- Email: [email protected]
Characterization of solutions to the Stieltjes–Wigert moment problem
Statistics and Probability Letters 79 (2009) 1337–1342
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Characterization of solutions to the Stieltjes–Wigert moment problem Marcos López-García ∗ Instituto de Matemáticas, Universidad Nacional Autónoma de México, México, D.F. C.P. 04510, Mexico
article
info
a b s t r a c t For β ∈ R, p > 0, and 0 < q < 1 fixed, it is well known that an integrable function R on (0, ∞) satisfying the functional equation f (qx) = qβ−1/2 (x + pq−1/2 )f (x), and f =
Article history: Received 2 September 2008 Received in revised form 11 December 2008 Accepted 18 February 2009 Available online 26 February 2009
1 q−β /2 (pqβ ; q)− ∞ , is a solution to the generalized Stieltjes–Wigert moment problem. In this work we characterize the functions satisfying both conditions, and we give some families of solutions to the generalized Stieltjes–Wigert moment problem not satisfying the previous functional equation. © 2009 Elsevier B.V. All rights reserved. 2
MSC: primary 60E99 44A60 secondary 35K05
1. Introduction For a measurable function f defined on R+ = (0, ∞), its nth moment is defined as sn (f ) = −(n+β)2 /2
a measurable function defined on R+ with moment sequence q solution to the generalized Stieltjes–Wigert moment problem.
pqβ ; q
R∞
−1
n
0
xn f (x)dx, n ∈ N. If f is
pqβ , q ∞ , n ∈ N, then we say that f is a
2 Throughout this paper we assume that β ∈ R, p > 0, and 0 < q < 1 has been fixed, so we can write q = e−σ for some σ > 0 fixed, and we will use that logq x = ln x/ ln q. We introduce the following notation,
(p; q)0 = 1,
(p; q)n =
n−1 Y
1 − pqk
for n ≥ 1,
and
k=0
(p; q)∞ =
∞ Y
1 − pqk .
k=0
The log-normal distribution has the following density function defined on R+ : dσ (x) = (2π σ 2 )−1/2 x−1 exp(−2−1 σ −2 (log x)2 ),
x > 0,
We recall that the so-called theta function is given by
θ (x, t ) = (4π t )−1/2
X
2 e−(x+n) /(4t ) ,
∀(x, t ) ∈ R2+ , (e.g. Cannon (1984, p. 59)).
n∈Z
The main result characterizes a class of solutions to the generalized Stieltjes–Wigert moment problem: Theorem 1. Let β ∈ R, p > 0, 0 < q < 1, and f ∈ L1 R+ . Then f satisfies the functional equation
f (qx) = qβ−1/2 x + pq−1/2 f (x), ∗
x > 0,
Tel.: +52 55 56 22 47 82. E-mail address: [email protected].
0167-7152/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2009.02.007
(1)
1338
and
M. López-García / Statistics and Probability Letters 79 (2009) 1337–1342
R∞ 0
−1 2 f (x)dx = q−β /2 pqβ ; q ∞ if and only if there exists a 1-periodic function ϕ ∈ L1 (0, 1) such that f (x) =
xβ (−pq1/2 x−1 ; q)∞ dσ (x) 1 + ϕ(β + logq x) and
1
Z
ϕ(x)θ (x, 2−1 σ −2 )dx = 0,
(2)
0
where σ 2 = − ln q. In any case,
−1 2 sn (f ) = q−(n+β) /2 pqβ ; q n pqβ ; q ∞
for all n ∈ N.
(3)
When β = α + 1/2, p = q1/2 , the corresponding functional equation has been analyzed before (e.g. Chihara (1979) and Christiansen (2003b)). López-García (in press) studied the case p = 0, which characterizes a class of solutions to the log-normal moment problem. Remark 1. If ϕ satisfies (2) then so does aReϕ , a ∈ R. Moreover, if Reϕ is bounded below (above), there is a ∈ R such that 1 + aReϕ ≥ 0. Hence in this case a probability density function f can be obtained by a standard normalizing procedure. Example 1. If ϕ (x) =
X
cn e−2σ
−2 π 2 n2
P
n∈Z
cn e2π nix ∈ L2 (0, 1), then ϕ satisfies (2) if and only if
= 0. (see (6)).
n∈Z
Example 2. By (7) we have that
ϕc (x) = −1 +
1
Z 0
θ (x, 2−1 σ −2 ) dx θ c − x, 2−1 σ −2
! −1
1
θ c − x , 2 −1 σ −2
(4)
is a 1-periodic, continuous function satisfying (2) for all c ∈ [0, 1). So, the corresponding function given by Theorem 1 can be written as follows (see (10)) fc (x) =
2 qβ c −β /2 (−pq1/2 x−1 ; q)∞ xc −1
,
Mc (q; q)∞ −qβ−c +1/2 x; q ∞ −qc −β+1/2 x−1 ; q ∞
and is a solution to the generalized Stieltjes–Wigert moment problem, where
c c − 21
M0 = − ln q,
Mc =
πq
sin (π c )
qc , q1−c ; q ∞ , (q; q)2∞
c ∈ (0, 1) .
