Dynamic approach to classical moment problem Текст научной статьи по специальности «Математика»

DYNAMIC APPROACH TO CLASSICAL MOMENT PROBLEM

A. S. Mikhaylov1'2, V. S. Mikahaylov 1>2,

1St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences,

191023, St. Petersburg 2 Saint Petersburg State University, 199034, St. Petersburg

UDC 517.984.54, 517.977.1, 517.927.25 DOI: 10.24411/9999-016A-2019-10054

We consider the problem of the construction of a measure supported on a real line from prescribed moments. The main idea is to use the auxiliary dynamical system with the discrete time associated with a semi-infinite Jacobi matrix. Then the inverse dynamic data for this system, so called response operator (discrete analog of a dynamic Dirichet-to-Neumann map) is given in terms of moments, and we can use ideas of the Boundary Control method to recover the spectral data, i.e. the measure of a truncated moments problem, from dynamic one. The remarkable fact is that in our procedure we do not use the Jacobi matrix itself. We also formulate the results on the uniqueness of the solution of Hamburger and Stieltjes moments problems. Keywords: moment problem, Boundary control method, Jacobi matrices.

Introduction

For a given a sequence of numbers sq, si, s2,... called moments, a solution of a Hamburger moment problem [1,13] is a Borel measure dp(A) on R such that

In the present paper we offered an approach to this problem based on the ideas of the Boundary control (BC) method [5,6]. We consider an initial boundary value problem (IBVP) for an auxiliary dynamical system with discrete time for a Jacobi matrix. The associated with this system is so-called response operator (discrete analog of a dynamic Dirichlet-to-Neumann map), which usually plays the role of inverse data in dynamic inverse problems [5,8]. The dynamical system of this type and inverse problems for this system were studied in [9-12], where the authors have shown that the kernel of response operator, which is called response vector, admits a spectral representation in terms of a spectral measure of the Jacobi matrix. Using the ideas of the BC method we extract the spectral data (the spectral measure, which is a solution of the moment problem!) from the dynamical one, i.e. from the response vector.

In the first section we set up the IBVP for a dynamical system associated with a finite Jacobi matrix, derive special representations of its solution, introduce operators of the BC method. In the second section we solve a truncated Hamburger moment problem by extracting spectral data from the dynamic one and derive the generalized spectral problem, whose solution is exactly the spectrum of truncated Jacobi matrix and controls of the dynamical system with special properties.

The work has been supported by the Russian foundation of basic research (grant numbers 18-01-00269, 17-01-00529, 17-01-00099).

(1)

ISBN 978-5-901548-42-4

1 Initial boundary value problem for a dynamical system associated with Jacobi matrix. Operators of the Boundary control method

For a given sequence of positive numbers {ao,ai,...} (in what follows we assume a0 = 1) and real numbers {bi,b2,...}, we denote by A the Jacobi operator, defined on l2, which has a matrix form:

A =

Ai

a\ 0

ai

b2 02

00

02 0 &3 a3

...

...

(2)

For N G N, by AN we denote the N x N Jacobi matrix which is a block of (2) consisting of the intersection of first N columns with first N rows of A. Fixing N G N we consider the dynamical system with discrete time associated with a finite Jacobi marix:

v„,t+i + vn,t-i - anvn+i,t - an-ivn-i,t - b„v„,t =0, t G N U {0}, n G l,...,N, v„,-i = v„, o = 0, n = 1, 2,...,N + 1, vo,t = ft, vN +i,t = 0, t G No,

(3)

where f = (fo,fi,...) is a boundary control. The solution to (3) is denoted by vf. Note that (3) is a discrete analog of dynamical system with boundary control for a wave equation on an interval [2,7].

The operator corresponding to a finite Jacobi matrix we also denote by AN. Operator AN is defined on RN, is given by

(A^)n = an^n+i + an-i^n-i + bn'^n, 2 < n < N - 1, (A^)i = bi^i + ai^2, n = 1,

and the Dirichlet condition at the "right end":

^N+i = 0.

We fix some positive integer T and denote by Tt the outer space of the system (3), the space of controls: Tt := Rt, f G Tt, f = (f0,..., fT-i), we use the notation = when control acts for all t ^ 1.

