Research On Differentiability Of FIF

This thesis consists of five parts.The first part of thesis is introduction.Firstly, we introduce some basic concepts on fractals and several mathematical research fields on fractals.Then, the concepts of fractal interpolation functions are mainly stated. Fractal interpolation functions (FIF for short) which were firstly defined by M. F. Barnsley in1986are based on Iterated Function System (IFS for short){K, wi:i=1,2,..., N}. There are two kinds of FIF’s based on the different two kinds of IFS’s:one is AFIF, which is defined by LIFS (1.2.3); the other is NLFIF, which is defined by NLIFS (1.2.5). Our topic is mainly based on a special kind of NLFIF ga, which is called an α-fractal function associated to g, where qi in (1.2.5) is defined by (1.2.6).Moreover, definitions of Walsh function system on [0,1) and the logical deriva-tives are given. Also the definitions and properties of local fields are stated. Finally, we introduce the p-type derivatives and integrals of functions defined on local fields.At the end of this part, some usual notations, definitions and basic results in Nevanlinna’s value distribution theory are shown.In the second part, we prove some conclusions on the differentiability of these two kind of FIF’s. We mainly study the non-differentiability of the a-fractal function ga, and have got some conditions under which ga is continuous everywhere but the set of its non-differentiable points is dense in its domain. Also we give some examples based on Weierstrass functions to support our conclusions. Firstly,several conditions under which AFIF f defined by(2.1.2) and (2.1.3) is nowhere differentiable in(0,1)are found: and there exists some j∈{1,2…,N)such that αj≠1/N,then f(x) is nowhere differentiable in(0,1). for each i∈{1,2,...,N),then f(x) is nowhere differentiable in (0,1).3.If|αi|≥1/N for each i∈{1,2…,N), and yi+1-yi-1≠2yi for each i∈{1,2,...,N-1),then f(x) is nowhere differentiable in(0,1).We get the hyperbolic FIF f*defined in§2.1by modifying some conditions. If αi≥0for each i∈{2,3…,N),or there exists some j∈{1,2…,N)such that αj=0,then f*(x)is almosrt everywhere differentiable.Moreover,our research on the smoothness of NLFIF consists of two parts:one is some conclusions on the modulus of continuity of NLFIF which is defined by NLIFS(2.2.1)；the other is on the differentiability of the α-fractal function gα.We have the following conclusions.Theorem2.2.2If there exists some i∈{1,2,...,N) such that αi=0,and g∈C1[0,1],then gα is almost everywhere differentiable in (0,1).Then,we improve Navascues’ conclusions(Theorem2.2.3),such that the result can be used more widely_Theorem2.2.4If g∈C1[0,1],|αi|≥1/N,i=1,2,...,N, and g’(x)does not agree with yN-y0in a non-empty open subinterval of I=[0,1],then the set of points at which gα is not differentiable is dense on I.Finally,we give some examples as applications of Theorem2.2.3and Theorem2.2.4.The main idea is based on Weierstrass function which is defined by IFSEspecially, when the scaling factors αi=2-λi,i=1,2are different,we con-struct a kind of so-callled "Weierstrass-like" function which is continuous everywhere but the set of its non-differentiable points is dense in its domain.In the third part,an IFS {L=B1×R,wi:i=1,2)on the p-series local field Kp is defined by with|αn|<1and φn:D→R is in Lipschitz class Lip(γn,Kp),where γn≥1.Thus,We have the following conclusions.Theorem3.2.1Let{L,wn:n=0,1,…,p-1)be the IFS defined in (3.2.1) with|αn|<1and φn∈Lip(γn,Kp) with γn≥1,n∈{0,1,…,p-1).Then there exists an ultra-metric d*L on L,such that the IFS in (3.2.1) is hyperbolic with respect to d*L.Moreover, there exists a unique nonempty compact set G(?)L such thatThe fractal interpolation