Difference operators and applications to the moment problem Текст научной статьи по специальности «Математика»

DIFFERENCE OPERATORS AND APPLICATIONS TO THE MOMENT _PROBLEM_

Strezhneva E. V.,

Associate professor, Ph.D of physical and mathematical sciences, Department of Special Mathematics, Kazan National Research Technical University named after A.N.Tupolev - KAI (KNRTU-KAI)

Falahieva E.R.,

second year student of the Institute of Radio Electronics and Telecommunications , Kazan National Research Technical University named after A.N.Tupolev - KAI (KNRTU-KAI)

ABSTRACT. We consider four-element difference equations with constant coefficients in the space of hol-omorphic functions outside the square and decaying at infinity. We are aimed at the properties of biorthogonal expansions induced by the corresponding difference operators. Applications to the moment problem for entire functions of exponential type are given.

Key words: Difference operator, method of regularization, biorthogonal expansion, moment problem.

1. Introduction

Let D be a square with the vertices at t1 = -t3 = 2-1(1 + i),t2 == -t4 = 2-1(1 - i) and with the sides lj, j = 1,4, counted in the positive direction along the boundary of the square (t E l1 ^ Imt = -2-1). Define the functions ai (z) = z + V to be generating transformations (or their inverse) of the doubly periodic group with the prime periods 1 and i.

Let us consider a linear difference equation with constant coefficients

(Vf)(z) = ZUWWj (z)] = g(z),z E D (1)

under the following assumptions:

1) The function f(z) is holomorphic outside the closure D of D,f(<x>) = 0, and its boundary value f-(t) satisfies the Holder condition in any compact avoiding the vertices. This class we denote by B.

2) The free term g(z) is holomorphic inside D and its boundary value is defined as g+ (t) E Hx(dD).

3) Aj ± 0 for all j.

4) At most logarithmic singularities are allowed at the vertices.

Condition 3) provides the nontriviality of the problem. Suppose, for instance, X1 = 0. The set C\ Ut^Pk1^) is connected, and the function g(z) must be analytically extendable from D to a neighborhood of infinity with g(m) = 0. It remains to apply the Borel transformation ([6], Chapter 1, Section 1) to the linear difference equation (1). Then, (1) implies h(z)F(z) = G(z) where F(z) and G(z) are the Borel upper functions associatied with the lower functions f(z) and g(z) respectively.

The function h(z) = X2 exp(z) + X3 exp(iz) + X4exp(-z)

is the characteristic quasipolynomial. The entire function G(z) must have zeros at least of the same multiplicity as the characteristic quasipolynomial. The function f(z) can be found explicitly under the above condition and we only must check that f E B. Therefore, we will normalize the equation (1) with the condition X1 = 1 in what follows. Condition 4) guarantees the finiteness of the fundamental system of solutions of the homogeneous equation. Otherwise, there would be (Vf)(z) = 0,zED^ (Vf(k))(z) = 0,zED and any system of derivatives of the function f(z) cannot be

linearly dependent because of the definition of the class B.

Two particular cases of the linear difference equation (1) (one is Vj Aj = 1, and the other one is X3 = 1,X2 = X4 = —1) were examined previously in [9]. The properties of biorthogonal decompositions related to such cases were formulated in [13, 15]. This paper consists of four parts. A method of regularization is proposed in Section 2 for the linear difference equation (1) with the additional restriction for the coefficients

X3 = X2X4. (2)

The complete study of the linear difference equation (1) with X2 = 1,X3 = X4 = —1 (case III) is given in Section 3. Unconditional solvability is established.

Various biorthogonal expansions related to case III are examined in Section 4. The case III is essentially different from from cases I and II as is observed in details in Remark 5).

Applications of (1) to the moment problem for entire functions of exponential type are given in Section5. 2. Method of regularization We look for a solution of the equation (1) in the form of the integral of Cauchy type

f(z) = (2ni)-1 JgD cp(T)(T — z)-1 dT,z£D (3) with unknown density that satisfies the Holder condition in the closure of each side of D. Then,

(1) ^ (A<p)(z) = (2ni)-1igD^(T)E(T — z) dT = g(z),z e D (4) where the kernel function is

E(u) = (u — i)-1 + X2(u + 1)-1 + X3(u + i)-1 +X4(u — 1)-1. (5) The functions Oj (z) induce the involutive and disconnected shift a(t) = [Oj(t),t e a(t): d+D ^ d-D at the vertices. Let us introduce an involutive operator in view of (2)

W: <p(t) ^ 4x^et<p[a(t)}, (6) where the piecewise constant is 9t = {1,t e I1; Xj, t e li,j = 22J}.

