New convolutions for the Hartley integral transform imbedded in the Banach algebras and convolution-type integral equations

Nguyen Minh Tuan; Bui Thi Giang; Quan Thai Ha

doi:10.1515/dema-2024-0087

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New convolutions for the Hartley integral transform imbedded in the Banach algebras and convolution-type integral equations

Nguyen Minh Tuan , Bui Thi Giang and Quan Thai Ha

Published/Copyright: February 26, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

This article presents two types of the new convolutions for the Hartley integral transform associated with the Hermite functions, gives rise to the identification of some commutative and non-commutative Banach algebras, and to the Young inequalities which, in a certain sense, can be seen as the exceptional Young inequalities. Moreover, the solvability of a class of the integral equations is investigated, and their solutions in a series form can be obtained by using the constructed convolutions.

Keywords: Hartley’s integral transform; generalized convolution; Hermite’s functions; Wiener’s algebra

MSC 2010: 42A38; 44A20; 44A35; 45E10

1 Introduction

The Hartley integral transform, first proposed by Ralph Vinton Lyon Hartley, in 1942, is defined as

( ℋ 1 f ) ( x ) = 1 2 π ∫ − ∞ ∞ cas ( x y ) f ( y ) d y ,

where f is a real- or complex-valued function defined on R , and the integral kernel, known as the cosine-and-sine or “cas” function, is defined as cas ( t ) ≔ cos ( t ) + sin ( t ) (see [1]). The transform ℋ 1 has concrete applied objectives and, in a sense, reflects the applied vision of Hartley, who did very significant research on electronics, having, e.g., invented the Hartley oscillator and being recognized as one of the founders of information theory. The Hartley integral transform is a spectral one, and it can be associated with the Fourier transform. However, the image of the Hartley integral transform of a real-valued function is again a real-valued one, rather than complex, presenting therefore a significant qualitative difference in comparison with the Fourier transform. Thus, the Hartley integral transform has some advantages over the Fourier transform, e.g., in the analysis of real signals since it avoids the use of complex arithmetic. In view of this, the use of the Hartley integral transform, in obtaining numerical solutions to problems, presents an advantage, as computers prefer real numbers.

In recent years, the Hartley integral transform has obtained increased importance in telecommunications, radio-sciences, signal processing, image reconstruction, and pattern recognition. In particular, there are comprehensive works involved in the analysis of the one-dimensional and two-dimensional Hartley integral transforms and in some of the practical problems where it can be useful (see [1–5] and references therein). Also, it is no surprise to realize that the research on variations of the Fourier transform and of the convolution-type equations has a long history and continues to be an object of deep research studies. However, there is a notable lack of systematic and theoretical studies covering the multi-dimensional Hartley integral transform, except for the interesting engineering book [1], involved in the two-dimensional Hartley integral transform and some practical problems.

Motivated by the above situation, we are here proposing a detailed investigation of a pair of multi-dimensional Hartley integral transforms ℋ 1 and ℋ 2 . Namely, in association with ℋ 1 , one can consider its paired transform ℋ 2 defined by

( ℋ 2 f ) ( x ) = 1 2 π ∫ − ∞ ∞ cas ( − x y ) f ( y ) d y .

Using the notation f ˇ ( x ) = f ( − x ) , we directly obtain that ( ℋ 1 f ) ( x ) = ( ℋ 2 f ) ∨ ( x ) and ( ℋ 2 f ) ( x ) = ( ℋ 1 f ˇ ) ( x ) . From our point of view, both ℋ 1 and ℋ 2 can be called the Hartley transforms. As we shall see in the following sections, the consideration of such a pair of integral transforms will be helpful to achieve some new related goals. In particular, there are some new convolutions associated with the two Hartley integral transforms, by which several Wiener-type algebras are generated, and the concrete application to the solvability of the corresponding integral equations is investigated.

The article is divided into four sections and organized as follows. Section 2 is very short, recalling the basic properties of the Hartley integral operators. Section 3 provides infinitely many generalized convolutions associated with the Hartley integral transforms. Namely, convolution (5) in Theorem 6 returns to the well-known convolution in [1] when h = 0 , and Theorem 7 provides not only a series of the new convolutions, but also interesting evidence: eigenfunctions of an integral transform have a closed relation with its generalized convolutions. Particularly, the new Young-type inequality (28) has been proved for convolutions (11)–(26), which are mainly concerned with the typical property of the Hermite functions. In Section 4, based on the constructed convolutions the Wiener-type algebras are identified, solvability of some classes of the integral equations is investigated in which the explicit solution of each one of those equations is determined.

2 Basic properties

This section presents the basic properties of the Hartley operators, which are useful in the following sections; actually, they are helpful to clarify the general structure of the Hartley integral operators. Due to the fact that there is no remarkable difference between the multi-dimensional and one-dimensional Hartley transforms in the proofs, the results below are proved for the multi-dimensional one. For this purpose, we write

( ℋ 1 f ) ( x ) ≔ 1 ( 2 π ) n 2 ∫ R n cas ( x y ) f ( y ) d y , ( ℋ 2 f ) ( x ) ≔ 1 ( 2 π ) n 2 ∫ R n cas ( − x y ) f ( y ) d y ,

where cas ( x y ) ≔ cas ( x 1 y 1 + … + x n y n ) , assuming the usual notation in multi-dimensional analysis z ≔ ( z 1 , … , z n ) . Conversely, let ( x y ) ≔ ⟨ x , y ⟩ denote the scalar product of x , y ∈ R n , and ∣ x ∣ 2 = ⟨ x , x ⟩ . Moreover, for a multi-indices α = ( α 1 , … , α n ) with α k ∈ N ( k = 1 , … , n ), we let ∣ α ∣ ≔ α 1 + … + α n . If x = ( x 1 , … , x n ) ∈ R n , the monomial x α is defined by x α ≔ x 1 α 1 … x n α n . Additionally, we use the standard notation

D x α ≔ ∂ α ∂ x 1 α 1 … ∂ x n α n .

Let S denote the Schwartz space of all rapidly decreasing functions on R n (see [6]).

