PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 86(100) (2009), 115–122 DOI: 10.2298/PIM0900115S

APPLICATION OF THE QUASIASYMPTOTIC BOUNDEDNESS OF DISTRIBUTIONS ON WAVELET TRANSFORM

Katerina Saneva

Communicated by Stevan Pilipović

Abstract. We analyze the boundedness of the wavelet transform W_{g}f of the quasiasymptotically bounded distribution f. Assuming that the distribution f ∈ S′(ℝ) is quasiasymptotically or r-quasiasymptotically bounded at a point or at infinity related to a continuous and positive function, we obtain results for the localization of its wavelet transform.

1. Introduction

Dependence of the localization properties of the continuous wavelet transform W_{g}f from the localization of the analyzing function f ∈ L^{2}(ℝ) and the wavelet g ∈ L^{2}(ℝ) in time and frequency space, as well as the opposite dependence is analyzed by Holschneider [2]. Working in the Fourier space, Pathak proved some Abelian theorems for the behaviour of the wavelet transform of L^{2} functions and tempered distributions [3]. Contrary to the approaches in [2, 3], that are based on classical estimations, we use the theory of asymptotic behaviour of distributions to the asymptotic analysis of the continuous wavelet transform. Several Abelian and Tauberian theorems for the wavelet transform are proved in [5, 6, 7] using the quasiasymptotics and the S-asymptotics of distributions. We refer to [12, 4, 1, 9, 10, 11] and references therein for the definitions, properties and application of these kinds of asymptotics of distributions.

In this article, we analyze the boundedness of the wavelet transform W_{g}f of the quasiasymptotically bounded distribution f. Assuming that the distribution f ∈ S′(ℝ) is quasiasymptotically bounded at 0 or infinity (respectively, at b_{0} ∈ ℝ) related to a continuous and positive function, we obtain novel results for the localization of its wavelet transform W_{g}f(b, a) (respectively, W_{g}f(b_{0}, a)), Theorem 3.1 and Theorem 3.2. Additionally, we define the notion of r-quasiasymptotic boundedness of distributions from the space S̃′_{r}(ℝ), r < 0, and assuming that f ∈ S̃′_{r}(ℝ) is r-quasiasymptotically bounded we prove the results for the boundedness of its wavelet transform, Theorem 3.3 and Theorem 3.4.

2000 Mathematics Subject Classification: 46F12; 42C40.

Key words and phrases: Wavelet transform, tempered distributions, quasiasymptotic boundedness.

2. Preliminaries and notations

The domain of functions considered in this article is ℝ. Therefore, we omit the suffix and write C^{∞} instead of C^{∞} (ℝ), S instead S(ℝ), and so on. The space of infinitely differentiable functions (smooth functions) is denoted by C^{∞} . The space of rapidly decreasing smooth functions defined on the real line, supplied with the usual topology is denoted by S. Its strong dual is the well known space of tempered distributions S′. We refer to [8, 13] for the properties of these spaces.

Let S̃_{r}, r < 0 be the space of functions φ ∈ C^{∞} for which all norms

(2.1)  ∥φ∥_{r,p} = sup_{x∈ℝ, α⩽p} (1 + |x|^{2})^{r/2} |φ^{(α)} (x)|, p = 0, 1, 2, …

are finite. The topology of a projective limit of the decreasing sequence of spaces

S̃_{r, 0} ⊃ S̃_{r,1} ⊃ ⋯⊃ S̃_{r,p} ⊃ ⋯,

where S̃_{r,p} is the completion of S in the norm ∥ · ∥_{r,p} defined by (2.1), is introduced in S̃_{r}. Each embedding S̃_{r,p+1} ⊂ S̃_{r,p} , p = 0, 1, 2, … is continuous.

The strong dual of S̃_{r}, r < 0 is denoted by S̃′_{r}. It is the inductive limit of the increasing sequence of spaces

S̃′_{r, 0} ⊂ S̃′_{r,1} ⊂ ⋯⊂ S̃′_{r,p} ⊂ ⋯,

where S̃′_{r,p} is the dual to the space S̃_{r,p} , p = 0, 1, 2, . . ..