Let us consider a holomorphic function a(z) in D with the boundary values from the same class, and let us require that

y + a+=Wy + Wa+. (7)

It is a non-homogeneous Carleman boundary value problem with respect to the unknown function a(z) with a piece-wise constant coefficient and of index zero. It is unconditionally solvable if

±m1 + m.2i; m^m.2 e L , where we assume a continuous branch of the logarithm at each side of D.

Related results can be found in [16] and their refinement in ([2], page 22). The latter condition can be satisfied always by selecting the branch of the root in (6). Let us fix the corresponding branch. If both problems (7) are unconditionally solvable, then the choice of the branch is irrelevant.

Since z g D in (3), we assume without loss of generality that

cp = W(p. (8)

Therefore, an analogue of the Sokhotskiy formula + A holds, where the singular integral operator A is obtained by a formal replacement of z e D byt e dD, understood as the principal Cauchy value. Therefore, the following formula

—2-14T3cp + Acp = g+. (9)

can be derived from (4).

Let us apply the operator W to the both sides of (9), and sum up the resulting expression and the original one. Finally, we have

Tv=g++ Wg+

(10)

(11)

in view of (8), where

T = —JT3] + AW + WA. Lemma 1. The kernel of the integral operator

(11)

T(p = 0 (13)

can be constructed in such a way that each entering function either satisfies the condition (8) or the reciprocal condition

<p = —Wcp. The adjoint equation

Tip = 0,

(14)

(15)

is defined by the operator T ' = 1J — AW1 — W1A, where the involuntary operator is

W1: 9(t)—//i-Teait)9[a(t)] . (16)

Recall that the branch of the root is already fixed. Clearly, 9a(t) = d-t, with an odd shift a(t). Therefore, the fundamental system of solutions to the ad joint equation (15) can be constructed so that each entering function either satisfies the condition

V = (17) or the reciprocal condition

V = —W1V. (18) Let us assume from the very beginning that both

non-homogeneous problems (7) are unconditionally solvable. Then, both respective homogeneous problems have only a trivial solution. Let the fundamental system of solutions to the adjoint equation (15) contain m functions {npj(t)},j = l,m, satisfying property (18), i.e., equation (10) has m solvability conditions

Lng+(t)xpj(t)dt = 0,j = 1,m.

(19)

K(t, t) = ^[dtEtu — a(tj) — 6aWE(a(t) — t)] (12)

is bounded, and therefore, the operators A and W are anticommutive modulo a compact term from L2(dD).

Proof. The boundness of the kernel (12) is established with a simple checking the various options of interrelated positions of the points t and t at the sides of the square taking into account condition (2).

We remark that the vertices are the points of discontinuity of jump type for the kernel (12) at each of the variables. The same holds for any partial derivative of the kernel.

Corollary 1. Equation (10) is the Fredholm equation.

Corollary 2. The operator A is regularizing to itself.

Indeed, Aa+ = 2-1^A^Wa+, and therefore, A2 = A[(A — 2-1JT3W) + +2-1JT3W] = —2-2A3J + 2-1JT3(WA + AW).

Let us remark that an abstract theory of such special operators was developed in [1].

Thus, the main result of this section follows.

Theorem 1. If condition (2) holds, then the linear difference equation (1) has at most finite number of solvability conditions.

Now, let us study the equivalence of the regulari-zation, i.e., let us verify that (10) implies (1).

The fundamental system of solutions to the homogeneous equation

Clearly, the right-hand side of equation (10) is orthogonal to the functions satisfying (17). If condition (19) holds, then there exists a solution to equation (10) satisfying property (8) (see, for example, [3]). Therefore, (10) ^ A+y + +WA+ç = g+ + Wg+ ^ (1). Let us assume now that only one of non-homogeneous problems (7) is unconditionally solvable, i.e., a+ = -Wa+ ^ a(z) = (exp(Az). The exponential a(-t) satisfies equalities (15) and (18), i.e., the number of the solvability conditions (19) equals exactly m - 1. However, now

(10) ^A+y + WA+(p = g+ + Wg+ ^

(A<p)(z) = g(z) + (a(z)

and this extra condition ensures that (J = 0. Let us write it as

(A<p)(0) = g(0). (20)

Thus we are able to formulate a theorem. Theorem 2. The linear difference equation (1) has the same number of solvability conditions as the number of the functions satisfying property (18) contained in the fundamental system of solutions to equation (15). The number of solvability conditions ism > 0, if both non-homogeneous Carleman problems (7) are unconditionally solvable, and m > 1 otherwise.