Let us first recall some known notions and results which are useful for our forthcoming proofs. The Hermite functions, denoted by Φ α , are defined by

(1) Φ α ( x ) ≔ ( − 1 ) ∣ α ∣ e 1 2 ∣ x ∣ 2 D x α e − ∣ x ∣ 2 (see [6,7]).

Theorem 1

(cf., e.g., [8]) Let ∣ α ∣ = r ( mod 4 ) , where r = 0 , 1 , 2 , 3 . Then,

ℋ 1 Φ α = Φ α , if r = 0 , 1 − Φ α , if r = 2 , 3 , and ℋ 2 Φ α = Φ α , if r = 0 , 3 − Φ α , if r = 1 , 2 .

Let ( C 0 ( R n ) , ‖ ⋅ ‖ ∞ ) stand for the supremum-normed Banach space of all continuous functions on R n that vanish at infinity. As about the Lebesgue L p ( R n ) space with p ∈ [ 1 , ∞ ) , it is endowed with the norm

‖ f ‖ p ≔ 1 ( 2 π ) n 2 ∫ R n ∣ f ( x ) ∣ p d x 1 p .

For saving space, let us use the unique notation ℋ 1 in some formulations, but all results in this section will be true for ℋ 2 . Having in mind the definition of ℋ 1 , the just mentioned norms and the circumstance that ∣ cas ( x y ) ∣ ≤ 2 , we directly obtain a Riemann-Lebesgue property for ℋ 1 , in the sense that ℋ 1 is a bounded linear operator, when acting from L 1 ( R n ) into C 0 ( R n ) , and

(2) ‖ ℋ 1 f ‖ ∞ ≤ 2 ‖ f ‖ 1 .

Proposition 2

(Inversion theorem; cf. [8])

If g ∈ S , then
g ( x ) = 1 ( 2 π ) n 2 ∫ R n ( ℋ 1 g ) ( y ) ( cos x y + sin x y ) d y .
ℋ 1 : S → S is an invertible linear operator, with ℋ 1 2 = I .
If f , ℋ 1 f ∈ L 1 ( R n ) , then
f ( x ) = 1 ( 2 π ) n 2 ∫ R n ( ℋ 1 f ) ( y ) ( cos x y + sin x y ) d y ,
for almost every x ∈ R n .

Corollary 3

(Uniqueness theorem) If f ∈ L 1 ( R n ) and ℋ 1 f = 0 in L 1 ( R n ) , then f = 0 in L 1 ( R n ) .

The operator is well-defined in L 2 ( R n ) in the sense of the following theorem.

Theorem 4

(Operator version of Plancherel’s theorem) There is a linear isometric operator ℋ ¯ 1 from L 2 ( R n ) into itself which is uniquely determined by the requirement that

ℋ ¯ 1 f = ℋ 1 f , for e v e r y f ∈ S .

Moreover, the extension operator possesses the identity ℋ ¯ 1 2 = I (where I denotes the identity operator in L 2 ( R n ) ).

Thanks to Theorem 4, instead of the notation ℋ ¯ 1 , we will simply write ℋ 1 , as there is no more danger of confusion. Note that the deduced identity in the previous proof gives the Parseval identity for ℋ 1 , which is now assembled in the following corollary.

Corollary 5

(Parseval’s identity) The operator ℋ 1 is unitary in the Hilbert space L 2 ( R n ) . In particular, ⟨ ℋ 1 f , ℋ 1 g ⟩ = ⟨ f , g ⟩ , for f , g ∈ L 2 ( R n ) .

Shortly, for ℋ 2 , we also have

‖ ℋ 2 f ‖ ∞ ≤ 2 ‖ f ‖ 1 ; f = ℋ 2 ( ℋ 2 f ) , provided f , ℋ 2 f ∈ L 1 ( R n ) ; ⟨ ℋ 2 f , ℋ 2 g ⟩ = ⟨ f , g ⟩ and ℋ 2 2 f = f , provided f , g ∈ L 2 ( R n ) .

3 Convolution transforms and Young’s inequalities

In this section, we construct a series of new convolutions associated with the multi-dimensional Hartley integral operators and their Young’s inequalities.

3.1 New convolutions

Convolutions were postulated early in the twentieth century and, since then, they have been studied and developed in a continuous way. In fact, each convolution can be seen as not only a new transform but also an object of study. From the practical point of view, convolutions can be useful in different ways. For instance, in signal processing, probably the most comprehensive identification is to see each convolution transform as a mathematical way of combining two signals to form a third signal – which is still the most important technique in digital signal processing. Anyway, the applications of convolutions and convolution-type operators are very diverse and can be found in a great variety of areas (cf., e.g., [3,7,9,10]).

For the sake of shortness of notation, we put θ ( x ) ≔ cas ( x h ) for an h ∈ R n fixed. Additionally, the following evident identities will be helpful for proving the next theorems in this work:

(3) 2 cas ( a ) cas ( b ) = cas ( a + b ) + cas ( a − b ) + cas ( − a + b ) − cas ( − a − b ) ,

(4) 2 cas ( a ) cas ( b ) cas ( c ) = cas ( − a + b + c ) + cas ( a − b + c ) + cas ( a + b − c ) − cas ( − a − b − c ) ,

for a , b , c ∈ R . We start by introducing simultaneously four convolutions concerned with the weight function θ ( x ) .

