑗+1 𝑦)󵄨󵄨󵄨 + 𝑀 ∑ 𝛼𝑗 󵄨 󵄨 𝑗=0

(56)

𝑗=𝑟

𝑘=0

𝑟−1 +∞ 󵄨 󵄨𝛾𝑥 ≤ ∑ 𝛼𝑗 󵄨󵄨󵄨󵄨𝜎𝑗+1 𝑥 − 𝜎𝑗+1 𝑦󵄨󵄨󵄨󵄨 𝑗 + 𝑀 ∑ 𝛼𝑗 𝑗=0

𝑗=𝑟

Discrete Dynamics in Nature and Society

9

5

2

4

1.5

3

1

2

0.5

1 0

0

−1

−0.5

−2

−1

−3

−1.5

−4 −5

0.5

0

1.5

1 s = 1.7

2

(a)

−2

0

0.5

1 s = 1.3

1.5

2

(b)

Figure 4: The graph of 𝑊(𝑥) in Example 21 with 𝑠 = 1.7 and 𝑠 = 1.3.

𝑟−1 󵄨 󵄨𝛾 ∞ ≤ max {1, 𝑀} ( ∑𝛼𝑗 󵄨󵄨󵄨󵄨𝜎𝑗+1 𝑥 − 𝜎𝑗+1 𝑦󵄨󵄨󵄨󵄨 + ∑𝛼𝑗 ) 𝑗=0

= max {1, 𝑀}

𝑗=𝑟

(1−𝑟)𝛾

𝑝

Denote by 𝐺(𝑊, 𝐷) the graph of the Weierstrass type (𝑠−2)𝑗 (−1)𝑥𝑗−1 on the dyadic series function 𝑊(𝑥) = ∑+∞ 𝑗=1 2

𝑟

field 𝐾2 defined in Example 21 and by 𝐺(𝑊⟨𝑚⟩ , 𝐷) the graph of 𝑚-order 𝑝-type derivative 𝑊⟨𝑚⟩ (𝑥) of 𝑊(𝑥); then [17] gives the following elegant results:

𝛾

(1 − 𝛼) − 𝛼 (𝑝 − 1) . (1 − 𝛼𝑝𝛾 ) (1 − 𝛼) (59)

The proof is complete. Example 21. The Weierstrass type function on the dyadic series field 𝐾2 is defined by [17] +∞

𝑊 (𝑥) = ∑ 2(𝑠−2)𝑗 Re 𝜒 (𝛽−𝑗 𝑥) , 𝑗=1

(60) Then 𝑊(𝑥) is 𝑚-order differentiable in the sense of (42) with 𝑚 < 2 − 𝑠. 𝑗 Since every 𝑥 ∈ 𝐷 can be expressed as 𝑥 = ∑+∞ 𝑗=0 𝑥𝑗 𝛽 , 𝑥𝑗 ∈ {0, 1}, 𝑗 = 0, 1, . . ., it follows that +∞

(61)

𝑗=1

Thus 𝑊(𝑥) is the FIF associated with the HIFS 𝛽𝑥 𝑥 𝑤1 ( ) = ( 1 2−𝑠 1 2−𝑠 ) , 𝑦 ( ) 𝑦+( ) 2 2 1 ⋅ 𝛽0 + 𝛽𝑥 𝑥 𝑤2 ( ) = ( 1 2−𝑠 1 2−𝑠 ) , 𝑦 ( ) 𝑦−( ) 2 2

dim𝐻𝐺 (𝑊, 𝐷) = dim𝐵 𝐺 (𝑊, 𝐷) = dim𝑃 𝐺 (𝑊, 𝐷) = 𝑠;

(63)

(2) the fractal dimensions of the graph 𝐺(𝑊⟨𝑚⟩ , 𝐷) satisfy for a.e. 𝑚 ∈ (1 − 𝑠, 2 − 𝑠)

𝑥 ∈ 𝐷, 1 ≤ 𝑠 < 2.

𝑊 (𝑥) = ∑ 2(𝑠−2)𝑗 (−1)𝑥𝑗−1 .

(1) the fractal dimensions of the graph 𝐺(𝑊, 𝐷) satisfy for a.e. 𝑠 ∈ (1, 2)

(62)

where 𝛼1 = 𝛼2 = (1/2)2−𝑠 , 𝜑1 (𝑥) = (1/2)2−𝑠 , and 𝜑2 (𝑥) = −(1/2)2−𝑠 . For every 𝛾 > 0, 𝜑1 , 𝜑2 ∈ Lip (𝛾, 𝐾2 ), by Theorem 20, we can get the result (see Figure 4).

dim𝐻𝐺 (𝑊⟨𝑚⟩ , 𝐷) = dim𝐵 𝐺 (𝑊⟨𝑚⟩ , 𝐷) = dim𝑃 𝐺 (𝑊⟨𝑚⟩ , 𝐷) = 𝑠 + 𝑚,

(64)

where dim𝐻𝐸, dim𝐵 𝐸, and dim𝑃 𝐸 are the Hausdorff, Box, and Packing dimension of 𝐸.

