Wavelets and Integration Operators

Abstract

A class of spline wavelet systems of Battle-Lemarié type is discussed with application to the study of norm related inequalities in weighted Besov spaces involving integration operators. B-spline, Battle-Lemarié wavelet system, wavelet basis, Muckenhoupt weight, Besov space, Riemann-Liouville operators, norm inequalities. VI International Conference Information Technologies and High-Performance Computing (ITHPC-2021),

1. Introduction

We are interested in a class of orthogonal spline wavelet systems of Battle-Lemarié type [ 1 ], [ 6 ]. Basing on the ideas taken from [ 8 ] and [ 9 ], the detailed description of such systems was established in [ 16 ] (see also [ 17 ] and [ 18 ]). Any Battle-Lemarié spline system of natural order n consists of a scaling function and a wavelet function - and - satisfying (2) and (3), respectively. The and are infinite linear combinations of B-splines (basic splines) of order n having supports’ lengths equal to n+1. In contrast, the Battle-Lemarié scaling and wavelet functions have unbounded supports. A type of localization property was discovered for and in [ 16 ]-[ 18 ]. The property is an algorithm resulting in new functions nBL and , which are particular finite linear combinations of integer shifts of and , respectively. The point is that and on ℝ. Moreover, similarly to the initial scaling and wavelet functions and , they form a Riesz basis in 2(ℝ). These two facts allowed to establish in [ 17 ] and [ 18 ] new decomposition theorems for weighted Besov and Triebel-Lizorkin function spaces expressed in terms of and . are compactly supported

This work is devoted to applications of Battle-Lemarié spline wavelet systems, as well as decomposition theorems mentioned above, to the study of norm related inequalities involving images and pre-images of integration operators of natural orders in weighted Besov spaces. Our results are based on the following differentiation (integration) property

′ (∙) = −1(∙) − −1(∙ −1), which binds B-splines of natural orders between each other along the smoothness scale. Recall that Bsplines

of order n are building blocks for the Battle-Lemarié scaling functions as well as for their compactly supported counterparts and .

and wavelet

Connections between images and pre-images of integral and differential operators in Besov and Triebel-Lizorkin function spaces have been studied in [13, Theorem 2.3.8], [ 2 ], [11, Theorem 2.20], [5, § 4], [16, p. 23], [ 7 ]. In particular, a norm related inequality involving a differentiation operator of natural order was established in [11, Theorem 2.20] (see also [5, § 4]). In § 4 we extend this result to arbitrary smoothness parameter s of Besov spaces (ℝ, ) with Muckenhoupt weights w (see

2020 Copyright for this paper by its authors. connect norms of images an pre-images of integration operators of natural orders in (ℝ, ).

Instruments for obtaining the results of our work are Battle-Lemarié spline wavelet systems. In § 2 we recall their exact formulae and describe their basic properties involving the localization one. Definition of weighted Besov spaces is given in § 3. New results are stated in § 4. For their proofs one can consult [ 19 ]. Characteristics of weighted discrete Hardy inequalities were also used for obtaining our statements in § 4. In settings of general measures they can be found in [10, § 1].

We use signs := and =: for determining new quantities. For positive functionals F and G we write ≪ if ≤ with some constant

> 0 depending, possibly, on irrelevant parameters only.

Relations of the type ≈ mean F ≪ G ≪ or = . 2. Battle-Lemarié spline wavelet systems

Battle-Lemarié scaling functions are polynomial splines with simple knots at ℤ obtained by orthogonalisation process of the B-splines. For ∈ ℕ the n-th order B-spline is defined recursively by ( ) = ( −1 ∗ 0)( ) = ∫ −1( − ) , where 0 = [0,1).

It is known [ 3 ] that supp = [0, + 1] and ( ) > 0 for all ∈ (0, + 1) (see Figure 1). 1 0 ∞ −∞

For a function ∈ 1(ℝ) its Fourier transform has the form ̂ ( ) = (2 )−1 ∫ − ( ) , ∈ ℝ.

The Battle-Lemarié scaling function

must satisfy the condition: ̂ ( ) = ̂ ( )( ∑ | ̂ ( + 2

2 )| ) − 1 2 .

∈ℤ The n-th order Battle-Lemarié wavelet is a function whose Fourier transform is ̂ ( ) = − − /2 ̅̂̅̅̅̅̅̅̅̅̅̅̅̅2̅̅̅̅)

( + ̅̂̅̅̅̅̅̅̅̅̅̅̅̅̅+̅̅̅̅̅) ( /2 ̂ ( /2).