We also need the definition of the spaces of highly localized function over the real line which are introduced in [2]. By S_{+} is denoted the set of functions s for which supp ŝ ⊂ [0, ∞) and for every localized exponent α > 0

∥s∥_{α} = sup_{x∈ℝ}k_{α}^{−1} (x)|s(x)| + sup_{ω⩾0}ϕ_{α}^{−1} (ω)|ŝ(ω)| < ∞,

where k_{α} (x) = (1+|x|^{2})^{−α/2} , ϕ_{α} (ω) = ω^{α} (1+ω)^{−α−1} and ŝ is the Fourier transform of s. The image of S_{+} under the parity operator is denoted by S_{−} , that is s ∈ S_{−} if and only if s(−x) ∈ S_{+} . The direct sum of S_{+} and S_{−} is denoted by S_{0}: S_{0} = S_{+} ⊕S_{−} . It is proved that the spaces S_{+} , S_{−} and S_{0} are closed subspaces of S; S_{+} (respectively, S_{−} ) consists of those functions in S whose Fourier transforms are supported by the positive (respectively, negative) frequencies only, and S_{0} consists of functions from S for which all the moments vanish [2, Theorem 19.1.3].

We refer to [2] for the definition and properties of the wavelet transform over the spaces L^{2} and S′. The wavelet transform of f ∈ L^{2} with respect to the wavelet g ∈ L^{2} is defined by

W_{g}f(b, a) := ∫_{−∞}^{+∞} f(t) \frac{1}{a} ḡ( \frac{t − b}{a} ) dt, b ∈ ℝ, a > 0 .

APPLICATION OF THE QUASIASYMPTOTIC BOUNDEDNESS OF DISTRIBUTIONS 117

The function g is usually called mother wavelet or analyzing wavelet. The functions g_{b,a}(·) = \frac{1}{a}g( \frac{·−b}{a} ), b ∈ ℝ, a > 0 which are obtained from the wavelet g by the operations of dilation and translation are called wavelets. The wavelet transform is a continuous linear transform from L^{2}(ℝ) into the space

L^{2}( ℝ^{2}, \frac{da db}{a^{2}}) = \{ F (b, a) : ∫_{0}^{+∞} \frac{da}{a^{2}} ∫_{−∞}^{+∞} |F (b, a)|^{2}db< ∞\} ,

which is a Hilbert space with the inner product

⟨F, G⟩ = ∫_{0}^{+∞} \frac{da}{a^{2}} ∫_{−∞}^{+∞} F (b, a) Ḡ(b, a) db, F, G ∈ L^{2}( ℝ^{2}, \frac{da db}{a^{2}}) .

The wavelet transform of the distribution f ∈ S′ with respect to the wavelet g ∈ S_{0} is the C^{∞} function over {(b, a) | b ∈ ℝ, a > 0} given by

(2.2)  W_{g}f(b, a) := ⟨f(t), ḡ_{b,a}(t)⟩, t ∈ ℝ

(see [2, Chapter 1, Section 25]). For every a > 0, W_{g}f(·, a) is a smooth function of polynomial growth since W_{g}f(b, a) = (f ∗ ˘ ḡ_{a})(b), b ∈ ℝ, where ˘ ḡ_{a}(t) = ḡ(−t/a)/a, t ∈ ℝ is a rapidly decreasing function.

The wavelet transform of the distribution f ∈ S̃′_{r}, r < 0 with respect to the wavelet g ∈ S_{0} is also given by formula (2.2).

We will give a definition of quasiasymptotic boundedness of distributions from S′ at a point and at infinity.

Definition 2.1. Let f ∈ S′and c(ε), ε ∈ (0, a) (respectively, c(k), k ∈ (a, ∞)),

a > 0 be a continuous positive function. We say that f is quasiasymptotically bounded at x_{0} (respectively, at infinity) related to c(ε) (respectively, c(k)) if there exist p ∈ ℕ_{0} and M > 0 such that

|⟨ \frac{f(x_{0} + εx)}{c(ε)} , φ(x) ⟩| ⩽ M∥φ∥_{p}, 0 < ε < 1
( respectively, |⟨ \frac{f(kx)}{c(k)} , φ(x) ⟩| ⩽ M∥φ∥_{p}, k > 1) ,

for every φ ∈ S.

We will also define quasiasymptotic boundedness of distributions from the space S̃′_{r}, r < 0, which we will call r-quasiasymptotic boundedness.

