Xidong Sun,Baifeng Liu,Xiliang Li,Yuliang Han

(Shandong Institute of Business and Technology,Yantai 264005,Shandong, E-mail:sunxd98@163.com(X.Sun))

DISCRETE ANALYTIC ALMOST PERIODIC FUNCTIONS IN A STRIP??

Xidong Sun,Baifeng Liu,Xiliang Li,Yuliang Han

(Shandong Institute of Business and Technology,Yantai 264005,Shandong, E-mail:sunxd98@163.com(X.Sun))

In this paper we study some properties of discrete analytic functions in a strip.In particular,we investigate some basic properties of discrete analytic almost periodic functions and show the existence of the discrete analytic almost periodic solutions to some discrete derivative equation in a strip.

discrete analyticity;almost periodic sequences in a strip;discrete derivative equation

2000 Mathematics Subject Classification 30A95;30G25;39A20

Ann.of Appl.Math.

31:2(2015),200-211

1　Introduction

Almost periodic sequence may be regarded as discrete version of(H.Boch)almost periodicity,which was introduced and studied in[1-3,5-8].These theories have received much attention for their importance in various fields,such as numerical methods of differential equations and dynamical systems,finite elements techniques,control theory and computer sciences[2,4-7,10].In the complex domain,H.Boch also studied the analytic almost periodic functions and established their principal properties[11].Some further results were obtained by A.S.Besicovitch,C.Corduneanu and others in[1,12].However,to the best of our knowledge,the sequence of analytic almost periodic functions is an untreated topic, which is the main motivation of this paper.

The discrete version of analytic functions is said to be discrete analytic functions,which was earliest studies by J.Ferrand in 1944[9].The properties of discrete analytic functions have been extensively investigated by R.J.Duffin[13,15,16],D.Zeilberger[14].In these works,it was shown that discrete analytic functions share many important properties of the classical continuous analytic functions in the complex domain,such as differentiation, integration,Cauchy’s integral formula,the maximum modulus principle,the classic theorems of Paley-Wiener and differential equations.Recently,the theory of discrete analytic functions has drawn a lot of attention and been extended under the influence of the statistical physics of lattice models,electrical networks[17-19].

In this paper,we first define and study the discrete analytic almost periodic function, which is regarded as a discrete version of the analytic almost periodic functions of a continuous complex variable,then we investigate some properties of such function sequences anddiscuss some discrete analytic almost periodic solutions of the discrete derivative equation. In Section 2,we present the discrete Cauchy’s integral formula in a strip.In Section 3,we give some properties of the discrete analytic almost periodic function.In the last section, we consider the existence of the discrete analytic almost periodic solutions to the discrete derivative equation.

Let Z be a set of integers.The points of complex plane with integer coordinate are called lattice points.We denote by M=Z+iZ the set of lattice points.If z=m+in is a lattice point,the points z,z+1,z+1+i,z+i are the vertices of a unit square associated with the point z=m+in.Regions are defined as the union of the unit square.A simple region R is a simply connected set,which is the union of a finite unit squares.We denote by(a,b)(respectively,[a,b])the strip a＜Rez＜b(respectively,a≤Rez≤b)(a,b∈Z)in the complex plane.

First,we recall some basic concepts and main results of discrete analytic function,which were introduced by R.J.Duffin in[13].

Definition 1.1 A complex-valued function f:M→C is said to be discrete analytic on a unit square if

where L denotes a linear operator.

The function f is said to be discrete analytic in region R if f is discrete analytic on every unit square in region R.

Let p=z0,z1,z2,···,zm=q(p,q∈M)denote a chain of lattice points.The linear integral of the function f:M→C from p to q is defined by

Note that if B is the boundary of a simple region R,then

Suppose that f is discrete analytic in a simple region R with boundary B,then

Let p and z be points of simple region R.It is also shown that if f is discrete analytic, then

is discrete analytic in R for integration path taken in R.R.J.Duffin symbolized such a relation by the notation

Let f and g be two discrete functions.Line integrals of two functions are defined as follows:

If B is a closed path,p=q and these summations may be reformed as follows:

where zm+1=z1.Let g′′be the second derivative of g,then it is shown that

Supposed that a given lattice function q(z)satisfy the following conditions

Then the following result is regarded as the discrete analog of the Cauchy’s integral formula.