(1) (2) (3) =1

Integer translations of analysis of 2(ℝ) generated by

form an orthonormal system within the multiresolution ([ 3 ], [ 20 ]). To establish explicit forms of elements of BattleLemarié spline wavelet systems { , } we fix ∈ ℕ. For each = 1, … , ( ) > 1 and define ( ) ≔ (2 ( ) − 1) − 2√ ( )( ( ) − 1) ∈ (0,1). we introduce The collection { ( )} and, respectively, the set of numbers { ( )} are uniquely defined dependent on n (see [16, § 2, p. 179] so that the sequence {− ( )}

is formed by the roots of Euler’s polynomial ∈ℤ

Relying on (4), we define the n-th order Battle-Lemarié scaling function = via its Fourier transform as follows (see [ 16 ] and [ 18 ]):

̂ ( ):= 2 (√1 +1( ) 11(( )))…… (1(+) ( )(̂ )() ). (5) The Fourier transform of a wavelet = related to has the form (see [16, § 2], [17, § 3.2]) √ 1( ) 1( )… ( ) ( ) ̂ ( ) ≔ 2 2 ( +1) [|1− /2 1( )|2 … |1 − /2 ( )|2]( /2 − 1) +1 ̂ ( /2) × (1 + − 1( ))(1 + 12( ))…(1 + − ( ))(1 + 2( )). (4) (6)

The Fourier pre-images of (5) and (6) can be viewed in [ 16 ]-[ 19 ] (see also Figures 2 and 3 below).

As it was already mentioned before, and for all ∈ ℕ have unbounded supports on ℝ. Therefore, one can operate, if applicable, with their localized analogs and , instead: ̂ ( ):= ̂ ( )(1 + 1( ))… (1 + ( ))

= 2 √ 1( ) 1( )… ( ) ( ) ̂ ( ), ̂ ( ) ≔ ̂ ( )(1 + − 1( ))(1 + 12( ))…(1 + − ( ))(1 + 2( )) = (−1) +1 1( )… ( )√ 1( ) 1( )… ( ) ( ) × |1 − − /2 1( )|2 …|1 − − /2 ( )|2( /2 − 1) +1 ̂ ( /2). (7) (8)

There exist some alternative versions of (7) and (8) (see [ 16 ]-[ 19 ]). They were applied in [ 17 ], [ 18 ] as dictionaries for decomposing elements of weighted Besov and Triebel-Lizorkin function spaces.

Weighted Besov function spaces

Muckenhoupt weights , ≤

≤ ∞ Let w be a locally integrable function, positive almost everywhere (a weight) on ℝ.

Let (ℝ), 0 < ≤ ∞, denote the Lebesgue space of all measurable functions f on ℝ quasinormed by ‖ ‖ (ℝ) ≔ (∫−∞∞| ( )| ′ ≔ /( − 1). Let ⊂ ℝ be a ball in ℝ, and | | stand for its volume.

with the usual modification if = ∞. For > 1 we put Definition 3.1. [12, Chapter V] (i) A weight w belongs to the Muckenhoupt class p, 1 < < ∞, if (ii) ∈ 1 if ( ) ≔ sup ⊂ℝ (| | ∫ ) ( 1 | |

1/ ′ ∫ 1− ′) < ∞; ( ) ≔ sup ⊂ℝ | | 1 ∫ ‖1/ ‖

∞( ) < ∞; 1/

1 1/ ) (iii) Muckenhoupt class ∞ is given by ∞ ≔ ⋃ ≥1 .

For basic properties and examples of weights from ∞ we refer to [ 12 ] (see also references there). 3.2.

Besov spaces with Muckenhoupt weights

For 0 < < ∞ and a weight w on ℝ we denote (ℝ) the weighted Lebesgue space quasi-normed by ‖ ‖ (ℝ) ≔ ‖ 1/ ‖ (ℝ)

with usual modification if = ∞.

For the definitions of unweighted Besov spaces (ℝ) we refer to [ 13 ] and [ 14 ]. Their weighted counterparts (ℝ, w) with w ∈ ∞ can be introduced in framework of the Schwartz space ′(ℝ).

Let (ℝ) be Schwartz space of all complex-valued rapidly decreasing, infinitely differentiable functions ∞(ℝ). By ′(ℝ) we denote its topological dual, the space of tempered distributions on ℝ. For

∈ (ℝ) the inverse Fourier transform ̌is given by the right hand side of (1) with i in place of –i. Both ̂ and ̌are extended to ′(ℝ) in the standard way.