Definition 2.2. Let f ∈ S̃′_{r}, r < 0 and c(ε), ε ∈ (0, a) (respectively, c(k), k ∈

(a, ∞)), a > 0 be a continuous positive function. We say that f is r-quasiasympto-tically bounded at x_{0} (respectively, at infinity) related to c(ε) (respectively, c(k)) if there exist p ∈ ℕ_{0} and M > 0 such that

|⟨ \frac{f(x_{0} + εx)}{c(ε)} , φ(x) ⟩| ⩽ M∥φ∥_{r,p} , 0 < ε < 1
( respectively, |⟨ \frac{f(kx)}{c(k)} , φ(x)⟩| ⩽ M∥φ∥_{r,p} , k > 1) ,

for every φ ∈ S̃_{r}.

3. Main results

In the following theorems we consider the boundedness of the wavelet transform W_{g}f(b, a) (respectively, W_{g}f(b_{0}, a)) assuming that distribution f ∈ S′ is quasiasymptotically bounded at 0 (respectively, at b_{0}) related to the continuous and positive function.

Theorem 3.1. Let f ∈ S′ and c(ε), 0 < ε < ε′ be a continuous positive function. If f is quasiasymptotically bounded at 0 related to c(ε), then there exist p ∈ ℕ_{0} and C > 0 such that

|W_{g}f(b, a)| ⩽ \frac{C}{a}( 1 + \frac{1}{a})^{p}(1 + |b|)^{p}, b ∈ ℝ, 0 < a < 1,

for every wavelet g ∈ S_{0}.

Proof. From the definition of the distributional wavelet transform we have

| \frac{W_{g}f(εb, εa)}{c(ε)}| = |⟨ \frac{f(x)}{c(ε)} , \frac{1}{εa} ḡ( \frac{x − εb}{εa} ) ⟩|.

After the change of variables x = εt we obtain

| \frac{W_{g}f(εb, εa)}{c(ε)}| = |⟨ \frac{f(εt)}{c(ε)} , \frac{1}{a} ḡ( \frac{t − b}{a} ) ⟩|.

Since f is quasiasymptotically bounded at 0 related to c(ε), and g ∈ S it follows that there exist p ∈ ℕ_{0} and M_{1} > 0 such that

(3.1)  | \frac{W_{g}f(εb, εa)}{c(ε)}| ⩽ M_{1} ∥ \frac{1}{a} ḡ( \frac{x − b}{a} ) ∥_{p},

for every 0 < ε < 1. In the following, we will use the condition 0 < a < 1, as well as the following elementary inequalities

1 + |x + y| ⩽ (1 + |x|)(1 + |y|), 1 + |xy| ⩽ (1 + |x|)(1 + |y|).
(3.2) ∥ \frac{1}{a} ḡ( \frac{x − b}{a} ) ∥_{p} = sup_{x∈ℝ, α⩽p} (1 + |x|^{2})^{p/2} |( \frac{1}{a} ḡ( \frac{x − b}{a} ) )^{(α)} |

= sup_{x∈ℝ, α⩽p} (1 + |x|^{2})^{p/2} \frac{1}{a^{α+1}} |ḡ^{(α)} ( \frac{x − b}{a} ) |

⩽ \frac{M_{2}}{a^{p+1}} sup_{x∈ℝ}(1 + |x|^{2})^{p/2}( 1 + | \frac{x − b}{a} |)^{−p}

⩽ \frac{M_{2}}{a^{p+1}} sup_{x∈ℝ}((1 + |x|)^{2})^{p/2}( 1 + | \frac{x − b}{a} |)^{−p}

= \frac{M_{2}}{a^{p+1}} sup_{x∈ℝ}(1 + |x|)^{p}( 1 + | \frac{x − b}{a} |)^{−p}

⩽ \frac{M_{2}}{a^{p+1}} sup_{x∈ℝ}(1 + |x − b|)^{p}(1 + |b|)^{p}( 1 + | \frac{x − b}{a} |)^{−p}

= \frac{M_{2}}{a^{p+1}} (1 + |b|)^{p} sup_{x∈ℝ}( 1 + a| \frac{x − b}{a} |)^{p}( 1 + | \frac{x − b}{a} |)^{−p}

⩽ \frac{M_{2}}{a^{p+1}} (1 + |b|)^{p} sup_{x∈ℝ}( 1 + | \frac{x − b}{a} |)^{p}(1 + a)^{p}( 1 + | \frac{x − b}{a} |)^{−p}
= M_{2} \frac{(1 + a)^{p}}{a^{p+1}} (1 + |b|)^{p},

where M_{2} is a positive constant.

