Transform Methods & Special Functions, Varna’96

Proceedings of Second International Workshop, 23 - 30 August 1996

FRACTIONAL INTEGRALS AND WAVELET TRANSFORMS

Boris Rubin^{∗}, Dmitri Ryabogin and Eliyahu Shamir

Abstract

Wavelet type representations of fractional integrals and derivatives are studied in the framework of L^{p}-spaces. These representations generalize the notion of Marchaud’s fractional derivative and are intimately connected with the Calderón reproducing formula. By choosing a relevant “wavelet” measure we give a unified representation of the following basic objects in fractional calculus on the real line: the Riemann-Liouville fractional integrals, the Riesz potentials, the conjugate Riesz potentials, the inverses and linear combinations of these operators.

Mathematics Subject Classification: 26A33

Key Words and Phrases: Fractional integrals and derivatives, continuous wavelet transforms.

Introduction

In the present article we announce some results related to representations of various fractional integrals and derivatives in the form

A^{α}f = 1 ; c_{ν}(α) ∫_{0}^{∞} f ∗ ν_{t} ; t^{1−α} dt = lim_{ε→0}_{ρ→∞} 1 ; c_{ν}(α) ∫_{ε}^{ρ} f ∗ ν_{t} ; t^{1−α} dt, α ∈ ℂ,  (0.1)

where ν_{t} is a suitable dilated measure (or distribution), c_{ν}(α) is a normalizing factor and the limit is understood in the L^{p}-norm or in the “almost everywhere”- sense. In the case α = 0, A^{α} coincides with the identity operator or with the

∗ Partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).

Typeset by AMS-TEX

Hilbert transform (depending on ν) and is connected with Calderón’s reproducing formula

f = 1 ; c_{u,v} ∫_{0}^{∞} f ∗ u_{t} ∗ v_{t} ; t dt,

(see e.g. [ 1 ], [ 4 ], [ 6 ], [ 7 ] and references therein). Various fractional integrals (and derivatives) can be written in the form (0.1). For example, by choosing ν = δ_{1}, (the Dirac unit mass at the point x = 1) and c_{ν}(α) = Γ(α) we obtain the well known Liouville fractional integral

A^{α}f = 1 ; Γ(α) ∫_{0}^{∞} f(x − t) ; t^{1−α}dt = 1 ; Γ(α) ∫_{−∞}^{x} f(y)dy ; (x − y)^{1−α} .

By putting ν = ∑_{j=0}^{ℓ} \binom{ℓ}{j}(−1)^{j}δ_{j}, Re α > −ℓ, with the unit Dirac masses at the

points j = 0, 1, ⋯, ℓ, one can readily see that A^{α}f is the Marchaud fractional derivative of f of order −α (up to a constant factor, cf. [ 5 ], [ 9 ]). By choosing different ν, this list of examples can be continued.

Our goal is to show that all basic operators in fractional calculus can be represented in the form (0.1). We also answer the following question: what classes of measures (or distributions) generate concrete types of fractional integrals and derivatives. Clearly, for Re α ≤ 0 the measure ν must enjoy some cancellation properties or, in other words, ν should be a “wavelet measure”. Owing to this phenomenon, the representations (0.1) may be called wavelet type representations of fractional integrals (or derivatives).

The proofs of the statements presented below can be found in [ 8 ].

Notation and Definitions

Below ℤ, ℕ, ℝ and ℂ denote the set of all integers, positive integers, real numbers and complex numbers respectively; [a] is the integer part of a ∈ ℝ. The notation C = C(ℝ), L^{p} = L^{p}(ℝ) for function spaces is standard; (f(x))_{±} = max{±f(x), 0}, Φ = Φ(ℝ) is the Lizorkin space of test functions, Φ′ = Φ′(ℝ) is the dual of Φ (see [ 3 ], [ 9 ]). For ω ∈ Φ and f ∈ Φ′ we denote

(f, ω) = ∫_{−∞}^{∞} f(x)ω(x)dx,

where the integral can be interpreted as the value of the distribution f at the test function ω(x). We write “f(x) = g(x) in the Φ′-sense” if (f, ω) = (g, ω) for all ω ∈ Φ. For g ∈ Φ′ and t ∈ ℝ the distribution g_{t} is defined by (g_{t}, ω) = (g(x), ω(tx)), ω ∈ Φ. In the following M denotes the set of all complex-valued finite Borel measures μ on the real line ℝ. For μ ∈ M the values μ({±∞}) are assumed to be zero.