Theorem 6

If f , g ∈ L 1 ( R n ) , then each one of the transforms below is a convolution satisfying an L 1 -norm inequality and respective factorization:

(5) ( f ∗ 1 g ) ( x ) = 1 2 ( 2 π ) n 2 ∫ R n [ f ( x − u + h ) + f ( x + u − h ) + f ( − x + u + h ) − f ( − x − u − h ) ] g ( u ) d u , ‖ f ∗ 1 g ‖ 1 ≤ 2 ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 1 g ) ( x ) = cas ( x h ) ( ℋ 1 f ) ( x ) ( ℋ 1 g ) ( x ) ;

(6) ( f ∗ 2 g ) ( x ) = 1 2 ( 2 π ) n 2 ∫ R n [ f ( x + u + h ) + f ( x − u − h ) + f ( − x − u + h ) − f ( − x + u − h ) ] g ( u ) d u , ‖ f ∗ 2 g ‖ 1 ≤ 2 ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 2 g ) ( x ) = cas ( x h ) ( ℋ 1 f ) ( x ) ( ℋ 2 g ) ( x ) ;

(7) ( f ∗ 3 g ) ( x ) = 1 2 ( 2 π ) n 2 ∫ R n [ − f ( x + u + h ) + f ( x − u − h ) + f ( − x + u − h ) + f ( − x − u + h ) ] g ( u ) d u , ‖ f ∗ 3 g ‖ 1 ≤ 2 ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 3 g ) ( x ) = cas ( x h ) ( ℋ 2 f ) ( x ) ( ℋ 1 g ) ( x ) ;

(8) ( f ∗ 4 g ) ( x ) = 1 2 ( 2 π ) n 2 ∫ R n [ f ( x + u − h ) − f ( x − u + h ) + f ( − x + u + h ) + f ( − x − u − h ) ] g ( u ) d u , ‖ f ∗ 4 g ‖ 1 ≤ 2 ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 4 g ) ( x ) = cas ( x h ) ( ℋ 2 f ) ( x ) ( ℋ 2 g ) ( x ) .

Proof

We start by proving the norm inequality and factorization identity for convolution (5). For any u ∈ R n fixed, by changing variables, we have

(9) 1 ( 2 π ) n 2 ∫ R n ∣ f ( ± x ± u ± h ) ∣ d x = 1 ( 2 π ) n 2 ∫ R n ∣ f ( x ) ∣ d x = ‖ f ‖ 1 .

Since f , g are L 1 -functions, we can change the order of integration to obtain

∫ R n ∣ ( f ∗ 1 g ) ( x ) ( x ) ∣ d x ≤ 1 2 ( 2 π ) n 2 ∫ R n ∫ R n { ∣ f ( x − u + h ) ∣ + ∣ f ( x + u − h ) ∣ + ∣ f ( − x + u + h ) ∣ + ∣ f ( − x − u − h ) ∣ } ∣ g ( u ) ∣ d u d x = 1 2 ( 2 π ) n 2 ∫ R n ∣ g ( u ) ∣ d u ∫ R n [ ∣ f ( x − u + h ) ∣ + ∣ f ( x + u − h ) ∣ + ∣ f ( − x + u + h ) ∣ + ∣ f ( − x − u − h ) ∣ ] d x = 2 ( 2 π ) n 2 ∥ g ∥ 1 ∥ f ∥ 1 < ∞ .

This implies that f ∗ 1 g ∈ L 1 ( R n ) and ∥ f ∗ 1 g ∥ 1 ≤ 2 ∥ f ∥ 1 ∥ g ∥ 1 .

We will directly prove the factorization identity. Indeed, by using (4) and changing variables s ≔ ± u ± v ± h in the integrals, we have

θ ( x ) ( ℋ 1 f ) ( x ) ( ℋ 1 g ) ( x ) = 1 ( 2 π ) n ∫ R 2 n θ ( x ) [ cas ( x u ) ] [ cas ( x v ) ] f ( u ) g ( v ) d u d v = 1 2 ( 2 π ) n ∫ R 2 n [ cas ( x ( − u + v + h ) ) + cas ( x ( u − v + h ) ) + cas ( x ( u + v − h ) ) − cas ( x ( − u − v − h ) ) ] f ( u ) g ( v ) d u d v = 1 ( 2 π ) n 2 ∫ R n cas ( x s ) d s 1 2 ( 2 π ) n 2 ∫ R n [ f ( s − v + h ) + f ( s + v − h ) + f ( − s + v + h ) − f ( − s − v − h ) ] g ( v ) d v = ℋ 1 ( f ∗ 1 g ) ( x ) .

As about the norm inequalities and factorization property for the other proposed convolutions, we will not present in here a detailed proof due to the limited space. Let us just mention that the norm inequalities can be proved in the same way as in the proof for (5), and the factorization identities can be obtained similar to the proof for (5), together with using identity (4) according to the kernels cas ( x u ) and cas ( − x v ) of ℋ 1 and ℋ 2 , respectively. The theorem is proved.□

Remark 1

Theorem 6 includes the convolutions appeared in almost materials concerning Hartley integral operators in the particular case of h = 0 , see [1, 2]. Conversely, h fixed in the function cas ( x h ) , and h appeared in f ( ± x ± u ± h ) within convolutions (5)–(8) may be distinct from each other. For instance, the h in convolution (5) is not necessarily the h in convolution (6). In signal processing, they can be constant shifts or delays in the time frequency.

For any α ∈ N n , we put

(10) N α ≔ 2 ‖ Φ α ‖ 1 = 2 ( 2 π ) n 2 ∫ R n ∣ Φ α ( x ) ∣ d x > 0 .

The following theorem contains infinitely many convolutions associated with the Hartley integral transforms ℋ 1 and ℋ 2 .

Theorem 7

Let ∣ α ∣ = r ( mod 4 ) , where r = 0 , 1 , 2 , 3 . If f , g ∈ L 1 ( R n ) , then each one of the following applications defines a convolution associated with ℋ 1 and ℋ 2 and Hermite weight Φ α , and they are followed by the corresponding norm inequality and the factorization identity:

• Case r = 0 .

(11) ( f ∗ 1 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ − Φ α ( x + u + v ) + Φ α ( x + u − v ) + Φ α ( x − u + v ) + Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 1 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 1 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 1 f ) ( x ) ( ℋ 1 g ) ( x ) ;

(12) ( f ∗ 2 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ Φ α ( x + u + v ) − Φ α ( x + u − v ) + Φ α ( x − u + v ) + Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 2 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 2 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 1 f ) ( x ) ( ℋ 2 g ) ( x ) ;

(13) ( f ∗ 3 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ Φ α ( x + u + v ) + Φ α ( x + u − v ) − Φ α ( x − u + v ) + Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 3 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 3 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 2 f ) ( x ) ( ℋ 1 g ) ( x ) ;

(14) ( f ∗ 4 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ Φ α ( x + u + v ) + Φ α ( x + u − v ) + Φ α ( x − u + v ) − Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 4 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 4 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 2 f ) ( x ) ( ℋ 2 g ) ( x ) .