Remark 1. The simplest cases I and II are examined as W = W^, the kernel function (5) is odd, the piecewise function 9t is even, and the kernel (12) is skewsymmetric K(t, t) = -K(t, t). In the case I, m = 1, and the exponential functions becomes a constant (X = 0). The unique solvability solution (20) can be

written in a nicer form as fdDg+(t)fo(t)dt = 0,

where f0 (z) is a non-trivial solution to the respective homogeneous linear difference equation. In the case II, we havem = 0.

Remark 2. The method of regularization is also applicable when the square is replaced with an arbitrary parallelogram.

Remark 3. Other methods of regularization were proposed in the case when the linear difference equation (1) with holomorphic coefficients Aj = Aj(z) in D, does not undergo the above method, see, e.g., [11,12,14].

Remark 4. The study of the linear difference equation (1) in the case when condition (2) does not hold is of particular interest. So far, we have not observed any progress in this direction.

3. Unconditional solvability

Let us focus on the case III of solvability of the linear difference equation (1), when A1 = A2 = 1, A3 = A4 = -1.

The case IV (A1 = A4 = 1,A2 = A3 = -1) is similar. These are the only two cases when W = -W1, the kernel function (5) is even, the piecewise function 9t is odd, and K(t,x) = K(r,t).

Both non-homogeneous Carleman problems (7) are unconditionally solvable, the choice of the radical branch is irrelevant, however, let us assume ^A-1 = i for definiteness.

Lemma 2. The homogeneous Fredholm equation (13) has no non-trivial solution in the case III (IV).

Proof. Given @(t) £ L2(dD) for any j we have (p(x) £ C(£j) because of Lemma 1 and (13). Let M = maxl(p(t)l,t £ 3D. Without loss of generality, let us assume that t £ l1 ^ a(t) = t + i,9t = 1. Let us estimate the modulus of the kernel (12) by examining four cases:

1) If t £ l1 ^ a(r) = t + i, 9a(T) = -1,r-t = u £ [-1,1], then K(t,x) = i[(u - 2i)-1 + (u + 1 -i)-1 — (u — 1 — i)-1 + (u + 1 + i)-1 -(u-1 +

1)-1 -(u + 2i)-1] = -4[(u2 + 4)-1 + (u2 -

2)(u4 + 4)-1i], i.e. IKI < V5.

2) If t£12^ a(r) = t — 1,9a(T) = -1, then we have

K(t,r) = (u- 2i)-1 + (u + 1 - i)-1 -(u + i-1)-1-(u-2)-1.

Obvious geometric reasons imply |(u - 2i)-1 -(u - 2)-1I < V2; |(u + 1 - i)-1 -(u + i- 1)-1| < V2, i.e., K < 2V2.

3) t££3^ K(t,x) = 0.

4) t £l4^ a(r) = t + 1, 9a(t) = 1.

Therefore, K(t,x) = (u- 2i)-1 -(u + 2)-1 +

(u + 1 + i)-1. Taking into account u = i(fi + 2-1) -(y + 2-1), where p,y £ [-2-1,2-1], we obtain IKI = 8lhI(Ih4 - 3ih2 - 6,25I)-1. Here we have h = ip-y. The numerator of this quotient does not exceed 4V2, while the denominator is at least 5, i.e., < 0.8V2. As 2,8V2+V5 < 2n, we conclude M = 0, which completes the proof.

The following theorem holds.

Theorem 3. The linear difference equation (1) is unconditionally solvable in the cases III and IV.

Remark 5. We have Tp = g o Tcp = g, in the simplest cases I and II, where $ = (p(-t),g = g(-t). This enables us to conclude whether the solution to the equation (10) is even or odd in relation to its right part. Indeed, the functions (p(t) and <p(-t) satisfy the same condition (8) or (14) simultaneously with the piecewise function 9t. This fact helps to identify the classes of solutions of the integral equation (10), known to be continuous on dD. However, this is not true in the cases III and IV. If either of these functions satisfies one of the conditions (8) or (14), then the second function satisfies the other condition.