(Hφ)(x) = p.v. 1 ; πi ∫_{−∞}^{∞} φ(y) ; x − y dy

is the Hilbert transform of φ. For φ ∈ L^{p}, 1 ≤ p ≤ ∞ and μ ∈ M

(φ ∗ μ_{t})(x) = ∫_{−∞}^{∞} φ(x − ty)dμ(y), t > 0.

In the following f̂ denotes the Fourier transform of the distribution f; f^{∨} is the inverse Fourier transform.

Definition 0.1. For ω ∈ Φ, α ∈ ℂ, the fractional integrals I_{±}^{α}ω, the fractional derivatives D_{±}^{α}ω, the Riesz potential I^{α}ω, the conjugate Riesz potential I_{s}^{α}ω, the Riesz fractional derivative D^{α}ω and the conjugate Riesz fractional derivative D_{s}^{α}ω are defined by:

(I_{±}^{α}ω)(x) = [(∓iξ)^{−α}ω̂(ξ)]^{∨}(x), (D_{±}^{α}ω)(x) = [(∓iξ)^{α}ω̂(ξ)]^{∨}(x), Re α ≥ 0,

\begin{aligned}(I^{α}ω)(x) & = (|ξ|^{−α}ω̂(ξ))^{∨}(x), (I_{s}^{α}ω)(x) = (|ξ|^{−α}sgnξ ω̂(ξ))^{∨}(x), Re α ≥ 0, \\ (D^{α}ω)(x) & = (|ξ|^{α}ω̂(ξ))^{∨}(x), (D_{s}^{α}ω)(x) = (|ξ|^{α}sgnξ ω̂(ξ))^{∨}(x), Re α ≥ 0. \end{aligned}

where (∓ix)^{−α} = e^{α log |x|∓(απi/2)sgnx}.

For the classical representations of these operators see [ 9 ].

Definition 0.2. For f ∈ Φ′ and α ∈ ℂ, the Φ′-distributions I_{±}^{α}f, D_{±}^{α}f, I^{α}f, D^{α}f, I_{s}^{α}f, D_{s}^{α}f are defined by duality as follows:

(I_{±}^{α}f, ω) = (f, I_{∓}^{α}ω̄), (D_{±}^{α}f, ω) = (f, D_{∓}^{α}ω̄), (I^{α}f, ω) = (f, I^{α}ω̄),

(D^{α}f, ω) = (f, D^{α}ω̄), (I_{s}^{α}f, ω) = (f, I_{s}^{α}ω̄), (D_{s}^{α}f, ω) = (f, D_{s}^{α}ω̄).

1. Basic relations involving Φ′-distributions

Given a Φ′-distribution ν and a complex number α, consider the formal integral

A^{α,ν} = ∫_{0}^{∞} ν_{t} ; t^{1−α} dt.  (1.1)

Our goal here is to give sense to this integral and to represent (1.1) as a linear combination of kernels arising in fractional calculus.

For α ∈ ℂ, consider distributions h_{±}^{α}, h^{α}, h_{s}^{α}, which have the following Fourier transforms in the Φ′-sense (see [ 9, p. 147 ], [ 2, p. 170 ]):

ĥ_{±}^{α}(ξ) = (∓iξ)^{−α} = |ξ|^{−α} exp (± απi ; 2sgn ξ), ĥ^{α}(ξ) = |ξ|^{−α}, ĥ_{s}^{α}(ξ) = sgn ξ|ξ|^{−α}.

Clearly, I_{±}^{α}f = h_{±}^{α} ∗ f, I^{α}f = h^{α} ∗ f, I_{s}^{α}f = h_{s}^{α} ∗ f for all α ∈ ℂ and f ∈ Φ′. Denote

A_{ε,ρ}^{α,ν} = ∫_{ε}^{ρ} ν_{t} ; t^{1−α} dt, 0 < ε < ρ < ∞.

Lemma 1.1. Let α ∈ ℂ, ν be a Φ′-distribution such that ν̂(ξ) ∈ L_{loc}(ℝ \ {0}).