functions on Kp(LFFIF) are defined by the above IFS(3.2.1).Furthermore,We get the explicit representation of LFFIF.Property3.2.1Let+∞f:D→R be the FIF associated with the HIFS in we have 0where we agree on∏αxk=1and the shift operator σ:D→D is defined by k=-1σx=(β-1x)|D, i.e.,Furthermore, the2-adic logical derivative of FIF defined by IFS (3.3.1) and the p-adic logical derivative of FIF defined by IFS (3.3.6) are calculated.Theorem3.3.1Let f(x) be the FIF corresponding to (3.3.1). ck, k=0,1, are the Walsh-Fourier coefficients of f(x). Then we have when where and 2n-1Theorem3.3.2Let f{x) be the FIF associated to IFS (3.3.1), then∑ckwk(x) κ=0is uniformly convergent to f(x) in [0,1) as n→∞, where Ck, k=0,1,... are defined by (3.3.4) and (3.3.5).Theorem3.3.3Let f(x) be the FIF corresponding to the IFS (3.3.6), then the Walsh-Fourier coefficients of f(x) satisfy when k=(k-s+1k-s+2...k0;00...), where∏define pn-1Theorem3.3.4Let f(x) be the FIF associated to IFS (3.3.6), then∑ckwk(x) k=0is uniformly convergent to f(x) in [0,1) as n→∞, where ck,k=0,1,... are defined by (3.3.8) and (3.3.9). pn-1Theorem3.3.5Let f(x) be the FIF associated to IFS (3.3.6). If limn→∞k=0∑kckwk(x) exists for every x∈I, then f(1)(x) exists, and where p≥2is a positive number.Moreover, we discuss the smoothness of LFFIF defined by IFS (3.2.1) and get the following important results.Lemma3.4.1Let f:D→R be the LFFIF associated with the HIFS where|a|<1,γ≥1and φ∈Lip(γ, Kp) with γ≠logp A, then f is continuous in D, moreover,Using the above lemma, we can have the following.Theorem3.4.1Let f:D→R be the LFFIF associated with the HIFS in (3.4.1), where|a|<1,γ≥1and φ∈Lip(γ,Kp) with γ≠logp1/|a|, then f is m-order differentiable in the sense of (1.3.5) with any m∈[0,min{γ, logp1/|a|}).Moreover, we consider in a general case.Theorem3.4.2Let f:D→R be the LFFIF associated with the HIFS in (3.2.1), where|an|<1,γn>1and φn∈Lip(γn, Kp) for each n∈{0,1,...,p-1}. Then f is m-order differentiable in the sense of (1.3.5) with any m∈[0, min{γ,logp1/α}).According to Theorem3.4.2, we can prove that The Weierstrass type function W(x) on the dyadic series field K2defined by is m-order differentiable in the sense of (1.3.5) with m<2-s.At the end of the third part, a comparison between the FIF on R and the LFFIF on Kp is given. The essential distinction is generated from the different structures of the base spaces.In this part,we use the relationship Lip(λ,K)=Cλ(K),λ∈(0,+∞)on Kp which only hold for Kp but not for R to get the smoothness of LFFI F.In the fourth part,we study on the degrees of growth of Meromorphic solutions of complex functional difference equations,and get the following conclusions.Theorem4.1.6Suppose that,(z)is a transcendental meromorphic solution of the functional difference equation where cv(v=1,…,n)are distinct complex constants,I={λ=(lλ,1,lλ,2,…,lλ,n) lλ,v∈N∪{0},v=1,2,…,n} and J={μ=(mμ,1,mμ,2,…,mμ,n):mμ,v,∈N∪{0},v=1,2,…,n} are two finite index sets,p(z)=pkzk+pk-1zk-1+…+p0∈C[z] of degree k>1,and Q(z,u) is a rational function in u of degu Q=g(>0). We also suppose that all the coefficients of (4.1.3)are small functions telative to f. Denoting Then qk≤σ, and whereTheorem4.1.7Suppose that f is a transcendental meromorphic solution of the equation where cv(v=1,…,n) are distinct complex constanst,I={λ=(lλ,1,lλ,2,…,lλ,n): lλ,v∈N∪{0},v=1,2,…,n} and J={μ=(mμ,1,mμ,2,…,mμ,n):mμ,v,∈N∪{0},v=1,2,…,n} are two finite index sets,a,b∈C and Q(z,u) is a rational function in u of degu Q=q(>0).We also suppose that all the coefficients of (4.1.3) are small functions relative to f.Denoting (ⅰ)If0<|a|<1,then we have (ⅱ)If|a|>1,then we have q≤σ and (ⅲ)If|a|=1and q>σ,then we have μ(f)=ρ(f)=∞In the last part of this thesis,some open problems are stated.