4. Biorthogonal expansions

Let us give some applications of the linear difference equation (1) in the case III to the expansion of analytic functions to biorthogonal series. Let gm(z) = (m\)-1(-z)m, and let us introduce a system of functions satisfying property (8), i.e., [ym(t)} ■ (Apm)(z) = gm(z),z £ D,m = 00. (21)

Clearly,

Vm,E(k)) = (2ni)(-1) JdDvm(T)E(kHj)dT = Smy,k = 0,oo. (22)

Let us examine four circular lunes bounded by the circle L:IzI = V2/2, circumscribed about the square and by four circles Lj: lz + iJ I = V2/2. It is clear that Vm = a+ + , where the functions am(z) are holomorphic within D.

Lemma 3. The functions am(z) are holomorphic inside L, and their boundary values at L have at most jump type discontinuity at the vertices tj.

In order to prove the lemma, it is sufficient to observe that we have

(21) o A-(pm - <pm = gm at dD. Let us consider the domain D0, bounded by arcs Lj(0 £ D0). The complementary set lying outside these circles we denote by Dm,o £ Dm. The condition of biorthogonality (22) is satisfied on dD0by virtue of Lemma 3. Therefore, we have

(ft" (r),E(k)(r)) = Sm,k;

t e dDn

(23)

for the integrals D.m(z) = (2ni) 1 fdDo(z-

Let us associate the biorthogonal series

H(z)~Y^=0amam(z) (24) to a function H(z) e A[CD0], see [16], where the coefficients am = (H,E(m)) are naturally defined.

Theorem 4. Series (24) represents its generating function locally uniformly in Ef^ c CD0.

Proof. Let t e dD, and let us consider the function h(t) = H(t) + a+(t), where a(z) is holomorphic in D is chosen so that the function h satisfies (8). Let us associate the series

h(t)~Y^=0am(pm(t) (25)

to the function h(t) with the coefficients am = (h,E(m)) = (-l)m(VH(m))(0).

By virtue of Lemma 2, 3 A > 0 ■ Vm\(pm(t)\ <

m

A(m\)(-1)2-~, where the radius of convergence of the Maclaurin series for the function (V H)(z) is at least -2/2. Therefore, series (25) converges uniformly, and (25) ^ h(t)= Z%=oam<Pm(t). (26)

Indeed, the difference between this function and its biorthogonal series equals 0 due to Lemma 2. Now let us use Lemma 3. Multiplying equality (26) by (t — z)-1 with z g D, and then, integrating it in 3D concludes the proof.

Let us take the function b(z) e [16] and

construct the following biorthogonal series

b(z)~Y%=opkE(k\z),zeDo (27) with the coefficients (Jk = (b,D.k). Theorem 5. The series (27) converges to the function according to which it is built, uniformly at each compact D2 c D0.

The proof follows directly from Theorem 4, if we represent the Cauchy kernel as a biorthogonal series (see, for example, ([18], ch. IV, par. 6, item 3)).

Theorem 6. Let the function b(z) be holomor-phic in D, and let b+ (t) e H^(dD). Then, the system of functions (21) is complete in the sense Vm(b, ym) = 0^ b(z) = 0.

Proof. Let us use the representation

—cp(t) + (ni)-1 JdD cp(T)E(T — t)dx = 2b+(t), (28)

where the density function (p satisfies (8). Let us multiply equality (28) by (21), and let us integrate it in dD, making use of the fact that the kernel function (5) is even. Then, (cp, tm) = 0. According to the Runge theorem, taking into account (8), we have (p = 0 ^ b = 0.

Remark 6. Let us assume b(z) = 1 in (27), and let us differentiate the resulting series, i.e., 0 = Y^=op2kE(2k+1)(z); 3k ■ p2k*0-,ze Do. (29) The system of derivatives of the kernel function (5) permits a nontrivial decomposition of zero. It is clear that the nontrivial decomposition of zero (29) cannot converge uniformly in dDo. A general theory of the nontrivial decompositions of zeros and representation systems was developed by Yu.F.Korobeynik [17].