5. A Comparison between the FIF on R and on 𝐾𝑝 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 𝐾𝑝 . Firstly, we note that, in R case, the differentiability of FIF 𝑔𝛼 (𝑥) could not be guaranteed by that of 𝑔(𝑥), even 𝑔(𝑥) has more higher order smoothness. See Example 7; the function 𝑔(𝑥) = cos 𝑥 + 1/(2𝜆 − 1) is infinitely often differentiable; however, the set of nondifferential points of 𝑊𝜆 (𝑥) is dense in [0, 2𝜋]. Compare with the case in 𝐾𝑝 ; the differentiability of FIF 𝑓(𝑥) has 𝑚-order smoothness depending on the 𝛼𝑛 and 𝛾𝑛 of IFS. Since we use the results in Lemmas 17 and 18 which only

10 hold for 𝐾𝑝 but not for R. The relationship Lip (𝜆, 𝐾) = 𝐶𝜆 (𝐾), 𝜆 ∈ (0, +∞) on 𝐾𝑝 helps us to get the smoothness of the Weierstrass type function in Example 21. For more examples see [1, 18]. Secondly, in the R case, the fractal dimensions of FIF have few research results. In the 𝐾𝑝 case, not only may we have the 𝑝-type smoothness of FIF but also may determine the space 𝐶𝛾 (𝐾𝑝 ) 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 𝑝-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 R case, the interpolation points are 𝑥0 < 𝑥1 < ⋅ ⋅ ⋅ < 𝑥𝑁, and 𝑥0 = 𝑎, 𝑥𝑁 = 𝑏, where 𝑁 is a natural number; one may choose it as one wants. However, in the 𝐾𝑝 case, 𝑥𝑛 ∈ 𝐷, 𝑛 ∈ {0, 1, . . . , 𝑝 − 1} with a prime 𝑝.

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).

References [1] W. Y. Su, Harmonic Analysis and Fractal Analysis Over Local Fields and Applications, Science Press, Beijing, China, 2011. [2] M. F. Barnsley, “Fractal functions and interpolation,” Constructive Approximation, vol. 2, no. 4, pp. 303–329, 1986. [3] M. Barnsley, Fractals Everywhere, Academic Press, New York, NY, USA, 1988. [4] M. F. Barnsley, J. Elton, D. Hardin, and P. Massopust, “Hidden variable fractal interpolation functions,” SIAM Journal on Mathematical Analysis, vol. 20, no. 5, pp. 1218–1242, 1989. [5] M. F. Barnsley and A. N. Harrington, “The calculus of fractal interpolation functions,” Journal of Approximation Theory, vol. 57, no. 1, pp. 14–34, 1989. [6] S. R. Chen, “The non-differentiability of a class of fractal interpolation functions,” Acta Mathematica Scientia B, vol. 19, no. 4, pp. 425–430, 1999. [7] Z. Sha and G. Chen, “Haar expansions of a class of fractal interpolation functions and their logical derivatives,” Approximation Theory and Its Applications, vol. 9, no. 4, pp. 73–88, 1993. [8] G. Chen, “The smoothness and dimension of fractal interpolation functions,” Applied Mathematics, vol. 11, no. 4, pp. 409–418, 1996. [9] G. Z. Wang, “Dimension and differentiability of a class of fractal interpolation functions,” Applied Mathematics B, vol. 11, no. 1, pp. 85–100, 1996. [10] H. D. Li, Z. L. Ye, and H. S. Gao, “On the continuity and differentiability of a kind of fractal interpolation function,”

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Applied Mathematics and Mechanics, vol. 23, no. 4, pp. 471–478, 2002. M. A. Navascu´es, “Non-smooth polynomials,” International Journal of Mathematical Analysis, vol. 1, no. 1–4, pp. 159–174, 2007. M. H. Taibleson, Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ, USA, 1975. W. X. Zheng, W. Y. Su, and H. K. Jiang, “A note to the concept of derivatives on local fields,” Approximation Theory and Its Applications, vol. 6, no. 3, pp. 48–58, 1990. W. Y. Su, “Pseudo-differential operators and derivatives on locally compact Vilenkin groups,” Science in China A, vol. 35, no. 7, pp. 826–836, 1992. W. Y. Su and G. X. Chen, “Lipschitz classes on local fields,” Science China A, vol. 37, no. 4, pp. 385–394, 2007. W. Y. Su and Q. Xu, “Fuction spaces on local fields,” Science in China A, vol. 35, no. 12, pp. 1373–1383, 2005. H. Qiu and W. Su, “The connection between the orders of 𝑝-adic calculus and the dimensions of the Weierstrass type function in local fields,” Fractals, vol. 15, no. 3, pp. 279–287, 2007. Z. X. Wang, “Chains of function spaces over Euclidian spaces and local fields,” Analysis in Theory and Applications, vol. 25, no. 1, pp. 92–100, 2009.

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Discrete Mathematics

Journal of

Volume 2014

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Discrete Dynamics in Nature and Society

Journal of

Function Spaces Hindawi Publishing Corporation http://www.hindawi.com

Abstract and Applied Analysis

Volume 2014

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Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Journal of

Stochastic Analysis

Optimization

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014