Gómez and López-García (2007, Section 3) obtained more examples of functions fulfilling (2). This work is organized as follows. Preliminaries are given in the following section, and the last section contains the proof of the main theorem and some families of solutions to the generalized Stieltjes–Wigert moment problem not satisfying the functional equation (1). 2. Notation and preliminaries Cannon (1984) studied the Gauss kernel and the theta function given as follows, 2 K (x, t ) = (4π t )−1/2 e−x /(4t ) ,
θ (x, t ) =
X
K (x + n, t ) =
n∈Z
X
(5) e
−4π 2 n2 t +2π nix
,
(x, t ) ∈ R+ . 2
(6)
n∈Z
The positive functions K , θ satisfy the heat equation on R2+ . Clearly θ (·, t ) is a 1-periodic continuous function for all t > 0. Moreover, 1
Z
θ (x, t )dx = 0
Z
K (x, t )dx = 1,
∀t > 0.
(7)
R
2 2 2 Since q = e−σ we can write dσ (x) = (2π σ 2 )−1/2 x−1 q(logq x) /2 . Notice that xβ dσ (x) = qβ−β /2 dσ qβ x , x > 0, so by making the change of variable y = β + logq x, we obtain
M. López-García / Statistics and Probability Letters 79 (2009) 1337–1342
∞
Z
2 xβ dσ (x) ψ logq x dx = q−β /2
Z
∞
Z
−∞ ∞
0
= q−β
2 /2
(2π σ −2 )−1/2 qy
2 /2
1339
ψ (y − β) dy
K (y, 2−1 σ −2 )ψ (y − β) dy −∞
= q−β
2 /2
1
Z
θ (y, 2−1 σ −2 )ψ (y − β) dy,
(8)
0
provided that ψ ∈ L1 (0, 1) is a 1-periodic function. For x ∈ R, we define Lq (x) =
X
2 qn /2 xn .
n∈Z
The value of the sum Lq (x) is known as Jacobi’s triple product identity (e.g. Gasper and Rahman (1990, p. 239)) Lq (x) = (q; q)∞ −q1/2 x; q ∞ −q1/2 x−1 ; q ∞ .
Since ln q = −σ 2 it is easy to check the identity
r
σ 2 x2 /2 q Lq q−x = θ x, 2−1 σ −2 , ∀x ∈ R. 2π For c ∈ [0, 1), Berg (1998) computed the following integral Z ∞ c c − 12 qc , q1−c ; q ∞ x c −1 πq Mc := dx = , c ∈ (0, 1) Lq (xq−c ) sin (π c ) (q; q)2∞ 0 and M0 = − ln q. By the Monotone Convergence Theorem and (9), we have Z 1 X Z 1 K (x + n, 2−1 σ −2 ) θ (x, 2−1 σ −2 ) dx = dx −1 σ −2 −1 σ −2 0 θ c − x, 2 n∈Z 0 θ c − x, 2 Z 2 qx /2 = dx 2
(9)
q(c −x) /2 Lq q−(c −x)
R
= q−c
2 /2
σ −2 Mc .
Let ϕc be the function given in (4). By (9) and the last equality we get that 1 + ϕc β + logq x =
2 qβ c −β /2 xc −1−β
Mc dσ (x) Lq qβ−c x
.
(10)
For 0 < q < 1, and p > 0 we introduce a sequence of functions (Πn )n∈Z defined on R+ as follows
Π0 ≡ 1,
Πn (x) =
|n| Y
n
x + pq
|n|
1 −j 2
! |n| n
,
n ∈ Z \ {0}.
j =1
It is easy to see that
Πn+1 (x) = x + pq−n−1/2 Πn (x) ,
∀n ∈ Z, x > 0.
(11)
Proposition 2. If f satisfies the functional equation (1) on R+ , then
2 f qn x = q(β−1)n+n /2 Πn (x) f (x) ,
∀n ∈ Z, x > 0.
Proof. Clearly the relation is valid for n = 0, 1. Let us suppose that the relation holds for some n > 1, then
f q(qn x) = qβ−1/2 qn x + pq−1/2 f (qn x) = qβ−1/2+n+(β−1)n+n
2 /2
2 /2
= q(β−1)(n+1)+(n+1)
x + pq−n−1/2 Πn (x) f (x)
Πn+1 (x) f (x) ,
x > 0.
On the other hand, by using the functional equation (1), we obtain that f (q−1 x) = q3/2−β x + pq1/2
−1
f (x),
x > 0.
(12)
1340
M. López-García / Statistics and Probability Letters 79 (2009) 1337–1342
So, the result is valid for n = −1. Let us suppose that the relation holds for some n < −1, then f q−1 (qn x) = q3/2−β qn x + pq1/2
= q3/2−β−n+(β−1)n+n
−1
x + pq1/2−n
2 /2
(β−1)(n−1)+(n−1)2 /2
=q
f (qn x)
−1
Πn−1 (x) f (x) ,
Πn (x) f (x)
x > 0.
Theorem 3. If f ∈ L1 R+ satisfies the functional equation (1) then ∞
Z
−β(n+m)−(n+m)2 /2
β
x Πn (x) f (x) dx = q m
pq ; q
0
∞
Z
m
f (x) dx,
(13)
0
for all m ∈ N, n ∈ Z. Proof. Using (12) we get that
2 qqn f qn (qx) = qβ(n+1)+(n+1) /2 Πn+1 (x) f (x) = qqβ n+n
2 /2
Πn (qx) f (qx) ,
∀n ∈ Z, x > 0.
It follows that ∞
Z
Πn+1 (x) f (x) dx = q−(n+β+1/2)
∞
Z
0
Πn (x) f (x) dx,
∀ n ∈ Z.
0
By iterating we can see that ∞
Z
Πn (x) f (x) dx = q−β n−n
∞
Z
2 /2
0
f (x) dx,
∀n ∈ Z.