Denote ν̂_{±}(ξ) = ν̂(ξ) ± ν̂(−ξ) ; 2 and let the integrals

a_{±} = ∫_{0}^{∞} ν̂_{±}(η) ; η^{1−α}dη

exist as the improper ones. Then A_{ε,ρ}^{α,ν} → a_{+}h^{α} + a_{−}h_{s}^{α} as ε → 0, ρ → ∞ in the Φ′-sense. If α ∉ ℤ , then this limit is also equal to c_{+}h_{+}^{α} + c_{−}h_{−}^{α} , where

c_{±} = 1 2[a_{+}/ cos(απ ; 2 ) ± a_{−}/ sin(απ ; 2 )].

Lemma 1.1 shows that the integral ∫_{0}^{∞}(ν_{t} ∗ f)(x)dt/t^{1−α} may be used for representation of the following operators:

(a) Riesz potentials and their inverses (a_{+} = 1, a_{−} = 0);

(b) conjugate Riesz potentials and their inverses (a_{+} = 0, a_{−} = 1);

(c) left-sided fractional integrals and derivatives (c_{+} = 1, c_{−} = 0);

(d) right-sided fractional integrals and derivatives (c_{+} = 0, c_{−} = 1);

(e) integrals and derivatives of integer order.

In a similar way one can represent compositions of mentioned operators with a Hilbert transform and linear combinations (with constant coefficients) of such operators.

2. Wavelet Type Representation of Fractional Derivatives (L^{p}-theory)

In this section we exhibit natural L^{p}-analogues of Lemma 1.1 and describe some classes of measures and distributions ν for which these analogues hold.

Theorem 2.1. Let Re α > 0, ν ∈ M. Assume that

∫_{−∞}^{∞} x^{j}dν(x) = 0 for all j = 0, 1, ⋯, [Re α] and ∫_{|x|>1} |x|^{β}d|ν|(x) < ∞

for some β > Re α. Let f ∈ L^{r}, 1 ≤ r < ∞, and one of the derivatives D_{±}^{α}f, D^{α}f, D_{s}^{α}f (in the Φ′-sense) belongs to L^{p}, 1 < p < ∞. Then the limit A^{α}f = lim_{ε→0} ∫_{ε}^{∞}(ν_{t} ∗ f)(x)dt/t^{1+α} exists in the L^{p}-norm and in the a.e. sense, and the following relations hold:

A^{α}f = γ_{+}D_{+}^{α}f + γ_{−}HD_{+}^{α}f = c_{+}D_{+}^{α}f + c_{−}D_{−}^{α}f = a_{+}D^{α}f + a_{−}D_{s}^{α}f

(in the second equality it is assumed α ∉ ℕ) where

γ_{+} = ∫_{−∞}^{∞} k^{+}(x)dx = \{ \begin{matrix}Γ(−α)[ ∫_{0}^{∞}x^{α}dν(x) + cos απ ∫_{−∞}^{0} |x|^{α}dν(x)], α ∉ ℕ, \\ (−1)^{ℓ} ; ℓ! ∫_{−∞}^{∞} x^{ℓ} log 1 ; |x|dν(x), α = ℓ ∈ ℕ, \end{matrix}

γ_{−} = −λ πi = − πi ; Γ(1 + α) ∫_{−∞}^{0} |x|^{α}dν(x).

c_{+} = Γ(−α) ∫_{0}^{∞}x^{α}dν(x), c_{−} = Γ(−α) ∫_{−∞}^{0} |x|^{α}dν(x).

For α ∉ ℕ,

a_{+} = Γ(−α) cos απ ; 2 ∫_{−∞}^{∞} |x|^{α}dν(x), a_{−} = −iΓ(−α) sin απ ; 2 ∫_{−∞}^{∞} |x|^{α}sgn x dν(x).

For α = ℓ ∈ ℕ,

a_{+} = \{ \begin{matrix}(−1)^{ℓ/2} ; ℓ! ∫_{−∞}^{∞} x^{ℓ} log 1 ; |x|dν(x), ℓ = 2k, k = 1, 2, ⋯, \\ π(−1)^{(ℓ−1)/2} ; ℓ! ∫_{−∞}^{0} x^{ℓ}dν(x), ℓ = 2k + 1, k = 0, 1, 2, ⋯; \end{matrix}

a_{−} = \{ \begin{matrix}i(−1)^{ℓ/2+1}π ; ℓ! ∫_{−∞}^{0} x^{ℓ}dν(x), ℓ = 2k, k = 1, 2, ⋯, \\ i(−1)^{(ℓ−1)/2} ; ℓ! ∫_{−∞}^{∞} x^{ℓ} log 1 ; |x|dν(x), ℓ = 2k + 1, k = 0, 1, 2, · · · . \end{matrix}

In the case of purely imaginary order operators D_{±}^{α}, D^{α}, D_{s}^{α} are well determined in Definition 0.1 on functions belonging to Φ. On L^{p}-functions, 1 < p < ∞, they are understood as the linear bounded (from L^{p} into L^{p}) multiplier operators extended from the dense subset Φ.