• Case r = 1 .

(15) ( f ∗ 1 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ Φ α ( x + u + v ) + Φ α ( x + u − v ) + Φ α ( x − u + v ) − Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 1 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 1 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 1 f ) ( x ) ( ℋ 1 g ) ( x ) ;

(16) ( f ∗ 2 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ Φ α ( x + u + v ) + Φ α ( x + u − v ) − Φ α ( x − u + v ) + Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 2 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 2 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 1 f ) ( x ) ( ℋ 2 g ) ( x ) ;

(17) ( f ∗ 3 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ Φ α ( x + u + v ) − Φ α ( x + u − v ) + Φ α ( x − u + v ) + Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 3 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 3 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 2 f ) ( x ) ( ℋ 1 g ) ( x ) ;

(18) ( f ∗ 4 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ − Φ α ( x + u + v ) + Φ α ( x + u − v ) + Φ α ( x − u + v ) + Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 4 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 4 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 2 f ) ( x ) ( ℋ 2 g ) ( x ) .

• Case r = 2 .

(19) ( f ∗ 1 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ Φ α ( x + u + v ) − Φ α ( x + u − v ) − Φ α ( x − u + v ) − Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 1 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 1 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 1 f ) ( x ) ( ℋ 1 g ) ( x ) ;

(20) ( f ∗ 2 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ − Φ α ( x + u + v ) + Φ α ( x + u − v ) − Φ α ( x − u + v ) − Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 2 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 2 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 1 f ) ( x ) ( ℋ 2 g ) ( x ) ;

(21) ( f ∗ 3 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ − Φ α ( x + u + v ) − Φ α ( x + u − v ) + Φ α ( x − u + v ) − Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 3 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 3 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 2 f ) ( x ) ( ℋ 1 g ) ( x ) ;

(22) ( f ∗ 4 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ − Φ α ( x + u + v ) − Φ α ( x + u − v ) − Φ α ( x − u + v ) + Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 4 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 4 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 2 f ) ( x ) ( ℋ 2 g ) ( x ) .

• Case r = 3 .

(23) ( f ∗ 1 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ − Φ α ( x + u + v ) − Φ α ( x + u − v ) − Φ α ( x − u + v ) + Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 1 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 1 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 1 f ) ( x ) ( ℋ 1 g ) ( x )

(24) ( f ∗ 2 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ − Φ α ( x + u + v ) − Φ α ( x + u − v ) + Φ α ( x − u + v ) − Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 2 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 2 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 1 f ) ( x ) ( ℋ 2 g ) ( x )

(25) ( f ∗ 3 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ − Φ α ( x + u + v ) + Φ α ( x + u − v ) − Φ α ( x − u + v ) − Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 3 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 3 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 2 f ) ( x ) ( ℋ 1 g ) ( x )

(26) ( f ∗ 4 Φ α g ) ( x ) = 1 2 ( 2 π ) n ∫ R n ∫ R n f ( u ) g ( v ) [ Φ α ( x + u + v ) − Φ α ( x + u − v ) − Φ α ( x − u + v ) − Φ α ( x − u − v ) ] d u d v , ‖ f ∗ 4 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 , ℋ 1 ( f ∗ 4 Φ α g ) ( x ) = Φ α ( x ) ( ℋ 2 f ) ( x ) ( ℋ 2 g ) ( x ) .

Proof

We shall start by the first case when r = 0 . Clearly, Φ α ∈ L 1 ( R n ) , and

1 ( 2 π ) n 2 ∫ R n ∣ Φ α ( x ± u ± v ) ∣ d x = 1 ( 2 π ) n 2 ∫ R n ∣ Φ α ( x ) ∣ d x = ‖ Φ α ‖ 1 = N α 2 .

For f , g ∈ L 1 ( R n ) , we have

∫ R n ∣ f ∗ 1 Φ α g ∣ ( x ) d x ≤ 1 2 ( 2 π ) n ∫ R n ∫ R n ∫ R n ∣ f ( u ) ∣ . ∣ g ( v ) ∣ . ∣ − Φ α ( x + u + v ) + Φ α ( x + u − v ) + Φ α ( x − u + v ) + Φ α ( x − u − v ) ∣ d x d u d v ≤ 2 ( 2 π ) n ∫ R n ∣ f ( u ) ∣ d u ∫ R n ∣ g ( v ) ∣ d v ∫ R n ∣ Φ α ( x ) ∣ d x = ( 2 π ) n 2 N α ‖ f ‖ 1 ‖ g ‖ 1 ,

which implies the norm inequality, and the fact that f ∗ 1 Φ α g ∈ L 1 ( R n ) .

We shall prove the factorization identity. By Theorem 1, ℋ 1 ( Φ α ) = Φ α . We also recall that Φ α ( x ) = Φ α ( − x ) for every x ∈ R n by (1). In view of these identities, applying (4) and changing appropriately the variables in the integrals, we have

Φ α ( x ) ( ℋ 1 f ) ( x ) ( ℋ 1 g ) ( x ) = 1 ( 2 π ) 3 n 2 ∫ R 3 n cas ( x u ) cas ( x v ) cas ( x t ) f ( u ) g ( v ) Φ α ( t ) d u d v d t = 1 2 ( 2 π ) 3 n 2 ∫ R 3 n [ cas x ( u + v − t ) + cas x ( u − v + t ) + cas x ( − u + v + t ) − cas x ( − u − v − t ) ] f ( u ) g ( v ) Φ α ( t ) d u d v d t

= 1 2 ( 2 π ) 3 n 2 ∫ R 3 n [ Φ α ( u + v − s ) + Φ α ( u − v + s ) + Φ α ( − u + v + s ) − Φ α ( − u − v − s ) ] f ( u ) g ( v ) cas ( x s ) d u d v d s = 1 2 ( 2 π ) n 2 ∫ R n cas ( x s ) d s 1 2 ( 2 π ) n 2 ∫ R 2 n [ Φ α ( u + v − s ) + Φ α ( u − v + s ) + Φ α ( − u + v + s ) − Φ α ( − u − v − s ) ] f ( u ) g ( v ) d u d v = 1 2 ( 2 π ) n 2 ∫ R n cas ( x s ) d s 1 2 ( 2 π ) n 2 ∫ R 2 n [ Φ α ( s − u − v ) + Φ α ( s + u − v ) + Φ α ( s − u + v ) − Φ α ( s + u + v ) ] f ( u ) g ( v ) d u d v = ℋ 1 ( f ∗ 1 Φ α g ) ( x ) .