0
Thus, (13) is valid for m = 0, n ∈ Z. Let us suppose that the relation holds for some m > 0. By (11) we get that ∞
Z
xm Πn+1 (x) f (x) dx = 0
∞
Z
qn+1/2
0
∞
Z
p
xm+1 Πn (x) f (x) dx +
xm Πn (x) f (x) dx, 0
therefore ∞
Z 0
2 xm+1 Πn (x) f (x) dx = q−β(n+1+m)−(n+1+m) /2 pqβ ; q m h
∞
Z
f (x) dx 0
× 1 − qβ(n+1+m)+(n+1+m)
2 /2−β(n+m)−(n+m)2 /2−n−1/2
= q−β(n+m+1)−(n+m+1)
2 /2
pqβ ; q
m
p
i
1 − pqβ qm s0 (f ).
2 Corollary 4. If f ∈ L1 R+ satisfies the functional equation (1) then sn (f ) = q−β n−n /2 pqβ ; q
R∞
n
0
f (x)dx, for all n ∈ N.
Corollary 5. If f ∈ L1 R+ satisfies the functional equation (1) then ∞
Z 0
f (x)
(−pq1/2 x−1 ; q)∞
dx = pqβ ; q ∞
∞
Z
f (x) dx.
(14)
0
Proof. By (13) its follows that, ∞
Z
xn Π−n (x) f (x) dx = pqβ ; q
0
We have that 0 ≤ xn Π−n (x) =
n
∞
Z
f (x) dx,
1 + pqj−1/2 x−1
Qn
j =1
1
lim xn Π−n (x) =
n −1
n→∞
lim
Q
n→∞ j=0
∀n > 0.
0
1 + pqj+1/2 x−1
−1
=
≤ 1, for n ≥ 1, x > 0, and 1
(−pq1/2 x−1 ; q)∞
,
therefore the result follows by the Dominated Convergence Theorem. Remark 2. If ψ is a 1-periodic function and f satisfies the functional equation (1), then so does the function f (x) ψ logq x .
M. López-García / Statistics and Probability Letters 79 (2009) 1337–1342
1341
Example 3. For β ∈ R, p > 0, 0 < q < 1, and σ 2 = − ln q, the positive function xβ (−pq1/2 x−1 ; q)∞ dσ (x) satisfies the functional equation (1). The last remark, (8) and (14) imply that ∞
Z
1 xβ (−pq1/2 x−1 ; q)∞ dσ (x)ψ(logq x)dx = (pqβ ; q)− ∞
∞
Z
xβ dσ (x)ψ(logq x)dx,
0
0
=
2 q−β /2
1
Z
(pqβ ; q)∞
σ −2 ψ(x − β)θ x, dx, 2
0
(15)
whenever ψ ∈ L1 (0, 1) is a 1-periodic function. In particular, we have that ∞
Z 0
−1 2 xβ (−pq1/2 x−1 ; q)∞ dσ (x)dx = q−β /2 pqβ ; q ∞ .
(16)
Remark 3. In q-calculus a function f defined on R+ is q-periodic if f (qx) = f (x) for all x > 0. If f is a q-periodic function defined on R+ then ψ(x) = f (qx ) is a 1-periodic function (in the usual sense) on R. So, we have that f is a q-periodic function if and only if there exists a 1-periodic function ψ such that f (x) = ψ(logq x). Remark 4. If f , g : R+ → R satisfy the functional equation (1) and g is a positive function then f /g (x) = f /g (qx), therefore f /g is a q-periodic function. By the previous remark there exists a 1-periodic function ψ such that f /g (x) = ψ(logq x). In particular, we can write f (x) = xβ (−pq1/2 x−1 ; q)∞ dσ (x)ψ(logq x), where ψ is a 1-periodic function. This observation was made in the literature before (e.g Chihara (1979, 1970) and Christiansen (2003a,b). 3. Proof of the main theorem Proof of Theorem 1. If f ∈ L1 R+ satisfies the functional equation (1), then there exists a 1-periodic function ψ such that f (x) = xβ (−pq1/2 x−1 ; q)∞ dσ (x)ψ(logq x), x > 0. The other part of the hypothesis and (15) imply that
2 q−β /2
pqβ ; q
= s0 (f ) =
∞
2 q−β /2
pqβ ; q
1
Z
∞
ψ(x − β)θ (x, 2−1 σ −2 )dx. 0
By (7) we have that ϕ(x) = −1 + ψ(x − β) is a 1 -periodic, integrable function satisfying (2). On the other hand, the function f (x) = xβ (−pq1/2 x−1 ; q)∞ dσ (x){1 + ϕ(β + logq x)} satisfies (1) if ϕ ∈ L1 (0, 1) is a 1-periodic function. From (15) with ψ (x) = ϕ(x + β), (16), and (2) we have that
R∞ 0
−1 2 f (x)dx = q−β /2 pqβ , q ∞ .
Another classes of solutions to the generalized Stieltjes–Wigert moment problem. Proposition 6. If h ∈ L1 (R) is bounded below (above) and
Z
K x, 2−1 σ −2 h (x + n) dx = 0,
∀n ∈ Z,
(17)
R
then the function f (x) = xβ (−pq1/2 x−1 ; q)∞ dσ (x) 1 + h(β + logq x) is a solution to the Stieltjes–Wigert moment problem. If h is not a 1-periodic function then f does not satisfy the functional equation (1).