From (3.1) and (3.2) it follows the estimation

(3.3)  | \frac{W_{g}f(εb, εa)}{c(ε)}| ⩽ M_{1}M_{2} \frac{(1 + a)^{p}}{a^{p+1}} (1 + |b|)^{p},

for 0 < ε < 1, 0 < a < 1 and b ∈ ℝ. We choose ε_{0} such that 0 < ε_{0} < 1 and put ε_{0}b = s and ε_{0}a = m (0 < m < 1) in (3.3). So, we obtain

|W_{g}f(s, m) | ⩽ \frac{M_{1}M_{2}}{ε_{0}^{p−1} m^{p+1}} (ε_{0} + m)^{p}(ε_{0} + |s|)^{p}c(ε_{0})

⩽ M_{1}M_{2} \frac{(1 + m)^{p}}{m^{p+1}} (1 + |s|)^{p} \frac{(1 + ε_{0})^{p}}{ε_{0}^{p−1}} c(ε_{0}).

Since c(ε) is a positive function it follows that there exists constant C > 0 such that

|W_{g}f(s, m)| ⩽ C \frac{(1 + m)^{p}}{m^{p+1}} (1 + |s|)^{p}, s ∈ ℝ, 0 < m < 1.  □

The proof of the next theorem is similar to the proof of Theorem 3.1.

Theorem 3.2. Let f ∈ S′ and c(ε), 0 < ε < ε′ be a continuous positive function. If f is quasiasymptotically bounded at b_{0}, b_{0} ∈ ℝ related to c(ε), then there exist p ∈ ℕ_{0} and C > 0 such that

|W_{g}f(b_{0}, a)| ⩽ C \frac{(1 + a^{2})^{p/2}}{a^{p+1}} , 0 < a < 1,

for every wavelet g ∈ S_{0}.

Remark 3.1. An analogous result could be obtained for the boundedeness of the wavelet transform W_{g}f(b, a) in the case when distribution f ∈ S′ is quasiasymptotically bounded at infinity.

Theorem 3.3. Let f ∈ S̃′_{r}, r < 0 and c(ε), 0 < ε < ε′ be a continuous positive

function. If f is r-quasiasymptotically bounded at 0 with respect to c(ε), then there exist p ∈ ℕ_{0} and C > 0 such that

|W_{g}f(b, a)| ⩽ C \frac{(1 + a)^{q}}{a^{p+1}} \frac{1}{|b|^{q/2}}, q = −r > 0, b ∈ ℝ, 0 < a < 1,

for every wavelet g ∈ S_{0}.

Proof. As in the proof of Theorem 3.1 we have

| \frac{W_{g}f(εb, εa)}{c(ε)}| = |⟨ \frac{f(x)}{c(ε)} , \frac{1}{εa} ḡ( \frac{x − εb}{εa} ) ⟩| = |⟨ \frac{f(εt)}{c(ε)} , \frac{1}{a} ḡ( \frac{t − b}{a} ) ⟩|.

Since f is r-quasiasymptotically bounded at 0 related to c(ε), and g ∈ S_{0} it follows that there exist p ∈ ℕ_{0} and M_{1} > 0 such that

(3.4)  | \frac{W_{g}f(εb, εa)}{c(ε)}| ⩽ M_{1} ∥ \frac{1}{a} ḡ( \frac{x − b}{a} ) ∥_{r,p}

for every 0 < ε < 1. Since 0 < a < 1 we have

∥ \frac{1}{a} ḡ( \frac{x − b}{a} ) ∥_{r,p} = sup_{x∈ℝ, α⩽p} (1 + |x|^{2})^{r/2} |( \frac{1}{a} ḡ( \frac{x − b}{a} ) )^{(α)} |

= sup_{x∈ℝ, α⩽p} (1 + |x|^{2})^{r/2} \frac{1}{a^{α+1}} | ḡ^{(α)} ( \frac{x − b}{a} ) |

⩽ \frac{1}{a^{p+1}} sup_{x∈ℝ, α⩽p} (1 + |x|^{2})^{r/2} | ḡ^{(α)} ( \frac{x − b}{a} ) |.

We put r = −q, q > 0 and get

(3.5)  ∥ \frac{1}{a} ḡ( \frac{x − b}{a} ) ∥_{r,p} ⩽ \frac{M_{2}}{a^{p+1}} sup_{x∈ℝ} \frac{1}{(1 + |x|^{2})^{q/2}} ( 1 + | \frac{x − b}{a} |)^{−q} ,

where M_{2} is a positive constant.