Remark 7. P.E.Appell [4] was apparently the first who considered the expansion of the functions analytic in the domain Do into the series of rational functions. A special case of expansion of holomorphic functions into the series of simple fractions in circular domains was proposed by P.E.Appell [5] in 1883, remarkable in its simplicity and generality. His results were exposed in a well-known monograph [20] by J.Walsh, however, with an inaccurate reference to the paper [4]. Meanwhile, the article [4] examines only two specific examples, one of which concerns the expansion of a function holomorphic in the domain Do. We would like to highlight two important differences of series (27) from the Appell series. First of all, the Appell series in the domain Do admits the form

b(z) = Y^=1Y4=1aj:n(z+V)-n, i.e., each function b( ) defines its own set of coefficients ajn, and different functions are expanded

into the series over different rational functions. But the series (27) with the coefficeints aj:n = AjPn constitute the functions b( ) expanded into the series over the same system of consecutive derivatives of the kernel function (5). The sum of series (27) cannot be zero there (see Introduction). Secondly, series (27) and the Appell series converge in the domain Dm. The sum of the Appell series equals zero in the neighborhood of the point at infinity because it follows from the decomposition of the Cauchy integral into a series. But the sum of the series (27) (with b(z) i 0) cannot be zero there. In fact, the difference equation (V f)(z) = 0 in the neighbourhood of infinity has only the trivial solution. It suffices to apply the Borel transformation.

Remark 8. All the results obtained in the previous section (except Theorem 6, where the evenness of the kernel function was used at a significant extent) can be extended to the case of an arbitrary kernel for which m = 0.

Let us define the function a(z) in the circular lunes as an analytic extension of a(t) from the respective side of the square. Let us introduce the functions hk = WE(k) and a system of integrals of the Cauchy type

\[2

Qk(z) = (2ni)-1 fL hk(T)(r — z)-1dT, \z\>-^,k = 0>. (30)

Then, by virtue of Lemma 3, we have (a+, Q-) =

Jm.k•

b(z)eÄ(lzl<^) [16]

Take the function

construct the following biorthogonal series

-2

and

b(z)~YZ=o8mam(z)M<-f (31)

with the coefficients 9m = (b+,Qm).

Theorem 7. Let b+ e C2(L) and b+(tj) = 0,j =

1.4. Then the series (31) converges to the generating function uniformly in the closed disk \z\ < —.

Proof. Additional requirements on the boundary values of the function b (z) ensure the uniform convergence of the series (31) on the circle L. It is sufficient to notice that 9m = —(b+, hm), and to apply the integration-by-parts formula twice considering the functions b+(t) and b+ (t) as the consecutive Leibniz terms. Then, the difference V(z) between the function b (z) and its biorthogonal series satisfies the conditions

fL V+(t)hk(t)dt = 0 ^ f dDo(W^+)(t)E(k)(t)dt = 0,

^fgo(ip+ + W^+)(t)E(t — z)dt = 0,ze D;k = 0^,

and in view of Lemma 2, we have = —W^>+ ^ ip(z) = 0, which concludesthe proof.

Remark 9. The points of the uniform and absolute convergence of the biorthogonal series in Theoremes

4.5, and 7 coincide.

5. Applications to the moment problem

We point out applications of the difference equation (1) to the moment problem for entire functions of exponential type. Let us consider the domain D as an

adjoint indicator diagram of the entire function F(z) Borel associated with f(z). Let us restrict ourselves to the case of an unconditionally solvable problem (1) (as an example of the case III examined in Section 2). Let us write it in the following form using the Borel transformation

fT] F(t)exp[-<rj(z)t]dt = g(z),z £ D, (32)

where T j is the ray argz = - ^j, as the points Oj(z) lie outside D at z £ D. Let us equal the Maclaurin coefficients in the left and the right sides of (32) in order to obtain the moment problem

L(F, k) = !T. F(t)(-1)ktk exp[-ht ■

t] dt = g(k)(0),k = 0777.

Here we have ht = { V, t £ T}. Let us now consider the moment problem

L(F,k) =

(33)

within the class of entire functions of exponential type F (z), with f(z) £ B.

Theorem 8. Let the radius R of convergence of the power series

œ k

9(z) = L~kr

j=0

be al least —. Then, the moment problem (33)

is unconditionally and uniquely solvable.

The moment problem (33) is a generalization of the classical Stieltjes moment problem in the case of four rays. Entire functions of exponential type F(z) has the indicator hF(6) = 2-1(cos 6 + sin 6), 6 £

[o,^] ; hF(d + = hF(6), i.e., the piecewise exponential weight exp [-ht • t] guarantees the convergence of the improper integrals (32). In some special cases, the equation (1) allows to explore the classical Stieltjes moment problem (see, for example, [8]) or a generalization of this problem to the case of two orthogonal rays. We can show that F(z) is an entire function of regular growth ([19], Ch. III) in the casefl > 1, exactly the same way as it was done in [10].

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