The proof of the factorization identity for convolution (11) is complete.

We emphasize that the proofs of all convolutions (12)–(24) can be processed in the same way as in the proof of (11), being based on the following main steps:

Deduction of the norm inequality, in an analogous manner to that of (11);
Noting that the Hermite functions Φ α for α ∈ N n are eigenfunctions of both transforms ℋ j ( j = 1 , 2 ) , with eigenvalues ± 1 , by Theorem 1;
Using the identity Φ α ( x ) = ( − 1 ) ∣ α ∣ Φ α ( − x ) for every x ∈ R n , by (1);
Building an adaptation of (4), so that it will be directly applicable to the specific proof. For instance, to prove (14), we may adapt (4), by substituting Q by − Q and R by − R , for having
2 cas ( P ) cas ( − Q ) cas ( − R ) = cas ( P − Q + R ) + cas ( P + Q − R ) + cas ( − P − Q − R ) − cas ( − P + Q + R ) .

Then, the proof of the factorization identity of convolution (14) can be done analogously to that of (11) as presented above.□

According to each one of r = 0 , 1 , 2 , 3 , the above constructed convolutions will be applied in the topics of the Wienner algebras and the convolution integral equations in Section 4.

3.2 Young’s inequality

This subsection proves that the convolutions in Theorems 6 and 7 fulfill the Young inequality. Let us recall Minkowski’s integral inequality

∫ Θ 2 ∫ Θ 1 F ( x , y ) d μ 1 ( x ) s d μ 2 ( y ) 1 s ≤ ∫ Θ 1 ∫ Θ 2 ∣ F ( x , y ) ∣ s d μ 2 ( y ) 1 s d μ 1 ( x ) ,

where ( Θ 1 , μ 1 ) and ( Θ 2 , μ 2 ) are measure spaces, F ( ⋅ , ⋅ ) : Θ 1 × Θ 2 ⟶ C is a measurable function, and s ≥ 1 .

Due to the great number of convolutions considered here, let us use the same symbol ⊛ for convolutions numbered from (5) to (8), and the symbol ⋆ for convolutions (11)–(26).

Theorem 8

The convolutions ⊛ are continuous bilinear maps between suitable L s ( R n ) spaces in the sense that if 1 ≤ p , q , r ≤ ∞ together satisfy 1 p + 1 q = 1 r + 1 , then

(27) ‖ f ⊛ g ‖ r ≤ C 1 ‖ f ‖ p ‖ g ‖ q , provided f ∈ L p ( R n ) , g ∈ L q ( R n ) ,

where C 1 is some positive constant.

For the convolutions ⋆ , it is possible to obtain even more, having the inequality

(28) ‖ f ⋆ g ‖ s ≤ C 2 ‖ f ‖ 1 ‖ g ‖ 1 for any s ≥ 1 , provided f , g ∈ L 1 ( R n )

for a positive constant C 2 .

Proof

We see that there are four terms in each one of the convolutions considered here. Applying the Minkowski inequality, it suffices to prove the inequalities for one term. In particular, we shall prove the inequalities for the terms:

M ( x ) ≔ ∫ R n f ( x + u + h ) g ( u ) d u , N ( x ) ≔ ∫ R 2 n f ( u ) g ( v ) Φ α ( x + u + v ) d u d v .

First, we shall prove (27). If f ∈ L p ( R n ) , g ∈ L q ( R n ) ,

∫ R n f ( x + u + h ) g ( u ) d u = ∫ R n f ( x − u + h ) g ˇ ( u ) d u = ( f ∗ ℱ g ˇ ) ( x + h ) ,

where ( ⋅ ∗ ℱ ⋅ ) stands for the usual Fourier convolution. Clearly, g ˇ ∈ L q ( R n ) . By the known Young convolution inequality [11], f ∗ ℱ g ˇ ∈ L r ( R n ) where 1 ⁄ p + 1 ⁄ q = 1 ⁄ r + 1 . Since h is a constant we deduce that M = ( f ∗ ℱ g ˇ ) ( ⋅ + h ) ∈ L r ( R n ) , which proves (27) for M ( x ) . For N ( x ) , changing the variable t ≔ − u − v , we have

N ( x ) = ∫ R 2 n f ( u ) g ( v ) Φ α ( x + u + v ) d u d v = ∫ R 2 n f ( − t − v ) g ( v ) Φ α ( x − t ) d t d v = ∫ R n Φ α ( x − t ) d t ∫ R n f ˇ ( t − v ) g ˇ ( v ) d v = [ Φ α ∗ ℱ ( f ˇ ∗ ℱ g ˇ ) ] ( x ) .

By the Young inequality, f ˇ ∗ ℱ g ˇ ∈ L r ( R n ) . Evidently, Φ α ∈ L s ( R n ) for any s ≥ 1 . Again, applying the Young inequality cited above, for the case 1 r + 1 1 = 1 r + 1 , and we derive that N ∈ L r ( R n ) . Second, we will prove (28). Since Φ α ∈ L s ( R n ) for any s ≥ 1 , and

1 ( 2 π ) n 2 ∫ R n ∣ Φ α ( x + u + v ) ∣ s d x 1 s = ‖ Φ α ‖ s ( u , v are fixed in R n ) ,

we can apply the Minkowski integral inequality mentioned above to receive

∫ R n ∫ R 2 n Φ α ( x + u + v ) f ( u ) g ( v ) d u d v s d x 1 s ≤ ∫ R 2 n ∫ R n ∣ Φ α ( x + u + v ) ∣ s ∣ f ( u ) ∣ s ∣ g ( v ) ∣ s d x 1 s d u d v = ∫ R 2 n ∫ R n ∣ Φ α ( x + u + v ) ∣ s d x 1 s ∣ f ( u ) ∥ g ( v ) ∣ d u d v = ( 2 π ) n 2 ‖ Φ α ‖ s ∫ R 2 n ∣ f ( u ) ∥ g ( v ) ∣ d u d v = ( 2 π ) n 2 ‖ Φ α ‖ s ‖ f ‖ 1 ‖ g ‖ 1 .

