Hausdorff Moments, Hardy Spaces and Power Series
Abstract.
In this paper we consider power and trigonometric series whose coefficients are supposed to satisfy the Hausdorff conditions, which play a relevant role in the moment problem theory. We prove that these series converge to functions analytic in cut domains. We are then able to reconstruct the jump functions across the cuts from the coefficients of the series expansions by the use of the Pollaczek polynomials. We can thus furnish a solution for a class of Cauchy integral equations.
E. De Micheli]
G.A. Viano]
1. Introduction
The problem of characterizing the analytic properties of the functions, in terms of the coefficients of their power expansions, is very old and goes back to a classical result due to Le Roy [13]:
Theorem (Le Roy).
If in the Taylor series the coefficients are the restriction to the integers of a function , holomorphic in the half–plane , and moreover there exist two constants and such that
(1) 
then the series converges to a function analytic in the unit disk , and furthermore admits a holomorphic extension to the cut plane .
Other similar results are due to Lindelöf [14] and Bieberbach [2]. More recently, Stein and Wainger [15] have reconsidered the problem in the framework of the Hardy space theory. More precisely, they assume that the coefficients are the restriction to the integers of a function holomorphic in the half–plane , and, in addition, is supposed to belong to the Hardy space with norm . Correspondingly, they consider the class of functions analytic in the complex –plane slit along the positive real axis from to (this domain is denoted by ). The space of functions analytic in for which is denoted by . These authors have proved the following result:
Theorem (Stein–Wainger).
Suppose that . Then if and only if , where . Moreover,
(2) 
where
(3) 
and
(4) 
being the jump function, i.e., ; and are the boundary values of and , respectively .
The main purpose of the present paper is the reconstruction of the jump function across the cut from the coefficients of the power expansion. This result is achieved by the use of the Pollaczek polynomials [1, 16], and it will be illustrated in Section 4. This reconstruction allows us to solve the Cauchy integral equation of the following type:
(5) 
when the Taylor coefficients , are supposed to be known. Unfortunately, in the numerical analysis and in the applications to physical problems only a finite number of Taylor coefficients are known, and, moreover, they are affected by noise or, at least, by round–off errors. Furthermore, the integral equations of first kind, like equations (5), give rise to the so–called ill–posed problems in the sense of Hadamard [8]: The solution does not depend continuously on the data. We shall briefly return on this important point in Section 4. In a separate paper we shall discuss in detail how to manage numerically the method presented here.
In the theorems of Leroy and Stein–Wainger the coefficients are required to be the restriction of a function holomorphic in the half–plane , and, in the case of the Stein–Wainger theorem, this function is also assumed to belong to the Hardy space . We prefer to start by requiring that the coefficients satisfy the so–called Hausdorff conditions [20], which guarantee that they can be regarded as Hausdorff moments in a sense that will be explained in Section 2. This approach is convenient for several reasons:

From the Hausdorff conditions and by the use of the Carlson theorem [3] we derive immediately the existence of a unique interpolation of the coefficients which is holomorphic in a half–plane.

By imposing to the coefficients Hausdorff conditions of various types, we can, correspondingly, obtain more specific properties of smoothness of the function which gives the jump across the cut.
The last point is particularly relevant. In fact, in the Stein–Wainger approach
one works essentially with the unitary equivalence between
and ; accordingly, the jump function belongs to
. On the other hand, more refined properties of continuity
and differentiability of the jump function are relevant in the mathematical
theory of the Cauchy integral equation, and particularly in the physical applications [5].
In agreement with this approach,
we shall prove in Section 3 theorems which are variations on the Stein–Wainger
result. The main mathematical tool used in our approach is the Watson
resummation method which leads, in a very natural way, to the Laplace
transform of the jump function. It turns out that this Laplace transform coincides exactly
with the Carlsonian interpolation of the coefficients.
In this way all what is necessary for extending the methods
and the results to expansions in terms of Legendre and ultraspherical polynomials is obtained.
In this extension, in fact,
the interpolating function coincides with the
spherical Laplace transform (in the sense of Faraut [4, 6, 7]),
that can be regarded as a composition of the classical Laplace
transform and the Abel–Radon transform.
This paper can be regarded as the completion and a large extension of a preliminary work by one of
the authors (G.A.V.) [17], where the Hausdorff moment problem has been
approached by the use of the Pollaczek polynomials.
2. Hausdorff Moments, Hardy Spaces and Markov Processes
2.1. Hausdorff Moments and Hardy Spaces
Given a sequence of (real) numbers , let denote the difference operator:
(6) 
Then, we have:
(7) 
(for every ); is the identity operator by definition. Now, suppose that there exists a positive constant such that:
(8) 
It can be proved [20] that condition (8) is necessary and sufficient in order to represent the sequence as follows:
(9) 
where belongs to .
We can prove the following Proposition.
Proposition 1.
If the sequence satisfies condition , then there exists a unique interpolation of this sequence, denoted by which belongs to the Hardy space , and satisfies the following properties:

is holomorphic in the half–plane ;

belongs to for any fixed value of ;

tends uniformly to zero as tends to infinity inside any fixed half–plane .
Proof.
If the sequence satisfies condition (8), then representation (9) holds true. If in this representation we put , then we obtain:
(10) 
Therefore the numbers can be regarded as the restriction to the integers of the following Laplace transform:
(11) 
Indeed, one has . Moreover, belongs to . Then, in view of the Paley–Wiener theorem [9] and of formula (11), we can conclude that belongs to the Hardy space , and properties (i), (ii) and (iii) follow. Thus, we can make use of the Carlson theorem [3], which guarantees that represents the unique interpolation of the sequence . ∎ , and therefore the function
Now, we can prove the following Proposition.
Proposition 2.
If the sequence , where , satisfies condition , then there exists a unique Carlsonian interpolation of the numbers , denoted by , which satisfies the following properties:

is holomorphic in ;

belongs to for any fixed value of ;

tends uniformly to zero as tends to infinity inside any fixed half–plane ;

belongs to for any fixed value of .
Proof.
Since the numbers
satisfy condition (8), then there exists a unique Carlsonian interpolation
of the sequence , denoted by ,
, which can be written as the product:
, where is the unique
Carlsonian interpolation of the numbers .
In view of condition (8) and Proposition 1 it follows that
belongs to . Therefore properties
(i), (ii) and (iii) follow immediately.
By applying the Schwarz inequality, and recalling that
for any fixed , we have:
(12) 
if , , . Finally, from inequality (12), and in view of the regularity and integrability of the function in the neighborhood of we can state in all generality that belongs to for any . ∎
2.2. Hausdorff Moments and Markov Processes
In this subsection we follow closely the paper of Watanabe [18]. Let be an abstract probability field. If is a sequence of random variables on which are mutually independent, and each one satisfies
(13) 
then it is called a Bernoulli sequence and denoted by . In the sequel we shall consider . Let be the set of all points such that . Next, we consider the Markov process attached to . Let us note that:
(14) 
where , , . The kernel is given by [18]:
(15) 
Now, consider an infinite sequence having no limit point in , and such that
(16) 
for a suitable . Then, using the Stirling formula, and taking into account equality (16), from (15) we obtain:
(17) 
Thus, one may consider that the Martin boundary [10] induced by the process coincides with the interval as a set, and, accordingly, the generalized Poisson kernel is given by . Finally, we note that for a function over the expectation is given by:
(18) 
Now, the following propositions due to Watanabe [18] can be stated.
Proposition 3.
Let be the Markov process attached to the Bernoulli sequence .

The Martin boundary induced by is equivalent to the interval with the ordinary topology;

The generalized Poisson kernel is:
(19) 
A function (belonging to the set of all the finite real valued functions over , vanishing at ) can be represented by means of a bounded signed measure on , (where is the Borel field consisting of all the ordinary Borel subsets in ), as follows:
(20) (for every ), if and only if is –harmonic, and is bounded in .
Proof.
See [18]. ∎
Proposition 4.
Let be the Markov process attached to the Bernoulli sequence . Given a sequence of real numbers such that:
(21) 
then the function defined by:
(22) 
is a –harmonic function, and can be represented by formula (20).
Proof.
See [18]. ∎
Notice that from representation (20) it follows:
(23) 
which can be compared with representation (9). Moreover, if the sequence satisfies inequality (8), then it satisfies also inequality (21). This can be proved easily by the use of the Cauchy inequality:
(24) 
In fact, if in inequality (24) we put: , , , we obtain:
(25) 
Therefore, from inequalities (8) and (25) we obtain:
(26) 
that coincides with inequality (21), if we put .
3. A Double Analytic Structure for a Class of Trigonometric and Power Series
In the complex plane of the variable () we
consider the following domains:
,
and
.
We introduce, correspondingly, the following cut domains:
, where
,
and
, where
(for a detailed description of these cut domains see [4]).
We will use the notation for every subset of
which is invariant under the translation group .
We can then prove the following theorem.
Theorem 1.
Let us consider the following series:
(27) 
and suppose that the set of numbers , satisfies condition , then:

series converges uniformly to a function analytic in ;

the function admits a holomorphic extension to the cut domain (see Fig. 1A);