Definition 1. For f,g G Twe define the convolution c = f * g G Tby the formula

ct = J2 f»St-», 1 G N U{0}.

s= o

The input i—> output correspondence in the system (3) is realized by a response operator: B^ : Tt ^ Rt defined by the rule

(RNNf)t = v{, t, t = 1,...,T.

This operator plays the role of inverse data, the corresponding IP were considered in [9,11]. The response vector-is a convolution kernel of a response operator, rN = (rN, rN,..., rN_i):

(Rtnf) = rN * l-i.

By choosing the special control f = S = (1, 0,0,...), the kernel of a response operator can be determined as

(Rtn S)t = = rN-!.

Let $„(X) be a solution to

I a„^„+i + a„-i^„-i + b„^„ = Uo = 0, fa = 1.

(4)

Denote by {Ak}N=i the roots of the equation <^N+i(A) = 0, it is known [1,13] that they are real and distinct. We introduce the vectors 4>n G RN by the rule := &(An), n,i = 1,... ,N, and define the numbers pk by

(j>k,j>l)= 5k i Pk,

where (•, •) is a scalar product in

Definition 2. The set of pairs

{Xk-Pk }f=i

is called spectral data of operator AN.

Let Tk (2A) be the Chebyshev polynomials of the second kind: i.e. they satisfy

Î7t+i + Tt-i - XTt =0, \7~0 = 0, 7Ï = 1.

In [9,11] the following representation for the solution v f was proved: Proposition 1. The solution to (3) admits a representation

'T.L i €, n = 1,...,N, ft, n = 0.

ck = — T (Xk) * f

Pk

(5)

The inner space of dynamical system (3) is HN := RN, h G %T, h = (hi,... ). By (5) we have that v^ T G HN. For the system (3) the control operator W^ : TT ^ HN is defined by the rule

f := <r,

The connecting operator C^ : TT ^ TT is defined via the quadratic form: for arbitrary f,g G TT we have that

{CTN f,g) = (vfT ,v°T) nN = (W* f,WSg) HN .

The spectral function of AN is introduced by the rule

»"(A) = S i-

{k | Afc<X}> k

(6)

v'L,t =

Then (5) implies the following representation formulas:

rTO t

YsW)ft-k MX) dpN (A),

1 k=i

/TO

Tt(X) dpN(A), t G N,

-TO

/TO

Tt-i(X)Tt-m(X) dpN (A), I,

TO

Another formula for CT which is valid if T ^ N (see [9,11]) says that:

rN rt-i

0,...,T - 1.

(7)

(8) (9)

CT =

fro + T2 + ... + T2T-2 + ... + T2T-3

ri + r-3 + ... + r-2T-3 ro + ... + r-2T-4

rT-3 + rT-i + rT+i ...

rT + rT-2 . . .

\ rT -i rT-2

rT + tt-2 tt- i\

r-Q + r-2 + r-4 ri + r-3

rT-2

n + T3 T2

TO + r-2 ri

n TO y/

(10)

The speed of a wave propagation in (3) is finite, which implies the following dependence of inverse data on coefficients {an, bn}: for M G N, M ^ N, the element v^2M_i depends on {aQ, ai,..., om-i} , {bi,..., bM}, on observing this we can make the following

Remark 1. The entries of response vector (rff,. .. 2)) depends on {ao,. .. {bi,. .. }, and

does not depend on the boundary condition at n = N +1, the entries starting from r^N-1 does "feel" the boundary condition at n = N + 1.

f

v

2 Truncated moment problem. Recovering Dirichlet spectral data

We observe the following: in the moment problem we are given the sequence of moments (1), and in the dynamic IP for the system (3) we are given a response vector [9,11], whose spectral representation has a form (8). Thus the knowledge of moments {so, ...} implies a possibility to calculate the response vector {ro,ri,...} by (8).

Definition 3. By a solution of a truncated moment problem of order N we call a Borel measure dp(X) on R such that equalities (1) with this measure hold for k = 0, 1, .. ., 2N.

Remark 2. The results from the previous section implies that from finite set of moments {so, si, .. ., S2N-2}, which is equivalent to the knowledge of {ro,ri,.. . ,r2N-2} it is possible to recover Jacobi 'matrix AN G RNxN whose elements can be thought of as a coefficients in dynamical system (3) with Dirichlet boundary condition at n = N +1, or N x N block in semi-infinite Jacobi matrix in (3) with no condition at the right end.