Since it holds

(1 + |b|)^{q} ⩽ (1 + |b − x| + |x|)^{q} ⩽ (1 + |x − b|)^{q} (1 + |x|)^{q}

= ( 1 + a | \frac{x − b}{a} |)^{q} (1 + |x|)^{q} ⩽ ( 1 + | \frac{x − b}{a} |)^{q} (1 + a)^{q}(1 + |x|)^{q},

we have

(3.6)  ( 1 + | \frac{x − b}{a} |)^{−q} ⩽ \frac{(1 + a)^{q}(1 + |x|)^{q}}{(1 + |b|)^{q}} .

From the inequalities (3.4), (3.5) and (3.6) we obtain

| \frac{W_{g}f(εb, εa)}{c(ε)}| ⩽ M_{1}M_{2} \frac{(1 + a)^{q}}{a^{p+1}} \frac{1}{(1 + |b|)^{q}} sup_{x∈ℝ} \frac{((1 + |x|)^{2})^{q/2}}{(1 + |x|^{2})^{q/2}}

= M_{1}M_{2} \frac{(1 + a)^{q}}{a^{p+1}} \frac{1}{(1 + |b|)^{q}} sup_{x∈ℝ}( 1 + \frac{2|x|}{1 + |x|^{2}} )^{q/2}

⩽ M_{1}M_{2}2^{q/2} \frac{(1 + a)^{q}}{a^{p+1}} \frac{1}{(1 + |b|)^{q}} .

If we choose 0 < ε_{0} < 1 and put ε_{0}b = s and ε_{0}a = m (0 < m < 1) in the above inequality, we get

| \frac{W_{g}f(s, m)}{c(ε_{0})} | ⩽ M_{1}M_{2}2^{q/2} \frac{ε_{0}^{p+1}}{m^{p+1}} \frac{(ε_{0} + m)^{q}}{(ε_{0} + |s|)^{q}} .

Since c(ε) is a positive function and from the following inequalities

(ε_{0} + m)^{q} ⩽ (1 + ε_{0})^{q}(1 + m)^{q}, (ε_{0} + |s|)^{q} = ((ε_{0} + |s|)^{2})^{q/2} ⩾ (2ε_{0}|s|)^{q/2}, we have

|W_{g}f(s, m)| ⩽ M_{1}M_{2}2^{q/2} \frac{ε_{0}^{p+1} (1 + ε_{0})^{q}}{2^{q/2}ε_{0}^{q/2}} c(ε_{0}) \frac{(1 + m)^{q}}{m^{p+1} |s|^{q/2}}

= C \frac{(1 + m)^{q}}{m^{p+1}} \frac{1}{|s|^{q/2}} ,  □

where s ∈ ℝ, 0 < m < 1 and C is a positive constant.

The result for the boundedness of the wavelet transform W_{g}f(b_{0}, a) for fixed b_{0} ∈ ℝ when f is r-quasiasymptotically bounded distribution at the point b_{0} is given in the following theorem.

Theorem 3.4. Let f ∈ S̃′_{r}, r < 0 and c(ε), 0 < ε < ε′ be a continuous positive function. If f is r-quasiasymptotically bounded at b_{0}, b_{0} ∈ ℝ related to c(ε), then there exist p ∈ ℕ_{0} and C > 0 such that

|W_{g}f(b_{0}, a)| ⩽ C \frac{(1 + a^{2})^{q/2}}{a^{p+1}} , q = −r > 0, 0 < a < 1,

for every wavelet g ∈ S_{0}.

Remark 3.2. An analogous result could be obtained assuming that f ∈ S̃′_{r}, r < 0 is r-quasiasymptotically bounded at infinity.

Remark 3.3. It is also possible to obtain an analogous result for the localization of the wavelet transform W_{g}f, assuming that the Fourier transform of distribution f ∈ S′ or f ∈ S̃′_{r}, r < 0 is quasiasymptotically or r-quasiasymptotically bounded, respectively.

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Department of Mathematics and Physics (Received 26 11 2008) Faculty of Electrical Engineering and Information Technologies (Revised 24 02 2009) University ‘Ss. Cyril and Methodius’ Skopje Macedonia

saneva@feit.ukim.edu.mk