Proof. As in (8), with β + n and h (· + β) instead of β and ψ , we have that (17) is equivalent to ∞
Z
xn+β dσ (x) h(β + logq x)dx = 0,
∀n ∈ Z.
0
Next, we use the well known identity (e.g. Gasper and Rahman (1990, p. 236)),
∞ X q k=0
k 2
zk
(q; q)k
= (−z ; q)∞ ,
z ∈ C,
the fact that xn+β (−pq1/2 x−1 ; q)∞ dσ (x) ∈ L1 R+ , n ∈ N, and the Monotone Convergence Theorem to get that
∞
Z
xn+β dσ (x) −pq1/2 x−1 ; q ∞ h(β + logq x)dx =
0
Z 2 ∞ X qk /2 pk k=0
(q; q)k
for all n ∈ N. The result follows from Example 3 and Theorem 1. For f ∈ L1 (R), we define its Fourier transform as
(Φ f )(ξ ) =
Z
∞
dx f (x)e−ixξ √ . 2π −∞
0
∞
xn−k+β dσ (x) h(β + logq x)dx = 0,
1342
M. López-García / Statistics and Probability Letters 79 (2009) 1337–1342
The following examples provide functions satisfying (17) which are not 1-periodic functions. Example 4. López-García (in press, Theorem 3) proved for h ∈ L2 (R) that the condition (17) is equivalent to
X
K (ξ + 2nπ , 2−1 σ 2 )(Φ h)(ξ + 2nπ ) = 0,
a.e. ξ ∈ R.
(18)
n∈Z
In particular, if γ ∈ L2 (R) has compact support and satisfies
X
γ (ξ + 2nπ ) = 0,
a.e. ξ ∈ R,
(19)
n∈Z
then the function h ∈ L2 (R) given by
Φ h = γ /K (·, 2−1 σ 2 ) satisfies (18). For instance, consider φ ∈ C ∞ (R) with supp φ ⊂ [0, 2π ], then γ (ξ ) = φ(ξ ) − φ(ξ − 2π ) fulfills (19), and the corresponding function h is a rapidly decreasing function. Example 5. Let α ∈ R \ 2π Z, λ ∈ C and g a 1-periodic function in L1 (0, 1). López-García (in press) proved that the function (λ + g (x))eiαx is not 1-periodic and fulfills (17) provided that
√
2π σ 2 K (α, 2−1 σ 2 )λ = −
Z
K (x, 2−1 σ −2 )g (x)eiα x dx. R
For instance, with g (x) = cos(2π x) we have that Re((λ + g (x))eiα x ) = (λ + cos(2π x)) cos(α x) satisfies (17) whenever
√
2π σ
2K
(α, 2 σ )λ = − −1
2
Z
K (x, 2−1 σ −2 ) cos(2π x) cos(α x)dx. R
Remark 5. For a density f defined on R+ and h ∈ L∞ (R+ ), khk∞ = 1, Stoyanov (2004) introduced the Stieltjes class with center f and perturbation h, given as follows S(f , h) = {f (x)(1 + ε h(x)) : x > 0, ε ∈ [−1, 1]},
R∞
provided that 0 xn f (x)h(x)dx = 0 for all n ∈ N. So, we have shown that any function satisfying the functional equation (1) belongs to the (generalized) Stieltjes class with center xβ (−pq1/2 x−1 ; q)∞ dσ (x). Pakes (2007) extended the results of Stoyanov for general distributions supported on R+ . Stoyanov and Tolmatz (2004, 2005) have found more examples of Stieltjes classes. Acknowledgement I thank the anonymous referee for carefully reading the original manuscript. References Berg, C., 1998. From discrete to absolutely continuous solutions of indeterminate moment problems. Arab. J. Math. Sci. 4, 1–18. Cannon, J.R., 1984. The One-Dimensional Heat Equation. Addison-Wesley, New York. Chihara, T.S., 1970. A characterization and a class of distribution functions for the Stieltjes–Wigert polynomials. Canad. Math. Bull. 13, 529–532. Chihara, T.S., 1979. On generalized Stieltjes–Wigert and related orthogonal polynomials. J. Comput. Appl. Math. 5 (4), 291–297. Christiansen, J.S., 2003a. The moment problem associated with the Stieltjes–Wigert polynomials. J. Math. Anal. Appl. 277 (1), 218–245. Christiansen, J.S., 2003b. The moment problem associated with the q-Laguerre polynomials. Constr. Approx. 19, 1–22. Gasper, G., Rahman, M., 1990. Basic Hypergeometric Series. In: Encyclopedia of Mathematics and its Applications, vol. 35. Cambridge University Press, Cambridge, UK. Gómez, R., López-García, M., 2007. A family of heat functions as solutions of indeterminate moment problems. Int. J. Math. Math. Sci. 2007, 11. doi:10.1155/2007/41526. Article ID 41526. López-García, M., Characterization of solutions to the log-normal moment problem. Theory Probab. Appl. (in press). Stoyanov, J., 2004. Stieltjes classes for moment indeterminate probability distributions. J. Appl. Probab. 41A, 281–294. Stoyanov, J., Tolmatz, L., 2004. New Stieltjes classes involving generalized gamma distributions. Statist. Probab. Lett. 213–219. Stoyanov, J., Tolmatz, L., 2005. Method for constructing Stieltjes classes for M-indeterminate probability distributions. Appl. Math. Comput. 165, 669–685. Pakes, A., 2007. Structure of Stieltjes classes of moment-equivalent probability laws. J. Math. Anal. Appl. 326, 1268–1290.