Proposition 1.1 Suppose that the contour B is the boundary of a simple region R and B is described in the counter-clockwise sense.If f is discrete analytic in a simple region R, z0lies in the interior of R,then

where q(z)satisfies(1.7).

Proposition 1.2[20](discrete maximum principle)Let R?M be a finite region and f be discrete analytic in the interior of R,then|f|takes its maximum at the boundary of R.

Now,we give the definition of discrete analytic almost periodic functions.

Definition 1.2 A function f:M→C(M in the strip)is called almost periodic in this strip if for any ε＞0,there exists an integer N(ε)＞0 such that among any N consecutive integers there exists an integer η on the imaginary axis,for which

The number η is called an ε-translation number of f.

Remark 1.1 The function f(z)=f(m+in)is almost periodic in n,uniformly with respect to m,a≤m≤b.

Definition 1.3 A function f:M→C is said to be discrete analytic almost periodic functions if it is discrete analytic and almost periodic in a strip.

Denote by DAAP(Z)the set of all such functions.

2　Discrete Analytic Functions

In this section,we generalize the discrete analog of Cauchy’s integral formula from a simple region to a strip.

Theorem 2.1 Let f be a discrete analytic function in a strip[a,b].Suppose that for each fixed z0∈(a,b),

where q(z)is defined as(1.7),then

Proof(see[13])Let R be the rectangle a≤m≤b,-r≤n≤r.If q(z)is replaced by q(z-z0),then it is followed from(1.8)that

where discrete contour RBis the boundary of the region R.A closed contour RBcontains four parts of interval of the parallel to the x axis(respectively the y axis).Hence,(2.3)can be written as

As r→∞,it follows from(2.1)that the integration of the contour on the interval of the Imz=r or Imz=-r tends to zero.Using(1.3)-(1.6),one has

The proof is complete.

Remark 2.1 Relation(2.2)is a discrete analog of the Cauchy’s integral formula in the stripe,which can be written as

where

The following result is useful in the proof of Theorem 3.4.

Theorem 2.2 Set

where ρ(x,y)=1+ieix-ieiy-eix+iy,z=m+in,then there exists a fixed integer N≥n0such that

Since the double integral of equation(2.4)is absolutely convergent,we can evaluate it as an iterated integral.The integration is first carried out with respect to y.

Set

We distinguish the following cases:

Combing Cases(1)and(2),the proof is complete.

3　Discrete Analytic Almost Periodic Functions

In this section,we consider some properties of the discrete analytic almost periodic function in a strip.

Theorem 3.1 If the function f(z)is a discrete analytic almost periodic function in a strip,then it is bounded in this strip.

Proof Let ε=1,N=N(1)be the corresponding consecutive integers in the definition of discrete analytic almost periodic function.Applying the maximum principle to the function f(z)in a rectangle a≤m≤b,0≤n≤N,we find the function f(z)is bounded. SetConsider now any integer n and a 1-ε translation number η of f(z)on the imaginary axis,which belong to sequence interval[-n,-n+N].It follows that 0≤n+η≤N,

Thus,the proof is complete.

Theorem 3.2A function f(z)=f(m+in)is almost periodic in n,uniformly with respect to m,a≤m≤b if and only if the family{f(z+ih)}is normal on M.

Proof In a strip[a,b],m∈M can only choose finite integer number a=m1,m2,···,mk=b. For every fixed mi,i=1,2,···,k,one prove this result in the same way as Theorem 2.4 in[5].

Theorem 3.3 A lattice points sequence Am,nis almost periodic in a strip if and only if there exists an almost periodic function f(z)such that Am,n=f(m+in).

Proof(see[1])If the function f(z)is an almost periodic function in a strip,then it follows that for any integer sequence nk,one can extract a subsequence f(z+in1k)converging uniformly on the straight line Rez=m.Consequently,the sequence f(m+i(n+n1k))is uniformly convergent with respect to n as k→∞.It shows that the function of an integer variable f(m+in)is normal,hence sequence Am,n=f(m+in)is almost periodic.