the jump function (which equals the discontinuity of across the cuts ) is a function of class (), and satisfies the following bound:
(28) where is the Carlsonian interpolation of the coefficients , and
(29) 
is the Laplace transform of the jump function : i.e.,
(30) 
the Plancherel equality holds true:
(31)
Proof.
Since the set satisfies condition (8), given an arbitrary number , there exists a real number such that for , . Therefore, we can write:
(32) 
The series at the r.h.s. of formula (32) is uniformly convergent for . Recalling the Weierstrass theorem on the uniformly convergent series of analytic functions, we can also conclude that the series converges uniformly to a function analytic in . On the other hand, series (27) can be rewritten as the following sum:
(33) 
where is a trigonometric polynomial analytic in . Therefore the first statement is proved.
In order to prove the other statements, let us introduce the following integral:
(34) 
where , is the unique Carlsonian interpolation of the sequence , which exists in view of the fact that the set satisfies condition (8), and the contour is contained in the half–plane and encircles the positive real semi–axis of the –plane (or a part of it) as is illustrated in Fig. 2A.
Now, let us consider the following inequalities:
(35)  
(36)  
(37) 
Let us recall that () tends uniformly to zero as inside any fixed half–plane , and belongs to for (see Proposition 2).
In view of these properties of , and by the use of bound (37), we can guarantee that the integral converges, and the contour can be deformed and replaced by the line , provided that the real variable is kept in (see Fig. 2A). Finally, by applying the Watson resummation method [19] we obtain for :
(38) 
(, integer; if , if ).
Proceeding in an analogous fashion for the integral (formula (34) with ), and distorting the contour integration in a similar way, we finally obtain for
(39) 
(, integer; if , if ).
Now, in the integral (38) we substitute for the complex variable , and we see that the obtained integral provides an analytic continuation of in the strip , continuous in the closure of the latter. Indeed we have:
(40) 
with
(41) 
then, in view of bound (37), and since for any fixed value of (see Proposition 2), the statement above is proved. Similarly, the analytic continuation of the function is defined in the strip . The discontinuity can be computed by replacing by in integrals (38) and (39), and subtracting Eq. (39) from Eq. (38). We then obtain
(42) 
Thus, we have proved that the function admits a holomorphic extension to the cut domain .
From formula (42) we derive the following bound for the jump function :
(43) 
where
(44) 
Using again formula (42), and recalling the Riemann–Lebesgue theorem, we can prove that is a function of class in view of the fact that belongs to for any (see Proposition 2).
Inverting formula (42) we obtain:
(45) 
which is, indeed, the Laplace transform of the jump function , and it is holomorphic for .
Finally, recalling that belongs to at any fixed value , we obtain the Plancherel equality (31) and, in particular:
(46) 
∎
Remarks.
(i) Theorem 1 proves a double analytic structure, connected with
series (27), in the following sense: To the functions ,
that interpolate the coefficients , and are holomorphic in the half–plane
, there corresponds the class of functions holomorphic
in the domain , and moreover
is the Laplace transform of the jump function across the cut.
(ii) Let us note that the Plancherel equality (formulae (31)
and (46)), as well as the analyticity property of the function ,
remain true under milder conditions on the coefficients . In fact, it is sufficient that
the ’s form a sequence of numbers that satisfies condition (8).
This is, indeed, the result contained in the theorem of Stein–Wainger [15] referred to
series (27). In conclusion, the more restrictive conditions, assumed in Theorem
1, are reflected by the smoothness property of the jump function, which is, however, a quite relevant
property playing an important role in the applications to physical problems.
By substituting the complex plane of the variable to the –periodic –plane, we can now give an equivalent presentation of the results of Theorem 1, in terms of properties of Taylor series and of Mellin transformation. To the cut at , it corresponds, in the –plane geometry, the cut located on the real axis () from up to . To the jump function it corresponds the function , which shall still be denoted hereafter simply by with a small abuse of notation which avoids, however, an useless proliferation of symbols. Adopting the same convention, we shall always denote the jump function, in the various geometries, with the same symbol: i.e., .
Theorem 2.
If in the Taylor series:
(47) 
the coefficients satisfy the assumptions required by Theorem 1, then:

the series converges uniformly to a function analytic in the unit disk ;

admits a holomorphic extension to the cut plane (see Fig. 1C);

the jump function , ()) is a function of class , and satisfies the following bound:
(48) where is the Carlsonian interpolation of the coefficients , and is given by formula ;

is the Mellin transform of the jump function: i.e.,
(49) 
the Plancherel formula associated with the Mellin transform gives
(50)
Remark.
We now present, without giving the proof, two variants of Theorems 1 and 2, which are relevant in the physical applications, and specifically in the theory of the thermal Green functions [5]. The proofs can be easily obtained, with small variations, from those of Theorems 1 and 2.
Proposition 5.
If in the following series
(52) 
the coefficients , in addition to the assumptions required by Theorem 1, satisfy also the following bound:
(53) 
then