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Characterization of solutions to the Stieltjes–Wigert moment problem Marcos López-García ∗ Instituto de Matemáticas, Universidad Nacional Autónoma de México, México, D.F. C.P. 04510, Mexico
article
info
a b s t r a c t For β ∈ R, p > 0, and 0 < q < 1 fixed, it is well known that an integrable function R on (0, ∞) satisfying the functional equation f (qx) = qβ−1/2 (x + pq−1/2 )f (x), and f =
Article history: Received 2 September 2008 Received in revised form 11 December 2008 Accepted 18 February 2009 Available online 26 February 2009
1 q−β /2 (pqβ ; q)− ∞ , is a solution to the generalized Stieltjes–Wigert moment problem. In this work we characterize the functions satisfying both conditions, and we give some families of solutions to the generalized Stieltjes–Wigert moment problem not satisfying the previous functional equation. © 2009 Elsevier B.V. All rights reserved. 2
MSC: primary 60E99 44A60 secondary 35K05
1. Introduction For a measurable function f defined on R+ = (0, ∞), its nth moment is defined as sn (f ) = −(n+β)2 /2
a measurable function defined on R+ with moment sequence q solution to the generalized Stieltjes–Wigert moment problem.
pqβ ; q
R∞
−1
n
0
xn f (x)dx, n ∈ N. If f is
pqβ , q ∞ , n ∈ N, then we say that f is a
2 Throughout this paper we assume that β ∈ R, p > 0, and 0 < q < 1 has been fixed, so we can write q = e−σ for some σ > 0 fixed, and we will use that logq x = ln x/ ln q. We introduce the following notation,
(p; q)0 = 1,
(p; q)n =
n−1 Y
1 − pqk
for n ≥ 1,
and
k=0
(p; q)∞ =
∞ Y
1 − pqk .
k=0
The log-normal distribution has the following density function defined on R+ : dσ (x) = (2π σ 2 )−1/2 x−1 exp(−2−1 σ −2 (log x)2 ),
x > 0,
We recall that the so-called theta function is given by
θ (x, t ) = (4π t )−1/2
X
2 e−(x+n) /(4t ) ,
∀(x, t ) ∈ R2+ , (e.g. Cannon (1984, p. 59)).
n∈Z
The main result characterizes a class of solutions to the generalized Stieltjes–Wigert moment problem: Theorem 1. Let β ∈ R, p > 0, 0 < q < 1, and f ∈ L1 R+ . Then f satisfies the functional equation
f (qx) = qβ−1/2 x + pq−1/2 f (x), ∗
x > 0,
Tel.: +52 55 56 22 47 82. E-mail address: [email protected].
0167-7152/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2009.02.007
(1)
1338
and
M. López-García / Statistics and Probability Letters 79 (2009) 1337–1342
R∞ 0
−1 2 f (x)dx = q−β /2 pqβ ; q ∞ if and only if there exists a 1-periodic function ϕ ∈ L1 (0, 1) such that f (x) =
xβ (−pq1/2 x−1 ; q)∞ dσ (x) 1 + ϕ(β + logq x) and
1
Z
ϕ(x)θ (x, 2−1 σ −2 )dx = 0,
(2)
0
where σ 2 = − ln q. In any case,
−1 2 sn (f ) = q−(n+β) /2 pqβ ; q n pqβ ; q ∞
for all n ∈ N.
(3)
When β = α + 1/2, p = q1/2 , the corresponding functional equation has been analyzed before (e.g. Chihara (1979) and Christiansen (2003b)). López-García (in press) studied the case p = 0, which characterizes a class of solutions to the log-normal moment problem. Remark 1. If ϕ satisfies (2) then so does aReϕ , a ∈ R. Moreover, if Reϕ is bounded below (above), there is a ∈ R such that 1 + aReϕ ≥ 0. Hence in this case a probability density function f can be obtained by a standard normalizing procedure. Example 1. If ϕ (x) =
X
cn e−2σ
−2 π 2 n2
P
n∈Z
cn e2π nix ∈ L2 (0, 1), then ϕ satisfies (2) if and only if
= 0. (see (6)).
n∈Z
Example 2. By (7) we have that
ϕc (x) = −1 +
1
Z 0
θ (x, 2−1 σ −2 ) dx θ c − x, 2−1 σ −2
! −1
1
θ c − x , 2 −1 σ −2
(4)
is a 1-periodic, continuous function satisfying (2) for all c ∈ [0, 1). So, the corresponding function given by Theorem 1 can be written as follows (see (10)) fc (x) =
2 qβ c −β /2 (−pq1/2 x−1 ; q)∞ xc −1
,
Mc (q; q)∞ −qβ−c +1/2 x; q ∞ −qc −β+1/2 x−1 ; q ∞
and is a solution to the generalized Stieltjes–Wigert moment problem, where
c c − 21
M0 = − ln q,
Mc =
πq
sin (π c )
qc , q1−c ; q ∞ , (q; q)2∞
c ∈ (0, 1) .