Conversely,suppose that Am,nis an almost periodic sequence.We define a function f in a strip by the following relations:

where m≤x＜m+1,n≤y＜n+1.

We notice that f(m+in)=Am,n.Next we show that f defined by(3.1)is almost periodic.Since f(m+in)is an almost periodic function of integer variable,for any ε＞0 there exists an integer N(ε)＞0 such that among any N consecutive integers there exists at least onetranslation-number of f(m+in).Consider atranslation number η.If m≤x＜m+1,n≤y＜n+1,then n+p≤y+p＜n+p+1.Since

Note that 0≤y-n＜1,m≤x＜m+1,we obtain

The proof is complete.

Let R1denote a lower half-strip a≤Rez≤b,Imz≤N.We define

which is called lattice point interval.The boundary of lower-half strip R1is composed of three parts of the lattice points interval,

Theorem 3.4 Suppose that the function f(z)is discrete analytic and satisfies condition (2.1)in the lower half strip R1.If f(z)is discrete analytic almost periodic on the lattice points interval B1,B2and B3,then f(z)is discrete analytic almost periodic in the lower half strip.

Proof The function q(z)is defined by(2.4).It is easy to show that q(z)satisfies the condition(1.7).Let the lattice points interval,then B1,B2,B3and B4is the close contour,which are regarded as the boundaries of the rectangle R2.To any z0∈R2,one has

The proof of the previous relation is similar to that of Theorem 2.1.So,we have

Set φ(z)=f(z+ip)-f(z).We obtain

By the condition of the theorem,we know that

From Theorem 2.2,one has

Similar to the analysis of(3.3),we obtain the following inequalities

Together(3.2)and(3.4)-(3.6),we obtain that|φ(z0)|＜ε,which is

According to Definition 1.2,the proof is complete.

4　Discrete Derivative Equations

In[15],the author investigated the discrete analytic continuation of solutions to difference equations.In this section,we consider whether the bounded solutions to this type of difference equations is discrete analytic almost periodic.

Let A be an n×n matrix of complex constants such that±2 and±2i are not eigenvalues of A.The function W(z)=(w1,w2,···,wn)Tand F(z)=(f1,f2,···,fn)Tare n-dimensional vector functions in a simple region R of the complex plane,define‖F(xiàn)‖= max{‖fi‖,1≤i≤n}.Consider the following non-homogeneous discrete derivative system

Proposition 4.1[15]Suppose that the function F(z)is a discrete analytic function in a simple region R containing the origin.Then system(4.1)has a discrete analytic solution

where C is a constant n-vector,and if R is simply connected,F(z)is single valued in R.

Now,we shall study the existence of the discrete analytic almost periodic solutions to equations(4.1).

Theorem 4.1Suppose that the function F(z)is a discrete analytic almost periodic function in a simple region R containing the origin.Then in a strip,equation(4.1)has a discrete analytic almost periodic solution.Moreover

where M ＞0 depends only on the matrix A,whose eigenvalues λ≠0.

Proof By the same discussion as in[1,Theorem 4.2],we could assume that the matrix A is triangular,that is,

First,let us consider the discrete derivative equation

whose general solution is

If the function f(z)is discrete analytic almost periodic and satisfies conditions of Theorem 4.1,we shall show that the bounded solution to equation(4.5)is discrete analytic almost periodic.

If e(z,λ)and f are analytic,the previous integral is independent of the path of integration.If the following path of integration is chosen as the lattice points intervalfrom z to∞,then

so,w0(z)is bounded.

For f(z)∈DAAP(Z),we show that w0(z)is almost periodic.

where t∈L.

When K＜1,similarly,we obtain that

is a bounded almost periodic solution to equation(4.5).It is analogous to show that

and w0(z)∈DAAP(Z).

Thus,the existence of discrete analytic almost periodic solution to equation(4.5)is proved.

