IJMMS 25:11 (2001) 709–715
PII. S0161171201005385
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON THE ANALYTIC FORM OF THE DISCRETE KRAMER
SAMPLING THEOREM
ANTONIO G. GARCÍA, MIGUEL A. HERNÁNDEZ-MEDINA,
and MARÍA J. MUÑOZ-BOUZO
(Received 15 March 2000)
Abstract. The classical Kramer sampling theorem is, in the subject of self-adjoint boundary value problems, one of the richest sources to obtain sampling expansions. It has become very fruitful in connection with discrete Sturm-Liouville problems. In this paper, a
discrete version of the analytic Kramer sampling theorem is proved. Orthogonal polynomials arising from indeterminate Hamburger moment problems as well as polynomials of
the second kind associated with them provide examples of Kramer analytic kernels.
2000 Mathematics Subject Classification. Primary 94A20, 44A60.
1. Introduction. The classical Kramer sampling theorem provides a method for
obtaining orthogonal sampling theorems [9, 11, 14]. The statement of this result is
as follows: let K(ω, λ) be a function, defined for all λ in a suitable subset D of R such
that, as a function of ω, K(·, λ) ∈ L2 (I) for every number λ ∈ D, where I is an interval
of the real line. Assume that there exists a sequence of distinct real numbers {λn } ⊂ D,
with n belonging to an indexing set I contained in Z, such that {K(ω, λn )} is a complete
orthogonal sequence of functions of L2 (I). Then for any F of the form
(1.1)
F (λ) = f (ω)K(ω, λ) dω,
I
where f ∈ L2 (I), we have
F (λ) = lim
N→∞
with
Sn (λ) =
I
F λn Sn (λ)
(1.2)
|n|≤N
K(ω, λ)K ω, λn dω
.
2
K ω, λn dω
(1.3)
I
The series in (1.2) converges absolutely and uniformly wherever K(·,λ)L2(I) is bounded.
In [6] the authors proved an extension of the Kramer sampling theorem to the case
when the kernel is analytic in the sampling parameter λ. Assume that the Kramer
kernel K is an entire function for any fixed ω ∈ I, and that the function h(λ) =
2
I |K(ω, λ)| dω is locally bounded on C. Then any function F defined by (1.1) is an
entire function as well as the sampling (interpolatory) functions (1.3). A kernel K satisfying the above additional conditions is called a Kramer analytic kernel.
A straightforward discrete version of Kramer’s theorem can be obtained [2, 8]. To
this end, let K(n, λ) be a kernel such that, as a function of the discrete variable n ∈
I ⊂ Z, the kernel K(·, λ) is in 2 (I) for any fixed λ ∈ D ⊂ R. Assume that, for a suitable
710
ANTONIO G. GARCÍA ET AL.
sequence {λn } in D, {K(·, λn )} is an orthogonal basis for 2 (I). Then any function of
the form F (λ) = n∈I f (n)K(n, λ), where f ∈ 2 (I), can be expanded by means of the
sampling series
F (λ) = F λn Sn (λ),
(1.4)
n
where the sampling functions are given by
1
K m, λn K(m, λ).
Sn (λ) = K ·, λn 2
m∈I
(1.5)
The main aim in this paper is to prove the analytic version of the Kramer sampling
theorem for the discrete case. The additional conditions required are in close parallel
with the ones assumed in [6] for the continuous case. In Section 3, we propose discrete
Kramer analytic kernels arising either from orthogonal polynomials associated with
indeterminate Hamburger moment problems, or from the second kind orthogonal