We thus obtain (28). The proof of Theorem 8 is complete.□

Remark 2

When p = q = r = 1 and s = 1 , we have the norm inequalities in Theorems 6 and 7.

4 Banach algebras and convolution-type integral equations

This section, is devoted to the applications of the constructed convolutions for structuring the Banach algebras for L 1 ( R n ) and investigating the solvability of some classes of the integral equations.

4.1 Banach algebras

In this subsection, we show that the Banach space L 1 ( R n ) , equipped with each one of the convolutions (or multiplications) in Section 3.1, and an appropriate norm, becomes a normed algebra. Moreover, some of them will be commutative. Thus, when endowed with the appropriate structure (which can be generated by using some of the above convolutions), L 1 ( R n ) can be seen as a specific commutative Banach algebra. There are infinitely many Banach algebras endowed with the mentioned convolutions. We refer further references, e.g., [12,13].

Theorem 9

The space X ≔ L 1 ( R n ) , equipped with any of the convolutions from (5) to (8) and from (11) to (26), becomes a non-unital complex algebra (which may be non-associative). Moreover:

The convolutions (5), (8), (11), (14), (15), (18), (19), (22), (23), and (26) are commutative.
The convolutions (6), (7), (12), (13), (16), (17), (20), (21), (24), and (25) are non-commutative.

Proof

The proof is divided into two steps.

Step 1: X has a normed ring structure. It is clear that X , equipped with each one of the convolutions listed above, has a ring structure. We prove the multiplicative inequalities for convolution (11) and omit the proof for the other cases (since they are similar). We have

∫ R n ∣ f ∗ 1 Φ α g ∣ ( x ) d x ≤ 1 2 ( 2 π ) n ∫ R n ∫ R n ∫ R n ∣ f ( u ) ∥ g ( v ) ∥ − Φ α ( x + u + v ) + Φ α ( x + u − v ) + Φ α ( x − u + v ) + Φ α ( x − u − v ) ∣ d u d v ≤ 2 ( 2 π ) n ∫ R n ∣ f ( u ) ∣ d u ∫ R n ∣ g ( v ) ∣ d v ∫ R n ∣ Φ α ( x ) ∣ d x .

Hence,

1 ( 2 π ) n 2 ∫ R n ∣ f ∗ 1 Φ α g ∣ ( x ) d x ≤ N α ( 2 π ) n 2 ∫ R n ∣ f ( u ) ∣ d u 1 ( 2 π ) n 2 ∫ R n ∣ g ( v ) ∣ d v = ‖ f ‖ 1 ‖ g ‖ 1 .

This implies ‖ f ∗ 1 Φ α g ‖ 1 ≤ N α ‖ f ‖ 1 ‖ g ‖ 1 .

Step 2: X has no unit. For the briefness of the proof, let us use the common symbol ∗ for all the convolutions exhibited in (11)–(24) and perform the proof just for these cases. Suppose that there exists an element e ∈ X such that f = f ∗ e = e ∗ f for every f ∈ X . Choosing δ ( x ) ≔ e − 1 2 ∣ x ∣ 2 we would obtain ℋ 1 δ = ℋ 2 δ = δ . By δ = δ ∗ e = e ∗ δ and the factorization identities of those convolutions, we would obtain ℋ i ( δ ) = Φ α ℋ k ( δ ) ℋ ℓ ( e ) , where ℋ i , ℋ k , ℋ ℓ ∈ { ℋ 1 , ℋ 2 } (note that it may be ℋ i = ℋ k = ℋ ℓ = ℋ 1 , etc.). We then would have δ = Φ α δ ℋ ℓ ( e ) . Since δ ( x ) ≠ 0 for every x ∈ R n , and we derive that Φ α ( x ) ( ℋ ℓ ( e ) ) ( x ) = 1 for every x ∈ R n . However, this contradicts the fact that

lim ∣ x ∣ → ∞ Φ α ( x ) ( ℋ ℓ ( e ) ) ( x ) = 0 ,

which is deduced from the Riemann-Lebesgue lemma. Hence, X has no unit.

Evidently, convolutions (11), (14), (15), (18), (19), (22), (23), and (26) are commutative. It suffices to prove the non-commutativity for (13), as that for (12), (17), (16), (21), (20), (25), and (24) can be proved analogously.

By Theorem 1 and Φ 0 ( x ) = e − 1 2 ∣ x ∣ 2 , Φ 1 ( x ) = − 2 x 1 e − 1 2 ∣ x ∣ 2 , we have

ℋ 1 ( Φ 0 ∗ Φ α ( 3 ) Φ 1 ) = Φ α Φ 0 Φ 1 , ℋ 1 ( Φ 1 ∗ Φ α ( 3 ) Φ 0 ) = − Φ α Φ 0 Φ 1 .

This implies that convolution (13) is non-commutative. The theorem is proved.□

It is well-known that the Wiener algebra W is defined as the Fourier transform of the Banach space L 1 ( R n ) , i.e., the image by the Fourier transform of L 1 ( R n ) . For the Hartley one, let us write W ≔ { ℋ 1 ( f ) : f ∈ L 1 ( R n ) } endowed with the norm ‖ ℋ 1 ( f ) ‖ W ≔ ‖ f ‖ 1 . Then, W can also be called the Wiener algebra associated with the Hartley transform.

Theorem 10

W , equipped with the pointwise multiplication of functions (and the just presented norm), is a commutative normed algebra.

The proof of this theorem is very simple, by using convolution (5) with the special case of h = 0 . Indeed, it suffices to prove that the usual multiplication is closed in W . Suppose F , G ∈ W . There exist f , g ∈ L 1 ( R n ) such that F = ℋ 1 f and G = ℋ 1 g . Having in mind the factorization identity of convolution (5) (with h = 0 ), it follows F ( x ) G ( x ) = ( ℋ 1 f ) ( x ) ( ℋ 1 g ) ( x ) = ℋ 1 ( f ∗ g ) ( x ) . As Theorem 6 indicates, f ∗ g ∈ L 1 ( R n ) . This implies F ⋅ G ∈ W . The commutativity of the pointwise multiplication in W is obvious, also due to the commutativity of the convolution multiplication: f ∗ g = g ∗ f .

