Dilation-and-modulation systems on the half real line
Li and Zhang Journal of Inequalities and Applications
Dilation-and-modulation systems on the half real line
Yun-Zhang Li
Wei Zhang
Translation, dilation, and modulation are fundamental operations in wavelet analysis. Affine frames based on translation-and-dilation operation and Gabor frames based on translation-and-modulation operation have been extensively studied and seen great achievements. But dilation-and-modulation frames have not. This paper addresses a class of dilation-and-modulation systems in L2(R+). We characterize frames, dual frames, and Parseval frames in L2(R+) generated by such systems. Interestingly, it turns out that, for such systems, Parseval frames, orthonormal bases, and orthonormal systems are mutually equivalent to each other, while this is not the case for affine systems and Gabor systems.
dilation-and-modulation system; frame; dual frame
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1 Introduction
Before proceeding, we recall some notions and notations. An at most countable sequence
{ei}i∈I in a separable Hilbert space H is called a frame for H if there exist < C ≤ C < ∞
such that
C f ≤
f , ei
≤ C f
for f ∈ H,
i∈I
where C, C are called frame bounds; it is called a Bessel sequence in H if the right-hand
side inequality in (.) holds, where C is called a Bessel bound. In particular, {ei}i∈I is called
a Parseval frame if C = C = in (.). Given a frame {ei}i∈I for H, a sequence {e˜i}i∈I is
called a dual of {ei}i∈I if it is a frame such that
f =
f , e˜i ei for f ∈ H.
i∈I
It is easy to check that {ei}i∈I is also a dual of {e˜i}i∈I if {e˜i}i∈I is a dual of {ei}i∈I . So, in this
case, we say {ei}i∈I and {e˜i}i∈I form a pair of dual frames for H. It is well known that {ei}i∈I
and {e˜i}i∈I form a pair of dual frames for H if they are Bessel sequences and satisfy (.).
The fundamentals of frames can be found in [–]. The Fourier transform of c ∈ l(Z) is
defined by cˆ(·) =
m∈Z c(m)e–πim·. For two sequences c and d on Z, the convolution c ∗ d
if it is well defined. The Kronecker delta is defined by δn,m = iiff nn == mm;. l(Z) denotes the
set of finitely supported sequences on Z. We denote by I the identity operator on l(Z),
and by χE its characteristic function for a set E. Write R+ = (, ∞). For a positive number
a > , a function h defined on R+ is said to be a-dilation periodic if h(a·) = h(·) on R+. For
a function f defined on [, a), we define the function f˜ on R+ by
f˜(·) = f a–l·
Z
on al, al+ for l ∈ ,
which is called the a-dilation periodization of f . Obviously, it is a-dilation periodic.
The translation operator Tx , the modulation operator Mx with x ∈ R, and the dilation
Dc with c > are, respectively, defined by
and
Tx f (·) = f (· – x),
Mx f (·) = eπix·f (·),
Dcf (·) = √cf (c·)
for f ∈ L(R). They are the basis of wavelet analysis. Affine systems of the form {Daj Tbkψ :
j, k ∈ Z} with ψ ∈ L(R) and a, b > , and Gabor systems of the form {EmbTnag : m, n ∈
Z} with g ∈ L(R) and a, b > have been extensively studied. However,
dilation-andmodulation systems of the form
{MmbDaj ψ : m, j ∈ Z}
with a, b >
{ψmDaj ψ : m, j ∈ Z}
with a > ,
where
ψm(·) = √
H(R) = f ∈ L(R) : fˆ(·) = a.e. on (–∞, ) ,
where the Fourier transform is defined by
fˆ(·) =
R
f (x)e–πix· dx for f ∈ L(R) ∩ L(R)
and extended to L(R) by the Plancherel theorem. Wavelet frames in H(R) of the form
{Daj Tmϕ : j, m ∈ Z} were studied in [, ], and some variations can be found in [–
]. By the Plancherel theorem, an H(R)-frame {Dj Tmϕ : j, m ∈ Z} leads to a
L(R+)frame
e–πi–jm·ϕˆ –j· : j, m ∈ Z .
In (.), e–πi–jm· is jZ-periodic with respect to additive operation, and the period
depends on the dilation factor j. However, ψm in (.) is a-dilation periodic and unrelated
to j. Therefore, frames of the form (.) are different from ones of the form (.) for L(R+)
and of independent interest. They are related to a kind of function-valued frames in [].
In [], numerical experiments were made to establish that the nonnegative integer shifts
of the Gaussian function formed a Riesz sequence in L(R+). In [], a sufficient condition
was obtained to determine whether the nonnegative translates form a Riesz sequence on
L(R+).
The rest of this paper is organized as follows. Section is devoted to characterizing
frames and dual frames for L(R+) with the structure of (.). Section is devoted to
Parseval frames and orthonormal bases for L(R+) of the form (.). It turns out that Parseval
frames, orthonormal bases, and orthonormal systems in L(R+) of the form (.) are
mutually equivalent to each other. It is worth noting that neither affine systems nor Gabor
systems have such a property.
2 Frame and dual frame characterization
This section characterizes L(R+)-frames and dual frames of the form (.). For this
purpose, we need some notations and lemmas. For f ∈ L(R+), we define
and
G(f , ·) = Daj+l f (·) j,l∈Z
F(·) = Dal f (·) l∈Z.
Lemma . For m, j ∈ Z, define ψm as in (.), (...truncated)