Gómez and López-García (2007, Section 3) obtained more examples of functions fulfilling (2). This work is organized as follows. Preliminaries are given in the following section, and the last section contains the proof of the main theorem and some families of solutions to the generalized Stieltjes–Wigert moment problem not satisfying the functional equation (1). 2. Notation and preliminaries Cannon (1984) studied the Gauss kernel and the theta function given as follows, 2 K (x, t ) = (4π t )−1/2 e−x /(4t ) ,
θ (x, t ) =
X
K (x + n, t ) =
n∈Z
X
(5) e
−4π 2 n2 t +2π nix
,
(x, t ) ∈ R+ . 2
(6)
n∈Z
The positive functions K , θ satisfy the heat equation on R2+ . Clearly θ (·, t ) is a 1-periodic continuous function for all t > 0. Moreover, 1
Z
θ (x, t )dx = 0
Z
K (x, t )dx = 1,
∀t > 0.
(7)
R
2 2 2 Since q = e−σ we can write dσ (x) = (2π σ 2 )−1/2 x−1 q(logq x) /2 . Notice that xβ dσ (x) = qβ−β /2 dσ qβ x , x > 0, so by making the change of variable y = β + logq x, we obtain
M. López-García / Statistics and Probability Letters 79 (2009) 1337–1342
∞
Z
2 xβ dσ (x) ψ logq x dx = q−β /2
Z
∞
Z
−∞ ∞
0
= q−β
2 /2
(2π σ −2 )−1/2 qy
2 /2
1339
ψ (y − β) dy
K (y, 2−1 σ −2 )ψ (y − β) dy −∞
= q−β
2 /2
1
Z
θ (y, 2−1 σ −2 )ψ (y − β) dy,
(8)
0
provided that ψ ∈ L1 (0, 1) is a 1-periodic function. For x ∈ R, we define Lq (x) =
X
2 qn /2 xn .
n∈Z
The value of the sum Lq (x) is known as Jacobi’s triple product identity (e.g. Gasper and Rahman (1990, p. 239)) Lq (x) = (q; q)∞ −q1/2 x; q ∞ −q1/2 x−1 ; q ∞ .
Since ln q = −σ 2 it is easy to check the identity
r
σ 2 x2 /2 q Lq q−x = θ x, 2−1 σ −2 , ∀x ∈ R. 2π For c ∈ [0, 1), Berg (1998) computed the following integral Z ∞ c c − 12 qc , q1−c ; q ∞ x c −1 πq Mc := dx = , c ∈ (0, 1) Lq (xq−c ) sin (π c ) (q; q)2∞ 0 and M0 = − ln q. By the Monotone Convergence Theorem and (9), we have Z 1 X Z 1 K (x + n, 2−1 σ −2 ) θ (x, 2−1 σ −2 ) dx = dx −1 σ −2 −1 σ −2 0 θ c − x, 2 n∈Z 0 θ c − x, 2 Z 2 qx /2 = dx 2
(9)
q(c −x) /2 Lq q−(c −x)
R
= q−c
2 /2
σ −2 Mc .
Let ϕc be the function given in (4). By (9) and the last equality we get that 1 + ϕc β + logq x =
2 qβ c −β /2 xc −1−β
Mc dσ (x) Lq qβ−c x
.
(10)
For 0 < q < 1, and p > 0 we introduce a sequence of functions (Πn )n∈Z defined on R+ as follows
Π0 ≡ 1,
Πn (x) =
|n| Y
n
x + pq
|n|
1 −j 2
! |n| n
,
n ∈ Z \ {0}.
j =1
It is easy to see that
Πn+1 (x) = x + pq−n−1/2 Πn (x) ,
∀n ∈ Z, x > 0.
(11)
Proposition 2. If f satisfies the functional equation (1) on R+ , then
2 f qn x = q(β−1)n+n /2 Πn (x) f (x) ,
∀n ∈ Z, x > 0.
Proof. Clearly the relation is valid for n = 0, 1. Let us suppose that the relation holds for some n > 1, then
f q(qn x) = qβ−1/2 qn x + pq−1/2 f (qn x) = qβ−1/2+n+(β−1)n+n
2 /2
2 /2
= q(β−1)(n+1)+(n+1)
x + pq−n−1/2 Πn (x) f (x)
Πn+1 (x) f (x) ,
x > 0.
On the other hand, by using the functional equation (1), we obtain that f (q−1 x) = q3/2−β x + pq1/2
−1
f (x),
x > 0.
(12)
1340
M. López-García / Statistics and Probability Letters 79 (2009) 1337–1342
So, the result is valid for n = −1. Let us suppose that the relation holds for some n < −1, then f q−1 (qn x) = q3/2−β qn x + pq1/2
= q3/2−β−n+(β−1)n+n
−1
x + pq1/2−n
2 /2
(β−1)(n−1)+(n−1)2 /2
=q
f (qn x)
−1
Πn−1 (x) f (x) ,
Πn (x) f (x)
x > 0.
Theorem 3. If f ∈ L1 R+ satisfies the functional equation (1) then ∞
Z
−β(n+m)−(n+m)2 /2
β
x Πn (x) f (x) dx = q m
pq ; q
0
∞
Z
m
f (x) dx,
(13)
0
for all m ∈ N, n ∈ Z. Proof. Using (12) we get that
2 qqn f qn (qx) = qβ(n+1)+(n+1) /2 Πn+1 (x) f (x) = qqβ n+n
2 /2
Πn (qx) f (qx) ,
∀n ∈ Z, x > 0.
It follows that ∞
Z
Πn+1 (x) f (x) dx = q−(n+β+1/2)
∞
Z
0
Πn (x) f (x) dx,
∀ n ∈ Z.
0
By iterating we can see that ∞
Z
Πn (x) f (x) dx = q−β n−n
∞
Z
2 /2
0
f (x) dx,
∀n ∈ Z.