polynomials associated with the former ones.
2. The discrete analytic Kramer sampling theorem. In this section, we state the
discrete version of the analytic Kramer sampling theorem. We use the index set Z both
for the discrete variable and for the sampling sequence {λn }. When the index set is N
or N0 = N ∪ {0}, the results are essentially the same after some minor changes in the
hypotheses.
Theorem 2.1. Let K : Z × C → C be a mapping satisfying the following properties
and conditions:
(1) K(·, λ) ∈ 2 (Z) for each λ ∈ C.
(2) K(n, ·) is an entire function for each n ∈ Z.
(3) There exists a sequence of real numbers {λm }m∈Z satisfying
(a) λm < λm+1 for all m ∈ Z.
(b) limm→±∞ λm = ±∞.
(c) The sequence {K(·, λm )}m∈Z is an orthogonal basis in 2 (Z).
(4) The function h(λ) = K(·, λ)22 (Z) is locally bounded on C.
Let Ᏼ be the set of functions F : C → C determined by
F (λ) =
∞
K(n, λ)f (n),
λ ∈ C,
(2.1)
n=−∞
where f ∈ 2 (Z). Then for all F ∈ Ᏼ, the following results hold:
(i) F is an entire function.
(ii) The functions Sm : C → C defined by
1
Sm (λ) = K ·, λm 22
K(·, λ), K ·, λm 2 (Z)
(2.2)
(Z)
are entire functions.
(iii) Every F ∈ Ᏼ admits the sampling expansion
F (λ) =
∞
n=−∞
F λn Sn (λ),
(2.3)
where the convergence of the series is absolute and uniform on compact subsets of C.
ON THE ANALYTIC FORM OF THE DISCRETE KRAMER . . .
711
Proof. First we prove that any function defined by (2.1) is an entire function.
Consider the function
∞
Fm (λ) =
K(n, λ)fm (n),
(2.4)
n=−∞
where fm (n) = f (n) if |n| ≤ m, and fm (n) = 0 if |n| > m. Clearly, Fm is an entire
function for each m. Moreover, by using the Cauchy-Schwarz inequality, we obtain
∞
F (λ) − Fm (λ) ≤
|K(n, λ)|f (n) − fm (n) ≤ K(·, λ)2 (Z) f − fm 2 (Z) . (2.5)
n=−∞
Hence, given an arbitrary compact subset Ω of C, we get
F (λ) − Fm (λ) ≤ C f − fm 2
(Z)
(2.6)
for every λ ∈ Ω and every m ∈ N. Then, Fm converges to F uniformly on compact sets
and consequently, F is an entire function. The proof that Sn (λ) is an entire function
for each n goes much in the same manner.
Now, expanding K(·, λ) with respect to the orthonormal basis
K n, λk
φk (n) = ,
(2.7)
K ·, λk 2
(Z) k∈Z
∞
we have K(n, λ) = k=−∞ ak (λ)φk (n), where
ak (λ) = K(·, λ), φk 2 (Z) = K ·, λk 2 (Z) Sk (λ), k ∈ Z.
(2.8)
∞
Therefore, for each λ ∈ C, the series k=−∞ Sk (λ)K(n, λk ) converges to K(n, λ) in
2 (Z). Now we prove the pointing convergence in (2.3). Indeed, by using the CauchySchwarz inequality
M
∞ M
K(n, λ) −
F (λ) −
F λm Sm (λ) ≤
K n, λm Sm (λ)f (n)
n=−∞
m=−M
m=−M
(2.9)
M
≤ K(n, λ) −
K n, λm Sm (λ)
f 2 (Z) .
2
∞
m=−M
(Z)
Hence, for each λ ∈ C, we have F (λ) = m=−∞ F (λm )Sm (λ).
By expanding f with respect to the orthonormal basis {φn } we obtain, from (2.3),
that {F (λn )/K(·, λn )2 (Z) }n∈Z belongs to 2 (Z). In the same way, from (2.8),
{K(·, λn )2 (Z) Sn (λ)}n∈Z belongs to 2 (Z). As a consequence, by using the CauchySchwarz inequality in
∞
∞
F λn K ·, λn 2 Sn (λ) ,
F λn Sn (λ) =
(2.10)
(Z)
K ·, λn 2
(Z)
n=−∞
n=−∞
we obtain that the series in (2.3) converges absolutely for each λ ∈ C. Finally, by
proceeding as in (2.10), we have
F (λ) −
F λm Sm (λ) = F λm Sm (λ)
|m|≤M
|m|>M