Note that equation(4.1)has the form of(4.5),therefore,there exists a discrete analytic almost periodic solution wnto(4.5),namely,if K＜1,

and if K＞1, Substituting wninto(4.1),for wn-1we get an equation form of(4.5).Since eigenvalues of A are not 0,±2 and±2i,from the above analysis it follows that wn-1is discrete analytic almost periodic.Hence,repeating the above process,we obtain that the bounded solution considered is discrete analytic almost periodic.

In what follows,we shall prove inequality(4.3).

Consider the following equation

The bounded solution wn-1of equation(4.10)is analogously expressed by the form(4.8)or (4.9).Hence,

where Kn-1=d(d|an-1,n|+1).Proceeding in same manner,we get

If we set M=max{Mi:1≤i≤n},then‖W‖≤M‖F(xiàn)‖.

The proof is complete.

Acknowledgement The author would like to express his gratitude to Professor Jialin Hong for his encouragement during the work and the anonymous referee for his/her helpful remarks and suggestions.Part of this work was done while the first author was visiting the Academy of Mathematics and Systems Science Chinese Academy of Sciences.

References

[1]C.Corduneanu,Almost Periodic Functions,Chelsea,New York,1989.

[2]J.Hong,C.Nuaez,The almost periodic difference equations,Math.Comput.Modelling, 28(1998),21-31.

[3]C.Zhang,Almost Periodic Type Functions and Ergodicity,Beijing:Science Press,2006.

[4]A.I.Alonso,J.Hong,R.Obaya,Almost periodic type solutions of some differential equations with piecewise constant argument via almost periodic type sequences,Appl.Math.Lett., 13(2000),131-137.

[5]Y.Han,J.Hong,Almost periodic random sequence in probability,J.Math.Anal.Appl., 336(2007),962-974.

[6]C.Niu,X.Chen,Almost periodic sequence solutions of a discrete Lotka-Volterra competitive system with feedback control,Nonlinear Anal.,10(2009),3152-3161.

[7]R.Yuan,J.Hong,The existence of almost periodic solutions for a class of differential equations with piecewise constant argument,Nonlinear Anal.,28(1997),1439-1450.

[8]H.Tornehave,On almost periodic movements,Mat.-Fys.Medd.Danske Vid.Selsk.,28(1954), 1-10.

[9]J.Ferrand,Fonctions préharmoniques et fonctions préholorphes,Bull.Sci.Math.,68(1944), 152-180.

[10]J.Hong,R.Obaya,A.Sanz,Almost periodic type solutions of some differential equations with piecewise constant argument,Nonlinear Anal.,45(2001),661-688.

[11]H.Bohr,Almost Periodic Functions,Chelsea,New York,1947.

[12]A.S.Besicovitch,Almost Periodic Functions,Dover,New York,1955.

[13]R.J.Duffin,Basic properties of discrete analytic functions,Duke Math.J.,23(1956),335-363.

[14]D.Zeilberger,H.Dym,Futher properties of discrete analytic functions,J.Math.Anal.Appl., 58(1977),405-418.

[15]R.J.Duffin,Discrete analytic continuation of solutions of difference equatons,J.Math.Anal. Appl.,9(1964),252-267.

[16]R.J.Duffin,C.S.Duris,A convolution product for discrete function theory,Duke Math.J., 31(1964),199-220.

[17]D.Alpay,P.Jorgense,R.Seager,D.Volok,On discrete analytic functions:products,rational functions and reproducing kernels,J.Appl.Math.Comput.,41(2013),393-426.

[18]C.Kiselman,Functions on discrete sets holomorphic in the sense of Ferrand,or monodiffric functions of the second kind,Sci.China,51A(2008),604-619.

[19]S.Smirnov,Discrete complex analysis and probability,Proceedings of the International Congress of Mathematicians,Hindustan Book Agency,New Delhi,1(2010),595-621.

[20]R.Felipe;M.G.Arroyo,The discrete Szegol kernel,J.difference Equ.Appl.,4(2008),367-380.

(edited by Liangwei Huang)

?The work was supported by NNSF of China(Nos.11171191,11201266)and NSF of Shandong Province(No.ZR2012AL01).

?Manuscript October 13,2014