0
Thus, (13) is valid for m = 0, n ∈ Z. Let us suppose that the relation holds for some m > 0. By (11) we get that ∞
Z
xm Πn+1 (x) f (x) dx = 0
∞
Z
qn+1/2
0
∞
Z
p
xm+1 Πn (x) f (x) dx +
xm Πn (x) f (x) dx, 0
therefore ∞
Z 0
2 xm+1 Πn (x) f (x) dx = q−β(n+1+m)−(n+1+m) /2 pqβ ; q m h
∞
Z
f (x) dx 0
× 1 − qβ(n+1+m)+(n+1+m)
2 /2−β(n+m)−(n+m)2 /2−n−1/2
= q−β(n+m+1)−(n+m+1)
2 /2
pqβ ; q
m
p
i
1 − pqβ qm s0 (f ).
2 Corollary 4. If f ∈ L1 R+ satisfies the functional equation (1) then sn (f ) = q−β n−n /2 pqβ ; q
R∞
n
0
f (x)dx, for all n ∈ N.
Corollary 5. If f ∈ L1 R+ satisfies the functional equation (1) then ∞
Z 0
f (x)
(−pq1/2 x−1 ; q)∞
dx = pqβ ; q ∞
∞
Z
f (x) dx.
(14)
0
Proof. By (13) its follows that, ∞
Z
xn Π−n (x) f (x) dx = pqβ ; q
0
We have that 0 ≤ xn Π−n (x) =
n
∞
Z
f (x) dx,
1 + pqj−1/2 x−1
Qn
j =1
1
lim xn Π−n (x) =
n −1
n→∞
lim
Q
n→∞ j=0
∀n > 0.
0
1 + pqj+1/2 x−1
−1
=
≤ 1, for n ≥ 1, x > 0, and 1
(−pq1/2 x−1 ; q)∞
,
therefore the result follows by the Dominated Convergence Theorem. Remark 2. If ψ is a 1-periodic function and f satisfies the functional equation (1), then so does the function f (x) ψ logq x .
M. López-García / Statistics and Probability Letters 79 (2009) 1337–1342
1341
Example 3. For β ∈ R, p > 0, 0 < q < 1, and σ 2 = − ln q, the positive function xβ (−pq1/2 x−1 ; q)∞ dσ (x) satisfies the functional equation (1). The last remark, (8) and (14) imply that ∞
Z
1 xβ (−pq1/2 x−1 ; q)∞ dσ (x)ψ(logq x)dx = (pqβ ; q)− ∞
∞
Z
xβ dσ (x)ψ(logq x)dx,
0
0
=
2 q−β /2
1
Z
(pqβ ; q)∞
σ −2 ψ(x − β)θ x, dx, 2
0
(15)
whenever ψ ∈ L1 (0, 1) is a 1-periodic function. In particular, we have that ∞
Z 0
−1 2 xβ (−pq1/2 x−1 ; q)∞ dσ (x)dx = q−β /2 pqβ ; q ∞ .
(16)
Remark 3. In q-calculus a function f defined on R+ is q-periodic if f (qx) = f (x) for all x > 0. If f is a q-periodic function defined on R+ then ψ(x) = f (qx ) is a 1-periodic function (in the usual sense) on R. So, we have that f is a q-periodic function if and only if there exists a 1-periodic function ψ such that f (x) = ψ(logq x). Remark 4. If f , g : R+ → R satisfy the functional equation (1) and g is a positive function then f /g (x) = f /g (qx), therefore f /g is a q-periodic function. By the previous remark there exists a 1-periodic function ψ such that f /g (x) = ψ(logq x). In particular, we can write f (x) = xβ (−pq1/2 x−1 ; q)∞ dσ (x)ψ(logq x), where ψ is a 1-periodic function. This observation was made in the literature before (e.g Chihara (1979, 1970) and Christiansen (2003a,b). 3. Proof of the main theorem Proof of Theorem 1. If f ∈ L1 R+ satisfies the functional equation (1), then there exists a 1-periodic function ψ such that f (x) = xβ (−pq1/2 x−1 ; q)∞ dσ (x)ψ(logq x), x > 0. The other part of the hypothesis and (15) imply that
2 q−β /2
pqβ ; q
= s0 (f ) =
∞
2 q−β /2
pqβ ; q
1
Z
∞
ψ(x − β)θ (x, 2−1 σ −2 )dx. 0
By (7) we have that ϕ(x) = −1 + ψ(x − β) is a 1 -periodic, integrable function satisfying (2). On the other hand, the function f (x) = xβ (−pq1/2 x−1 ; q)∞ dσ (x){1 + ϕ(β + logq x)} satisfies (1) if ϕ ∈ L1 (0, 1) is a 1-periodic function. From (15) with ψ (x) = ϕ(x + β), (16), and (2) we have that
R∞ 0
−1 2 f (x)dx = q−β /2 pqβ , q ∞ .
Another classes of solutions to the generalized Stieltjes–Wigert moment problem. Proposition 6. If h ∈ L1 (R) is bounded below (above) and
Z
K x, 2−1 σ −2 h (x + n) dx = 0,
∀n ∈ Z,
(17)
R
then the function f (x) = xβ (−pq1/2 x−1 ; q)∞ dσ (x) 1 + h(β + logq x) is a solution to the Stieltjes–Wigert moment problem. If h is not a 1-periodic function then f does not satisfy the functional equation (1).