Theorem 2.2. Let Re α = 0, ν ∈ M,

ν(ℝ) = 0, ∫_{|x|>1} |x|^{β}d|ν|(x) < ∞, ∫_{|x|<1} |x|^{−δ}d|ν|(x) < ∞

for some β > 0 , δ ∈ (0, 1). If f ∈ L^{p}, 1 < p < ∞, then the limit

A^{α}f = lim_{ε→0}_{ρ→∞} ∫_{ε}^{ρ}(ν_{t} ∗ f)dt/t^{1+α}

exists in the L^{p}-norm and in the a.e. sense, and the following relations hold:

A^{α}f = γ_{+}D_{+}^{α}f + γ_{−}HD_{+}^{α}f = c_{+}D_{+}^{α}f + c_{−}D_{−}^{α}f = a_{+}D^{α}f + a_{−}D_{s}^{α}f

(in the second equality it is assumed that α ≠ 0). Here γ_{±} are defined by

γ_{+} = \begin{cases}Γ(−α)[ ∫_{0}^{∞}x^{α}dν(x) + cos(απ) ∫_{−∞}^{0} |x|^{α}dν(x)], α ≠ 0, \\ ∫_{−∞}^{∞} log 1 ; |x|dν(x), α = 0; \end{cases}

γ_{−} = − πi ; Γ(1 + α) ∫_{−∞}^{0} |x|^{α}dν(x),

c_{±} are the same as in Theorem 2.1 and a_{±} can be evaluated by the following formulae:

a_{+} = \begin{cases}Γ(−α) cos(απ/2) ∫_{−∞}^{∞} |x|^{α}dν(x), & α ≠ 0 \\ ∫_{−∞}^{∞} log 1 ; |x|dν(x), & α = 0. \end{cases}

a_{−} = \{ \begin{matrix}−iΓ(−α) sin(απ/2) ∫_{−∞}^{∞} |x|^{α}sgn x dν(x), & α ≠ 0 \\ πi ; 2 ∫_{−∞}^{∞} sgn x dν(x), & α = 0. \end{matrix}

Consider some generalizations. Assume that Re α ≥ 1 and ν belongs to a certain class of Φ′-distributions which satisfies the following condition.

Condition 2.3. If α = ℓ + α_{0}, ℓ ∈ ℕ, 0 ≤ Re α_{0} < 1, then ν_{0} ≡^{(Φ′)} I_{+}^{ℓ}ν is a finite Borel measure (i.e. ν_{0} ∈ M) such that

ν_{0}(ℝ) = 0 and ∫_{|x|>0} |x|^{β}d|ν_{0}|(x) < ∞ for some β > Re α_{0}.

If Re α_{0} = 0, we assume additionally, that

∫_{|x|<1} |x|^{−δ}d|ν_{0}|(x) < ∞ for some δ ∈ (0, 1).

Example 2.4. Let Re α_{0} ≠ 0,

ν(x) = δ(x − 1) − ∑_{j=0}^{ℓ} (−1)^{j} ; j! δ^{(j)}(x),

δ(x) being the Dirac δ-function. It is easy to see that ν satisfies Condition 2.3. Furthermore, for such ν and f ∈ Φ,

∫_{0}^{∞} ν_{t} ∗ f ; t^{1+α} dt = ∫_{0}^{∞} [f(x − t) − ∑_{j=0}^{ℓ} (−t)^{j} ; j! f^{(j)}(x)] dt ; t^{1+α} .