4.2 Convolution-type integral equations

We can state that for each one of the proposed convolutions from (5) to (8) and from (11) to (26), a remarkable number of the corresponding integral equations can be solved explicitly. Namely, using the notations ⊛ and ⋆ used in Section 3.2, there are two typical classes of such convolution integral equations:

(29) λ φ ( x ) + ( k ⊛ φ ) ( x ) = p ( x ) ,

(30) λ φ ( x ) + ( k ⋆ φ ) ( x ) = p ( x ) .

In the above equations, the complex number λ together with the two L 1 -functions k and p are given, and the unknown function φ is assumed to be in the Banach space L 1 ( R n ) . The main difference between those classes of the integral equations can be classified as follows: the second term on the left-hand side of (29) is a single integration, in contrast that of (30) is a double integration in which there is an essential contribution of the Hermite function Φ α for some α ∈ N n .

Taking into consideration convolution (21), we consider the following integral equation in L 1 ( R n ) :

(31) λ φ ( x ) + ( k ∗ 3 Φ α φ ) ( x ) = p ( x ) ,

where λ ∈ C is given, and k and p are given in L 1 ( R n ) . Let us investigate equation (31) in detail by means of our proposed approach and omit other equations for saving space, since the others in (29)–(30) can be treated in the same way.

For shortness of notation, put G ( x ) ≔ λ + Φ α ( x ) ( ℋ 2 k ) ( x ) . By (2), the function Φ α ( x ) ( ℋ 2 k ) ( x ) is uniformly continuous and bounded on whole the space R n with sup x ∈ R n ∣ Φ α ( x ) ( ℋ 2 k ) ( x ) ∣ < ∞ , and lim x → ∞ ∣ G ( x ) ∣ = λ .

Theorem 11

If equation (31) has a solution in L 1 ( R n ) , then the necessary condition is that ( ℋ 1 p ) ( x ) = 0 whenever G ( x ) = 0 . Assume that this condition is fulfilled and ℋ 1 p ⁄ G ∈ L 1 ( R n ) . Then, the unique L 1 -solution is given by
(32) φ ( x ) ≔ ℋ 1 ( ℋ 1 p ⁄ G ) ( x ) a . e . ( for a l m o s t e v e r y x ∈ R n ) .
If G ( x ) ≠ 0 for every x ∈ R n and ℋ 1 p ∈ L 1 ( R n ) , then equation (31) is solvable in L 1 ( R n ) , where the solution is given by formula (32).
Particularly, if ∣ λ ∣ > max x ∈ R n { ∣ Φ α ( x ) ( ℋ 2 k ) ( x ) ∣ } , then equation (31) is solvable in L 1 ( R n ) , and its solution can be expressed in a series form.

Proof

(1) Note that two very different cases of λ = 0 and λ ≠ 0 are being considered simultaneously in this situation. Suppose φ 0 ∈ L 1 ( R n ) is an L 1 -solution of (31). Applying ℋ 1 to both sides of (31) yields G ( x ) ( ℋ 1 φ 0 ) ( x ) = ( ℋ 1 p ) ( x ) . Since both sides of this identity define the uniformly continuous functions on R n , we derive ( ℋ 1 p ) ( x ) = 0 , provided G ( x ) = 0 . Moreover, by the assumption ℋ 1 p ⁄ G ∈ L 1 ( R n ) we can apply the inversion theorem to obtain φ 0 ( x ) = ℋ 1 ( ℋ 1 p ⁄ G ) ( x ) (a.e). Conversely, using the inversion theorem for ℋ 1 , it is easy to prove that the function given by (32) satisfies (31) for almost every x ∈ R n , provided ℋ 1 p ⁄ G ∈ L 1 ( R n ) .

(2) Since the Hermite function has some zero-points, ∣ λ ∣ > 0 . Arguing similar to the above, we see that if φ 0 is a solution of (31), then ( ℋ 1 φ 0 ) ( x ) = ( ℋ 1 p ) ( x ) ⁄ G ( x ) . Since G ( x ) is uniformly continuous and bounded on R n with lim x → ∞ ∣ G ( x ) ∣ = λ ≠ 0 , we deduce so the function G ( x ) − 1 is, by which M < ∣ G ( x ) ∣ − 1 < N for some M , N > 0 . Therefore, ℋ 1 p ⁄ G ∈ L 1 ( R n ) if and only if so ℋ 1 p is. We now meet the case of Item (1) proved above.

(3) The assumption implies λ ≠ 0 . Applying ℋ 1 to both sides of (31), we obtain

(33) ( ℋ 1 φ ) ( x ) = ( ℋ 1 p ) ( x ) λ 1 − δ ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) λ + δ ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) ,

where δ = 1 or − 1 with r = 0 , 3 or r = 1 , 2 , respectively (by Theorem 1). We set

(34) ( f ⊙ g ) ( x ) ≔ 1 2 ( 2 π ) n 2 ∫ R n [ f ( x − u ) + f ( x + u ) + f ( − x + u ) − f ( − x − u ) ] g ( u ) d u .

Actually, f ⊙ g is a special case of convolution (5) when h = 0 . Consider G ( z ) ≔ z ( λ + z ) − 1 which is an analytic function on the domain O λ ≔ { z ∈ C : ∣ z ∣ < ∣ λ ∣ } . By Theorem 1 and the assumption, it yields

(35) ∣ δ ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) ∣ = ∣ Φ α ( x ) ( ℋ 2 k ) ( x ) ∣ < ∣ λ ∣ for all x ∈ R n .