Proof. As in (8), with β + n and h (· + β) instead of β and ψ , we have that (17) is equivalent to ∞
Z
xn+β dσ (x) h(β + logq x)dx = 0,
∀n ∈ Z.
0
Next, we use the well known identity (e.g. Gasper and Rahman (1990, p. 236)),
∞ X q k=0
k 2
zk
(q; q)k
= (−z ; q)∞ ,
z ∈ C,
the fact that xn+β (−pq1/2 x−1 ; q)∞ dσ (x) ∈ L1 R+ , n ∈ N, and the Monotone Convergence Theorem to get that
∞
Z
xn+β dσ (x) −pq1/2 x−1 ; q ∞ h(β + logq x)dx =
0
Z 2 ∞ X qk /2 pk k=0
(q; q)k
for all n ∈ N. The result follows from Example 3 and Theorem 1. For f ∈ L1 (R), we define its Fourier transform as
(Φ f )(ξ ) =
Z
∞
dx f (x)e−ixξ √ . 2π −∞
0
∞
xn−k+β dσ (x) h(β + logq x)dx = 0,
1342
M. López-García / Statistics and Probability Letters 79 (2009) 1337–1342
The following examples provide functions satisfying (17) which are not 1-periodic functions. Example 4. López-García (in press, Theorem 3) proved for h ∈ L2 (R) that the condition (17) is equivalent to
X
K (ξ + 2nπ , 2−1 σ 2 )(Φ h)(ξ + 2nπ ) = 0,
a.e. ξ ∈ R.
(18)
n∈Z
In particular, if γ ∈ L2 (R) has compact support and satisfies
X
γ (ξ + 2nπ ) = 0,
a.e. ξ ∈ R,
(19)
n∈Z
then the function h ∈ L2 (R) given by
Φ h = γ /K (·, 2−1 σ 2 ) satisfies (18). For instance, consider φ ∈ C ∞ (R) with supp φ ⊂ [0, 2π ], then γ (ξ ) = φ(ξ ) − φ(ξ − 2π ) fulfills (19), and the corresponding function h is a rapidly decreasing function. Example 5. Let α ∈ R \ 2π Z, λ ∈ C and g a 1-periodic function in L1 (0, 1). López-García (in press) proved that the function (λ + g (x))eiαx is not 1-periodic and fulfills (17) provided that
√
2π σ 2 K (α, 2−1 σ 2 )λ = −
Z
K (x, 2−1 σ −2 )g (x)eiα x dx. R
For instance, with g (x) = cos(2π x) we have that Re((λ + g (x))eiα x ) = (λ + cos(2π x)) cos(α x) satisfies (17) whenever
√
2π σ
2K
(α, 2 σ )λ = − −1
2
Z
K (x, 2−1 σ −2 ) cos(2π x) cos(α x)dx. R
Remark 5. For a density f defined on R+ and h ∈ L∞ (R+ ), khk∞ = 1, Stoyanov (2004) introduced the Stieltjes class with center f and perturbation h, given as follows S(f , h) = {f (x)(1 + ε h(x)) : x > 0, ε ∈ [−1, 1]},
R∞
provided that 0 xn f (x)h(x)dx = 0 for all n ∈ N. So, we have shown that any function satisfying the functional equation (1) belongs to the (generalized) Stieltjes class with center xβ (−pq1/2 x−1 ; q)∞ dσ (x). Pakes (2007) extended the results of Stoyanov for general distributions supported on R+ . Stoyanov and Tolmatz (2004, 2005) have found more examples of Stieltjes classes. Acknowledgement I thank the anonymous referee for carefully reading the original manuscript. References Berg, C., 1998. From discrete to absolutely continuous solutions of indeterminate moment problems. Arab. J. Math. Sci. 4, 1–18. Cannon, J.R., 1984. The One-Dimensional Heat Equation. Addison-Wesley, New York. Chihara, T.S., 1970. A characterization and a class of distribution functions for the Stieltjes–Wigert polynomials. Canad. Math. Bull. 13, 529–532. Chihara, T.S., 1979. On generalized Stieltjes–Wigert and related orthogonal polynomials. J. Comput. Appl. Math. 5 (4), 291–297. Christiansen, J.S., 2003a. The moment problem associated with the Stieltjes–Wigert polynomials. J. Math. Anal. Appl. 277 (1), 218–245. Christiansen, J.S., 2003b. The moment problem associated with the q-Laguerre polynomials. Constr. Approx. 19, 1–22. Gasper, G., Rahman, M., 1990. Basic Hypergeometric Series. In: Encyclopedia of Mathematics and its Applications, vol. 35. Cambridge University Press, Cambridge, UK. Gómez, R., López-García, M., 2007. A family of heat functions as solutions of indeterminate moment problems. Int. J. Math. Math. Sci. 2007, 11. doi:10.1155/2007/41526. Article ID 41526. López-García, M., Characterization of solutions to the log-normal moment problem. Theory Probab. Appl. (in press). Stoyanov, J., 2004. Stieltjes classes for moment indeterminate probability distributions. J. Appl. Probab. 41A, 281–294. Stoyanov, J., Tolmatz, L., 2004. New Stieltjes classes involving generalized gamma distributions. Statist. Probab. Lett. 213–219. Stoyanov, J., Tolmatz, L., 2005. Method for constructing Stieltjes classes for M-indeterminate probability distributions. Appl. Math. Comput. 165, 669–685. Pakes, A., 2007. Structure of Stieltjes classes of moment-equivalent probability laws. J. Math. Anal. Appl. 326, 1268–1290.