2 1/2
 F
λ
m
≤
K ·, λ 2  K(·, λ)2 (Z) .

m
(Z)
|m|>M
(2.11)
712
ANTONIO G. GARCÍA ET AL.
Since K(·, λ)2 (Z) is locally bounded on C, we obtain the uniform convergence in
(2.3) on compact subsets of C.
As a straightforward consequence of the above theorem, we derive the following
interpolation result.
Corollary 2.2. Assuming all conditions in Theorem 2.1, let {cn }n∈Z be a sequence
of complex numbers such that
2
∞ cn
< ∞.
(2.12)
K ·, λn 2 (Z) n=−∞
Then there exists a unique f ∈ 2 (Z) such that the corresponding F (λ), given by (2.1),
satisfies
(2.13)
F λn = cn , for each n ∈ Z.
Proof. Since {φk }k∈Z is an orthonormal basis for 2 (Z), by using (2.12) and the
Riesz-Fischer theorem, there exists a unique sequence h in 2 (Z) such that
∞
c̄n = h, K ·, λn 2 (Z) =
h(k)K k, λn .
(2.14)
k=−∞
Therefore, cn = h, K(·, λn )2 (Z) = h̄, K(·, λn )2 (Z) . By defining f = h̄, we have the
desired result for the function F (λ) = k∈Z K(k, λ)f (k).
3. Orthogonal polynomials as discrete analytic Kramer kernels. We consider an
indeterminate Hamburger moment sequence s = {sn }∞
n=0 , and we denote by Vs the set
of positive Borel measures satisfying it, that is,
∞
Vs = µ 0 : sn =
x n dµ(x), n 0 .
(3.1)
−∞
We consider the associated polynomials of first and second kind, {Pn }∞
n=0 and
∞
{Qn }∞
an orthonormal system with respect to the inn=0 . The sequence {Pn }n=0 forms
∞
ner product given by x n , x m = −∞ x n+m dµ(x), where µ ∈ Vs . The Pn ’s are uniquely
determined by the additional condition that their leading coefficients are positive [1].
The sequence {Qn }∞
n=0 is given by
∞
Pn (x) − Pn (t)
Qn (x) =
dµ(t),
(3.2)
x −t
−∞
where µ is any measure in Vs .
In the case of an indeterminate moment problem, the series
∞
Pn (z)2 ,
∞
Qn (z)2 ,
n=0
n=0
(3.3)
converge uniformly on compact subsets of the complex plane.
Moreover, in [3, 7, 8], it is proved that the existence of sequences {λm } of distinct
real numbers, such that
∞
,
(3.4)
P0 λm , P1 λm , P2 λm , P3 λm , . . .
m=0
ON THE ANALYTIC FORM OF THE DISCRETE KRAMER . . .
713
is an orthogonal basis for 2 (N0 ). The same occurs for the sequence of polynomials
of second kind {Qn } by considering the so-called shifted moment problem [8, 12]. As
a consequence, we can state the following result: let {Pn } and {Qn } be the sequences
of first and second kind polynomials associated with an indeterminate Hamburger moment problem. Then by defining either K(n, λ) = Pn (λ) or K(n, λ) = Qn (λ), we obtain
discrete Kramer analytic kernels.
The existence of the sequences {λm } can be explained as follows: the set Vs of
solutions to an indeterminate Hamburger moment problem can be parametrized with
the one-point compactification ᏼ ∪{∞} of the Pick (or Herglotz) functions set ᏼ [1, 13].
When the parameter is restricted to constant functions taking values in R ∪ {∞}, we
obtain the set of N-extremal measures {µt } which satisfy
∞
A(z)t − C(z)
dµt (x)
=−
, z ∈ C \ R,
(3.5)
x
−
z
B(z)t − D(z)
−∞
where A, B, C, and D are the entire functions forming the so-called Nevanlinna matrix associated with the moment problem. See [4, 13] for explicit formulas of the
Nevanlinna matrix. It is known that, for each t ∈ R ∪ {∞}, µt is the discrete measure
µt = z∈Zt mz δz , where