To give a definition of Besov spaces with Muckenhoupt weights we fix 0 = ∈ (ℝ) such that supp ⊂ { ∈ ℝ: | | < 2}

and ( ) = 1 if | | ≤ 1, and let ( ) = (

Definition 3.2. Let 0 < < ∞, 0 < ≤ ∞, ∈ ℝ, w ∈ resolution of unity. Weighted Besov space Bpsq(ℝ, w) is the collection of all f ∈ ′(ℝ) such that ∞ and { }∞

=0 be a smooth dyadic ‖ ‖ (ℝ, ) ≔ (∑ ∞ =0 2 ‖( ̌ ̂)‖ (ℝ)

1/ ) (9) (with the usual modification if = ∞) is finite. (9). The theory and properties of (ℝ, ) can be found in [ 4 ].

Definition of the above space is independent of the choice of , up to equivalence of quasi-norm

4. Inequalities for integration operators

For ∈ 1 (ℝ) we consider the left- and the right-hand side Riemann-Liouville operators and Moreover, + ( ) ≔ − ( ) ≔ (11) /2 ∫ /2 −1 of positive orders . For natural

= the + and − are integration operators of natural orders.

The main results of this work are inequalities between norms of images and pre-images of operators (10) and (11) in Bpsq(ℝ, w). For simplicity we assume that f ≡ 0 outside of suppI±αf. ℕ let + be defined by (10). Suppose that ( ) ≡ 0 for < 0. (i) Assume that for the both = and = it holds Theorem 4.1. [ 19 ] Let 1 < < ∞, 0 < ≤ ∞, ∈ ℝ, weights , ∈ ∞ and ∈ 1 (ℝ). For ∈ ≈ ( /2 )/2 ∈ ℕ ∈ ℤ.

(12) Then + ∈ (ℝ, ) if ∈

+ (ℝ, ) provided ≥

1 ≥

0≤ ≤ + ≔ ≥0 ( ∑ ( − + 1) (2 −1) ( )) ( ∑ [ ( )]1− ′) + ≥0 ( ∑ ( )) ( ∑ ( − + 1) ′(2 −1)[ ( )]1− ′)

< ∞. 1 1 0≤ ≤ ≤ ≤0 1 ′ 1 ′ 1 ′ 1 ′ ‖ + ‖ (ℝ, ) ≪ ‖ ‖ + (ℝ, ),

(ii) If + ∈ (ℝ, ) then ∈ − (ℝ, ), besides, where := +.

‖ ‖ − (ℝ, ) ≪ ‖ + ‖ (ℝ, ).

Analogous result is valid for the left-hand side Riemann-Liouville operator I−α. ℕ let − be defined by (11). Suppose that ( ) ≡ 0 for > 0.

Theorem 4.2. [ 19 ] Let 1 < < ∞, 0 < ≤ ∞, ∈ ℝ, weights , ∈ ∞ and ∈ 1 (ℝ). For ∈ (i) Assume that (12) holds for = and = . Then − ∈ (ℝ, ) if ∈ + (ℝ, ) provided ≤

1 ≤

≤ ≤0 − ≔ ≤0 ( ∑ ( − + 1) (2 −1) ( )) ( ∑ [ ( )]1− ′) + ≤0 ( ∑ ( )) ( ∑ ( − + 1) ′(2 −1)[ ( )]1− ′) < ∞.

Moreover, ‖ − ‖ (ℝ, ) ≪ ‖ ‖ + (ℝ, ),

(ii) If − ∈ (ℝ, ) then ∈ − (ℝ, ), besides, where := +.

‖ ‖ − (ℝ, ) ≪ ‖ − ‖ (ℝ, ).

The results of this section can be generalized to some other cases of integration operators and various types of weighted function spaces (see [19, Theorems 5.2, 5.3 and Remark 5.4]).

5. Acknowledgements

The research work of the first author related to Section 4 was performed at Steklov Mathematical Institute of Russian Academy of Sciences under financial support of the Russian Science Foundation (project 19-11-00087).

The studies were carried out using the resources of the Center for Shared Use of Scientific Equipment "Center for Processing and Storage of Scientific Data of the Far Eastern Branch of the Russian Academy of Sciences", funded by the Russian Federation represented by the Ministry of Science and Higher Education of the Russian Federation under project No. 075-15-2021-663.

6. References

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