On some classes of difference equations of infinite order

We consider a certain class of difference equations on an axis and a half-axis, and we establish a correspondence between such equations and simpler kinds of operator equations. The last operator equations can be solved by a special method like the Wiener-Hopf method.


Introduction
Difference equations o f finite order arise very often in various problems in mathemat ics and applied sciences, for example in mathematical physics and biology. The theory for solving such equations is very full for equations with constant coefficients [1,2], but fully incomplete for the case o f variable coefficients. Some kinds o f such equations were obtained by the second author by studying general boundary value problems for mode elliptic pseudo differential equations in canonical non-smooth domains, but there is no solution algorithm for all situations [3][4][5]. There is a certain intermediate case between the two mentioned above, namely it is a difference equation with constant coefficients of infinite order. Here we will briefly describe these situations.
The general form o f the linear difference equation o f order n is the following [1,2]: where the functions ak(x), k = 1,...,n , v(x) are defined on M and given, and u(x) is an unknown function. Since n e N is an arbitrary number and all points x,x + 1,...,x + n, Vx e M , should be in the set M , this set M may be a ray from a certain point or the whole R.
A more general type o f difference equation o f finite order is the equation k=0 where в }n=0 c R.
Further, such equations can be equations with a continuous variable or a discrete one, and this property separates such an equation on a class o f properly difference equations Springer and discrete equations. In this paper we will consider the case o f a continuous variable x, and a solution on the right-hand side will be considered in the space L 2 (R ) for all equations.
The function p n (i ) is called a symbol o f a difference operator on the left-hand side (3) (cf. [6]). I f pn(i ) = 0, V i e R, then (3) can easily be solved, u(x) = F-\ x (P-1 ( i )v ( i ) ).

D ifference equ ation o f infinite order w ith constant coefficients
The same arguments are applicable for the case o f an unbounded sequence {вк} + TO. Then the difference operator with complex coefficients Proof The proof o f this assertion can be obtained immediately. □ I f we consider the operator (4) for x e Z only -TO then its symbol can be defined by the discrete Fourier transform [7,8] o, d( f ) = ^akelfikf, f e [ -п , п ].

Difference and discrete equations
Obviously there are some relations between difference and discrete equations. Particu larly, if {fik} -" = Z, then the operator (5) is a discrete convolution operator. For studying discrete operators in a half-space the authors have developed a certain analytic technique For studying this equation we will use methods o f the theory o f multi-dimensional sin gular integral and pseudo differential equations [3,6,12] which are non-usual in the theory o f difference equations. Our next goal is to study multi-dimensional difference equations, and this one-dimensional variant is a model for considering other complicated situations.
This approach is based on the classical Riemann boundary value problem and the theory o f one-dimensional singular integral equations [13][14][15].

Background
The first step is the following. W e will use the theory o f so-called paired equations [15] of in the space L 2 (R), where a, b are convolution operators with corresponding functions a(x), b(x), x e R, P± are projectors on the half-axis R ± . M ore precisely, Applying the Fourier transform to (7) we obtain [12] the following one-dimensional sin gular integral equation [13][14][15]: [9][10][11]. Below we will try to enlarge this technique for more general situations.

General difference equations
W e consider the equation the type (8) where P, Q are two projectors related to the Hilbert transform Equation (8) is closely related to the Riemann boundary value problem [13,14] where G ( f ), g (f ) are given functions on R.
There is a one-to-one correspondence between the Riemann boundary value problem (9) and the singular integral equation (8), and

Topological barrier
W e suppose that the symbol G ( f ) is a continuous non-vanishing function on the compactification R ( G ( f ) = 0 , Vf e ]R) and The last condition (10), is necessary and sufficient for the unique solvability o f the prob lem (9) in the space L 2(R ) [13,14]. Moreover, the unique solution o f the problem (9) can be constructed with a help o f the Cauchy type integral Ф+ t = G +(t)P(G-\t)g(t)), where G± are factors o f a factorization for the G(t) (see below), G+(t) = exp(P(ln G (t))), G-(t) = exp(Q(ln G (t))).

Difference equations on a half-axis
Equation (6) can easily be transformed into (7) in the following way. Since the right-hand side in (6) is defined on R+ only we will continue v(x) on the whole R so that this con tinuation If e L 2(R). Further we will rename the unknown function u+(x) and define the Thus, we have the following equation: which holds for the whole space R.
A fter the Fourier transform we have where a ( f ) is called a symbol o f the operator D.

D efin ition A factorization for an elliptic symbol is called its representation if it is in the
where the factors о+, о_ admit an analytic continuation into the upper and lower complex half-planes C ± , and o^1 e L c (R).
Example 1 Let us consider the Cauchy type integral It is well known this construction plays a crucial role for a decomposition L 2( orthogonal subspaces, namely on two where A ± (R ) consists o f functions admitting an analytic continuation onto C±.
The boundary values o f the integral Ф^) satisfy the Plemelj-Sokhotskii formulas [13,14], and thus the projectors P and Q are corresponding projectors on the spaces o f analytic functions [15].
The simple example we need is exp(u) = exp(Pu) • exp(Qu).
Th eorem 2 Let о (S) e C (R ), Ind о = 0. Then (6) has unique solution in the space L 2(R+) for arbitrary right-hand side v e L 2(R+), and its Fourier transform is given by the formula Proof

General solution
Since a ( f ) e C (R ), and Ind a is an integer, we consider the case ж = Ind a e N in this section.
Th eorem 3 Let Ind a e N. Then a general solution of ( 6) in the Fourier image can be writ ten in the form and it depends on ж arbitrary constants.
Taking into account our notations we have and we conclude from the last that the left-hand side and the right-hand side also are a polynomial P*-1( f ) o f order ж -1. It follows from the generalized Liouville theorem [13,14] because C orollary 4 Let v(x) = 0, ж e N. Then a general solution of the homogeneous equation (6) is given by the formula U + (f ) = ( f -i)-*a+71( f )P e-1 (f).

Solvability conditions
Th eorem 5 Let -Ind a e N. Then (6) has a solutionfrom L 2(R+) iff the following conditions hold: Proof W e argue as above and use the equality (14); we write it as Since we work with L 2(R ) both the left-hand side and the right-hand side are equal to zero at infinity, hence these are zeros, and