We deduce that G ( z ) is analytic along the curve γ ≔ G ( δ ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) ) ∈ C . Since G ( 0 ) = 0 , we rely on the Wiener-Levy theorem for the Fourier transform (e.g., see [14]) to derive that there is a function P ∈ L 1 ( R n ) such that

G ( δ ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) ) = δ ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) λ + δ ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) = ( ℱ P ) ( x ) = 1 2 ( ℋ 1 + ℋ 2 ) P − i 2 ( ℋ 1 − ℋ 2 ) P = ℋ 2 1 + i 2 P + 1 − i 2 P ˇ ≔ ( ℋ 2 Q ) ( x ) ,

where Q ( x ) ≔ 1 + i 2 P ( x ) + 1 − i 2 P ˇ ( x ) . Combining this and (33) gives

( ℋ 1 φ ) ( x ) = ( ℋ 1 p ) ( x ) λ [ 1 − ( ℋ 2 Q ) ( x ) ] = ℋ 1 p λ − p ⊙ Q ˇ .

Thanks to the uniqueness theorem for ℋ 1 , we obtain

(36) φ ( x ) = p ( x ) λ − ( p ⊙ Q ˇ ) ( x ) ( a.e. ) .

Conversely, it is easy to prove that φ given by (36) fulfills the equation (a.e.), by a simple calculation.

Actually, the implicit solution (36) can be expressed in a series. Indeed, write

( Φ α ⊙ k ) ⊙ ( j ) ≔ ( Φ α ⊙ k ) ⊙ … ⊙ ( Φ α ⊙ k ) ︸ j times ( Φ α ⊙ k ) , j ≥ 1 .

Since Φ α is a Schwartz function (a rapidly decreasing smooth one), we can prove inductively that

( Φ α ⊙ k ) ⊙ ( j + 1 ) ∈ L 1 ( R n ) , [ ℋ 2 ( Φ α ⊙ k ) ] j + 1 ∈ L 1 ( R n ) , and [ ℋ 2 ( Φ α ⊙ k ) ] j + 1 = ℋ 2 ( Φ α ⊙ k ) ⊙ ( j + 1 ) ( x ) for all j ≥ 0 .

By the facts: G ( z ) is analytic on O λ , the functions [ ℋ 2 ( Φ α ⊙ k ) ( x ) ] j + 1 with j ≥ 0 are uniformly continuous on R n by the Lebesgue-Riemann lemma (see Theorem 7.5 in [6]), and by (34)–(35), we have a uniformly and absolutely convergent Taylor expansion

(37) G ( δ ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) ) = δ ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) λ + δ ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) = 1 λ δ ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) 1 + δ ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) λ = ∑ j ≥ 0 ( − 1 ) j [ δ ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) ] j + 1 λ j + 1 = ∑ j ≥ 0 ( − 1 ) j δ j + 1 [ ℋ 2 ( Φ α ⊙ k ) ( x ) ] j + 1 λ j + 1 = ℋ 2 ∑ j ≥ 0 ( − 1 ) j δ j + 1 ( Φ α ⊙ k ) ⊙ ( j + 1 ) ( x ) λ j + 1 ( for all x ∈ R n ) .

In the last equality, one can interchange the integral transform ℋ 2 and the infinite summation, since the series is uniformly and absolutely convergent, each term of the series is a Lebesgue integral, and

‖ ℋ 2 ( Φ α ⊙ k ) ‖ ∞ = ‖ ℋ 2 Φ α . ℋ 2 k ‖ ∞ = ‖ Φ α . ℋ 2 k ‖ ∞ < λ .

The uniform and absolute convergence of series (37) means that G ( z ) is analytic along the curve γ as previously emphasized, and the Wiener-Levy theorem can be applied here. The desired conclusion follows at once by inserting the series (37) into (33). From our point of view, the above arguments on the power series work, because ( ℋ 2 Φ α ) ( x ) ( ℋ 2 k ) ( x ) is an L 1 -function, in contrast ( ℋ 2 f ) ( x ) ( ℋ 2 k ) ( x ) is not so for the general case that f , k ∈ L 1 ( R n ) . The theorem is proved.□

Thus, infinitely many integral equations generated by the Hermite functions can be solved effectively by the introduced convolutions (further details, see [2]).

Remark 3

(a) We here meet the characteristic fact of the Hartley transform as: by the assumption ∣ λ ∣ > max x ∈ R n { ∣ Φ α ( x ) ( ℋ 2 k ) ( x ) ∣ } in Item (3), the solution can be expressed in a series form, which is different from that given by (32) derived from the simple assumption G ( x ) ≠ 0 in Item (2). For instance, if λ ∈ C \ R (which is a complex number) and k given is real-valued, then G ( x ) ≠ 0 for every x ∈ R n , since Φ α ( x ) ( ℋ 2 k ) ( x ) ∈ R n . Hence, if ∣ λ ∣ = ∣ Φ α ( x ) ( ℋ 2 k ) ( x ) ∣ for some x ∈ R n , then series (37) may fail. About the series solution, we refer the reader to the book [15].

(b) Since the mentioned convolutions are still valid for f , g ∈ L 2 ( R n ) with the same properties: the respective norm inequalities and factorizations (but for almost every x ), one can obtain similar results for equations (29)–(30) by some appropriate adaptation. Namely, all assumptions should hold for almost every x . We here omit the formulation of the cases.

(c) The sufficient condition in Theorem 11 is realistic. Indeed, if ℋ 1 p ∈ L 1 ( R n ) and if λ given is large enough, then ℋ 1 p ⁄ G ∈ L 1 ( R n ) . This is absolutely suitable with the well-known knowledge of the Neumann series and the particular series solution as shown in Item (3). This can be seen as a new approach compared with the other published works on convolution integral equations (see [1, 3,9,16,17]).

Observe that the integral equation (29) contains the two Toeplitz and Hankel kernels. For the Wiener-Hopf equations and the Wiener-Hopf plus Hankel equations, we refer the reader to the works [18–20].

# These authors contributed equally to this work.

Funding information: This work is partially supported by the project entitled The Fourier oscillatory integral transform and some relative issues coded QG.22.47, Viet Nam National University, Ha Noi, Viet Nam.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: Data sharing is not applicable to the article as no datasets were generated or analyzed during the current study.

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Received: 2023-10-05

Revised: 2024-06-09

Accepted: 2024-07-02

Published Online: 2025-02-26

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2024-0087

Keywords for this article

Hartley’s integral transform; generalized convolution; Hermite’s functions; Wiener’s algebra

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