In [11] the authors proved the following

Theorem 1. The vector (ro,ri,r2,. .. ,r2N-2) is a response vector for the dynamical system (3) if and only if the matrix Ct (with T = N) defined by (9), (10) is positive definite.

This theorem and formulas for the entries of Jacobi matrix obtained in [11] implies the following procedure of solving the truncated moment problem:

1) Calculate (r0,ri, r2,..., r2N-2) from si,..., S2N-2} by using (8).

2) Recover N x N Jacobi matrix AN using formulas for ak, bk from [11]

3) Recover spectral measure for finite Jacobi matrix AN prescribing arbitrary selfadjoint condition at n = N +1. Or one can do

3') Extend Jacobi matrix AN to finite Jacobi matrix AM, M > N, prescribe arbitrary selfadjoint condition at n = M + 1 and recover spectral measure of AM.

3") Extend Jacobi matrix AN to infinite Jacobi matrix A, and recover spectral measure of AM.

Every measure obtained in 3), 3'), 3'') gives a solution to the truncated moment problem. Below we propose a different approach: we recover the spectral measure corresponding to Jacobi matrix directly from moments (from the operator CN), without recovering the Jacoi matrix itself.

Agreement 1. We assume that controls f G TN, f = (/0, .. ., fn-i) are extended: f = (f-i, fo, .. ., fn-i, In), where fi = Jn = 0.

We introduce the special space of controls TN = {/ G Tt | /0 = 0} and the operator D : Tt ^ Tt acting by

(Df )t= ft+i + ft-i. The following statements can be easily proved using arguments from [11]:

Proposition 2. The operator WN maps TN isomorphically onto 'HN and TN maps isomorphically onto 'HN-i. Proposition 3. On the set Tt the following relation holds:

WNDf = DWnf, f gTN . (11)

Taking f,g G T^ we can evaluate the quadratic form, bearing in mind (11):

(CN Df,g)^N = (WN Df,WN g)HN = (DWn f,WN g)uN_1 = (An-ivf ^ )%N_1 . (12)

The latter expression in (12) means that only AN-i block from the whole matrix AN is in use. Then it is possible to perform the spectral analysis of AN-i using the classical variational approach, the controllability of the system (3) (see Proposition 2) and the representation (12), see also [4]. The spectral data of Jacobi matrix AN-i with Dirichlet boundary condition at n = N can be recovered by the following procedure:

1) The first eigenvalue is given by

\N-1 — min (CN Df,f) . (13)

1 f et«, (cNf,f )rN=iK J,J'TN K '

2) Let f1, be the minimizer of (13), then

Pi = (CN f 1,f ^ r N •

3) The second eigenvalue is given by

\N-1 — ^ min (°NDf>f)t- ■ (14)

f et» ,(cNf,f )pN = 1K (CNf,fiK N =0

4) Let f2, be the minimizer of (14), then

^ — (nT f2 f2 ,

>tt ■

P2 — (CT f 2 J2),

Continuing this procedure, we recover the set |AN 1, Pk}N=i and construct the measure dpN 1 (A) by (6).

Remark 3. The measure, constructed by the above procedure solves the truncated moment problem for the set of moments {so, si, • • •, S2N-a}-

2.1 Euler-Lagrange equations

In this section we derive equations which can be thought of as a Euler-Lagrange equations for the problem of the minimization of a functional (CNDf,f) ■FN in with the constrain (CT f,f) -FN = 1 described in the previous section. Similar method of deriving equations which can be used for recovering of spectral data was used in [3]. By fk, k = 1, • • •, N we denote the control that drive system (3) to prescribed state (see (4)):

WT fk = <t>k, k = 1,...,N. Due to Proposition 2, such a control exists and is unique for every k. We introduce the operator

Dn:fN^fN, (DNf)n = fn-i, n =1,...,N - 1, (DNf\ = 0,

and denote by P : PN+1 ^ PN the embedding. Then P* : PN ^ PN+1 extends vector by zero: (P* f )N = 0.