 z ∈ C | B(z)t − D(z) = 0
if t ∈ R,
Zt =  z ∈ C | B(z) = 0
if t = ∞,
(3.6)
mz =
A(z)t − C(z)
B (z)t − D (z)
for z ∈ Zt .
Recall that the zeros of the entire function B(z)t − D(z) or B(z) are real and simple, and they form, precisely, one of the sequences {λtm } we are looking for. The
N-extremal measures are precisely those measures µ ∈ Vs for which the polynomials
are dense in L2 (µ). In the above cases we can identify the set Ᏼ determined by (2.1)
with the space L2 (µt ).
Another equivalent formulation can be given either in terms of the self-adjoint
extensions of a semi-infinite Jacobi matrix, or in terms of the discrete Weyl’s limit
point/limit circle theory for the self-adjoint extensions of a discrete Sturm-Liouville
boundary value problem. Given the sequence {sn }∞
n=0 of Hamburger moments, we can
∞
find two sequences {bn }∞
and
{a
}
of
real
and
positive numbers, respectively,
n n=0
n=0
namely bn = xPn2 (x)dµ(x) and an = xPn (x)Pn+1 (x)dµ(x), so that the moment
problem is associated to self-adjoint extensions of the semi-infinite Jacobi matrix


b0 a0
0
0 ···


a0 b1 a1
0 · · ·



.. 
0 a
.
b
a


1
2
2
Ꮽ=
(3.7)



.
.
0

.
0 a2 b3

 .

..
..
..
..
..
.
.
.
.
with its domain D(Ꮽ) being the set of sequences of finite support. The uniqueness of
the Hamburger problem depends on the existence of a unique self-adjoint extension
714
ANTONIO G. GARCÍA ET AL.
of the operator defined by Ꮽ [13, page 86]. Associated with the Jacobi matrix Ꮽ is the
three-term recurrence relation
λXn = an Xn+1 + bn Xn + an−1 Xn−1 ,
n
0.
(3.8)
By taking initial conditions X−1 = 0 and X0 = 1 we obtain the sequence {Pn (λ)}∞
n=0 ,
and for X−1 = −1 and X0 = 0 we obtain the second kind associated polynomials
{Qn (λ)}∞
n=0 . Following B. Simon [13, page 94], the indeterminacy of the Hamburger
moment problem can be thought as the discrete Weyl’s limit point/limit circle theory
for the self-adjoint extensions of the discrete Sturm-Liouville problem associated with
the former three-term recurrence relation. When the moment problem is indetermit
(z)
nate we are in the limit-circle at infinity, and the Weyl-Titchmarsh functions m∞
are given precisely by (3.5). Its poles, that is, the zeros of the denominator in (3.5)
are the eigenvalues of the self-adjoint extension, and {P0 (λtm ), P1 (λtm ), . . . }∞
m=0 is the
2
corresponding sequence of complete orthogonal eigenvectors in (N0 ).
Furthermore, in [5, 8], it is proved that the sampling (interpolatory) functions, in
this particular case, are nothing more than analytic interpolation functions, that is,
they are of the form
G(λ)
,
(3.9)
Sm (λ) = G λm λ − λm
where G is an entire function having its only simple zeros at the sequence {λm }. The
corresponding sampling expansion can be written as a Lagrange-type interpolation
series
∞
G(λ)
,
F (λ) =
(3.10)
F λm G
λ
λ − λm
m
m=0
where the convergence of the series is absolute and uniform on compact subsets of
the complex plane.
Finally, in [10] one can find an example, the q-Hermite polynomials, where all the
necessary ingredients for the analytic discrete Kramer sampling theorem are explicity
computed.
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Antonio G. García: Departamento de Matemáticas, Universidad Carlos III de Madrid,
Avda de la Universidad, 30, 28911 Leganés-Madrid, Spain
E-mail address: agarcia@math.uc3m.es
Miguel A. Hernández-Medina: Departamento de Matemática Aplicada, E.T.S.I.T. Universidad Politécnica de Madrid, Avda Complutense s/n, 28040 Madrid, Spain
E-mail address: mahm@mat.upm.es
María J. Muñoz-Bouzo: Departamento de Matemáticas Fundamentales, Facultad de
Ciencias, U.N.E.D., Senda del Rey, 9, 28040 Madrid, Spain
E-mail address: mjmunoz@mat.uned.es