This coincides with the Gelfand-Shilov regularization of the divergent integral ∫_{0}^{∞} t^{−α−1}f(x − t)dt (cf. [ 2, p. 48 ]). Theorem 2.5. Let α = ℓ + α_{0}, ℓ ∈ ℕ, Re α_{0} ∈ [0, 1). Assume that ν satisfies Condition 2.3. Given f ∈ Φ′, let one of the derivatives D_{±}^{α}f, D^{α}f, D_{s}^{α}f (in the Φ′-sense) belong to L^{p}, 1 < p < ∞. If Re α ∉ ℕ, then the truncated integral ∫ε^{∞}(ν_{t} ∗ f)(x)dt/t^{1+α} coincides (in the Φ′-sense) with the L^{p}-function that tends to

γ_{+}D_{+}^{α}f + γ_{−}HD_{+}^{α}f = c_{+}D_{+}^{α}f + c_{−}D_{−}^{α}f = a_{+}D^{α}f + a_{−}D_{s}^{α}f

in the L^{p}-norm and in the a.e. sense. Here γ_{±}, c_{±}, a_{±} have the same form as in Theorems 2.1 and 2.3 in which α and ν should be replaced by α_{0} and ν_{0} = I_{+}^{ℓ}ν respectively. If Re α ∈ ℕ, then the above statement is valid for the integral

∫ε^{ρ}(ν_{t} ∗ f)(x)dt/t^{1+α} with ε → 0, ρ → ∞.

3. Wavelet type representation of fractional integrals

In this section we are concerned with integrals which have the form I^{α}(ν, φ) = ∫_{0}^{∞}(φ ∗ ν_{t})(x) dt/t^{1−α}, Re α > 0. They represent fractional integrals (and their linear combinations) and can be regarded as solutions to the corresponding differential or, more generally, pseudo-differential equations.

Given a Φ′-distribution f that agrees with a certain locally integrable function, we denote the latter by [f]_{Φ}. Theorem 3.1 Let Re α > 0, α = ℓ + α_{0}, ℓ is a nonnegative integer , 0 ≤ Re α_{0} < 1. Assume that ν is an integrable function with the following property: the generalized derivative (in the Φ′-sense) ν_{0} = (d/dx)^{ℓ}ν belongs to M and satisfies Theorem 2.2. If φ ∈ L^{p}, 1 ≤ p < ∞ and one of the Φ′-distributions I_{±}^{α}φ, I^{α}φ, I_{s}^{α}φ agrees with a certain L^{r}-function, 1 < r < ∞, then

lim ; ε →0_{ρ →∞} ∫{ρ \atop ε} φ ∗ ν_{t} ; t^{1− α} dt = γ̃_{+}[Iα_{+}φ]_{Φ} + γ̃_{−}H[Iα_{+}φ]_{Φ} =

= c̃_{+}[Iα_{+}φ]_{Φ} + c̃_{−}[Iα_{−}φ]_{Φ} (α ∉ ℕ)

= ã_{+}[Iαφ]_{Φ} + ã_{−}[Iα_{s}φ]_{Φ}

(the limit being understood in the L^{r}-norm and in the a.e. sense) where

γ̃_{+} = \{ \begin{matrix}Γ(α_{0}) [∫_{0}^{∞} x−α^{0}dν_{0}(x) + cos(α_{0}π) ∫_{−∞}^{0} x−α^{0}dν_{0}(x)] , & α_{0} ≠ 0, \\ ∫_{−∞}^{∞} log 1 ; |x|dν_{0}(x), & α_{0} = 0, \end{matrix}

γ̃_{−} = − πi ; Γ(1 − α_{0}) ∫_{−∞}^{0} |x|−αdν_{0}(x);

c̃_{+} = Γ(α_{0}) ∫_{0}^{∞} x−α^{0}dν_{0}(x),  c̃_{−} = (−1)^{ℓ}Γ(α_{0}) ∫_{−∞}^{0} |x|−α^{0}dν_{0}(x);
ã_{+} = (c̃_{+} + c̃_{−}) cos(απ/2),  ã_{−} = i(c̃_{+} − c̃_{−}) sin(απ/2), α ∉ ℕ,

ã_{+} = \{ \begin{matrix}(−1)^{k} ∫_{−∞}^{∞} log 1 ; |x|dν_{0}, & α = 2k, \\ (−1)^{k−1}π ∫_{−∞}^{0} dν_{0}, & α = 2k − 1, k ∈ ℕ, \end{matrix}

ã_{−} = \{ \begin{matrix}(−1)^{k+1}πi ∫_{−∞}^{0} dν_{0}, & α = 2k, \\ (−1)^{k−1}i ∫_{−∞}^{∞} log 1 ; |x|dν_{0}, & α = 2k − 1, k ∈ ℕ. \end{matrix}

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Institute of Mathematics

The Hebrew University

Jerusalem, Israel