Theorem 2. The spectrum of AN and (non-normalized) controls fk, k = 1,...,N are the spectrum and the eigenvectors of the following generalized spectral problem:

(P (DN+1)* CN+1P * + CN Dn) fk = Xk CN fk, k =1,...,N. (15)

Proof- For h € PT we always assume that h-1 = hT = 0 (see Agreement 1). For a fixed k = 1,.. •, N we take fk € PN such that WN fk = v^kN = (pk, then for arbitrary g € PN we can evaluate:

(Xk CN fk ,g)TN = (Ak v^ ^N) = (Ak 4>k ^NV n = = {(AN v*) y,N)

g

Nl ^N \ ,N'J UN ' \ —1' J ^N

We note that

jn 9 vg N — ( (T) „g

-.N + 1

U

— ( HN — «W^N )) + (^N-I^n) hn (16)

9 vgN — (<N(T),...yN,N), nv g — (vg vg o i

That is why we can rewrite the first summand in the right hand side of (16) as

9 \ ( h D-+1S\

№n+1,<n) hn — (^W^1/) — (CN+1fk,D"+1g)TN+i

(17)

Analogously:

U N fk I fk fk A

3 v-,N-1 — [V1,N-\T--,vN-1,N-1, 0

WN VD-fk - (vfk Vfk 0)

n 3 ,N — \yv1, N-1, ■ ■ ■ , VN-1,N-1, 0I ■

So we can rewrite the second summand in the right hand side of (16) as

«V-1,-U) HN = (VDlfk = (CNDN fk .

(18)

' nN

Finally from (16), (17) and (18) we deduce that

(\kCNh,9)^n = (CN+1h,DN+1 g)^+1 + (CnDnfk,g)tn . (19)

Using operators P, P* we can rewrite (19) in the form

(P (DT_ + 1 )* CT + 1P* + CTDT_^ fk = \kCTfk.

Thus the pair [fk, Xk} gives the solution to (15). Now let the pair {f, A} be the solution to (15) with f G FN f = fk, A = Ak for any k = 1 ..., N. Then WNf = vf N = J2k=1 ak4>k for some ak G R. We can evaluate for arbitrary g G FN :

0= ((p(£N+1)*CN+1P* + CnDn) f-ACNf,g) = (CN+1P*f,DN+1P*g)^N+1 + (CNDN f,g)TN -A (yftN, <N )HN = (<Nf+1,vDNT9) nN+i + (<Nf, <N) A(vf,N ,v9,N)un = KN+1, v9,N+^uN + (vf,NT1,V9,NT^UN -A(vf,NT1,V%T1

HN

(N N

ANJ2ak<Pk -Aj2ak<Pk,WNg k=1 k=1

(j>k(Ak -A)4>k,WNg^J

]ak(Ak -A)4>k,WNg i

' HN

From the above equality and Proposition 2 it follows that all ak except one are equal to zero, and for such aj, A = Aj, which completes the proof. □

Having found spectrum and non-nomalized controls from (15) we can recover the measure of AN with Dirichlet boundary condition at n = N + 1 by the following procedure:

1) Normalize controls by choosing (CN fk, fk)JzN = 1,

2) Observe that WNfk = ak4>k for some ak G R, where the constant is defined by ak = (Rfk)N.

3) The norming coefficients are given by pk = aj,, k = 1,... ,N.

4) Recover the measure by (6).

2.2 Conclusion

Denote by MN the subset of Borel measures on R, such that dv(X) G MN is a solution of the truncated moment problem (1) of the order N. We use the Boundary control method to construct the special solution of a truncated Hamburger moment problem: for N G N the set of moments {so, s1,..., s2N} determines the measure dpN(A) G MN, where the constructed measure is a spectral measure of a finite Jacobi operator with Dirichlet condition at the right end. We point out that in our procedure we do not use the Jacobi matrix, but rather special matrix, constructed from moments. It is not hard to see (from (9) and results from [11]) that MN is a convex set, and MNl Ç MN2 when N1 > N2. Taking N to infinity we deduce that the set of solutions of the moment problem (1) either convex, or it consists of one element dpN (A) —yNdp(A) (see also [1,13]).

It is possible to apply the approach proposed to studying the special cases: Stieltjes and Hausdorff moment problems, and also to the questions on the uniqueness of solution to moment problems, that will be the subjects of forthcoming publications.

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Mikhaylov Alexander Sergeevich — PhD, senior researcher of the St. Petersburg Department of

V.A. Steklov Institute of Mathematics RAS e-mail: [email protected]; Mikhaylov Victor Sergeevich — PhD, senior researcher of the St. Petersburg Department of

V.A. Steklov Institute of Mathematics RAS e-mail: [email protected]; Received — April 30, 2019