New Convolutions for an Oscillatory Integral Operator on the Half-line

The main purpose of this work is to present three new convolutions for oscillatory integral operators defined on the positive half-line and in the framework of L1 Lebesgue spaces. Therefore, such new functions introduced by the new convolutions will have factorization properties when considering the oscillatory integral operators under consideration. Moreover, some fundamental and operational properties of the mentioned integral operator are also studied in the first part of the paper. INTRODUCTION Convolutions are very important operations in Functional Analysis, Integral Operators and Integral Transforms and in their applications. In fact, probably the most important property of a convolution – besides generating new functions upon previously given two or more functions – is to satisfy a factorization property which is typically intrinsically associated with one or more than one integral operators. In most of the cases, such factorization property is fundamental to solve consequent integral equations which can be characterized by those convolutions. The properties of the convolutions may also be very diverse up to the concrete definition of each convolution (cf. [1, 2, 7, 10, 15, 23, 24, 25]). The integral operators which are associated with each one of the know convolutions may have also a very distinct nature (see, e.g., [3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 21, 22]). Moreover, the range of possible applications of convolutions to other sciences is also wide (cf. [16, 17, 18, 19, 20, 26]). Here, we will introduce some new convolutions associated with oscillatory integral operators on the positive half-line. We start by defining those operators. Let L(R+) denote the Lebesgue space of all absolutely integrable complex-valued functions on R+. For any function f ∈ L(R+), we define the integral operator S ±,± by (S ±,± f )(x) = 1 √ 2π ∫ +∞ 0 (± cos(xy) ± sin(xy)) f (y)dy, x ∈ R+, (1) respectively. The main aim of this work is to introduce new convolutions for the operators just presented. That is to say, in the present case, to introduce new functions, generated by two previously given functions, which will exhibit certain factorization identities when considering the above operators applied to all those functions. This will achieved in the final section of this work. FUNDAMENTAL PROPERTIES OF THE INTEGRAL OPERATOR S +,+ In the present section, we will be concentrated in studying some fundamental properties of the operator S +,+: this will be the case of some mapping properties. The Integral Operator S +,+ in the Framework of the Schwartz Space Let us denote by S(R+) the Schwartz space on R+, and by C∞ 0 (R+) the Banach space of all continuous and infinitely differentiable functions on R+ that vanish at infinity. In this subsection, the operator S +,+ is studied in the framework of the Schwartz space. We prove that if f ∈ S(R+) then, S +,+ f ∈ S(R+). The following lemma shows some basic properties of the integral operator S +,+ in the Schwartz space S(R+). Lemma 1 Suppose that f ∈ S(R+). Then, (i) S +,+ f is continuous; (ii) ( dk dxk S +,+ f ) (x) =  ik+3S +,−(y f )(x), if k is odd ikS +,+(y f )(x), if k is even , k ∈ N; (iii) S +,+ ( dk dyk (y α f ) ) (x) =  ik+1xkS +,−(y f )(x), if k is odd ik xkS +,+(y f )(x), if k is even , k ∈ N, α ∈ Z; (iv) ||S +,+ f ||∞ ≤ 1 π || f ||1. Proof. (i) For any x ∈ R+, we have lim h→0 (S +,+ f )(x + h) = lim h→0 1 √ 2π ∫ ∞ 0 [cos((x + h)y) + sin((x + h)y)] f (y)dy = 1 √ 2π ∫ ∞ 0 f (y) lim h→0 [cos((x + h)y) + sin((x + h)y)]dy = (S +,+ f )(x). (ii) Let us compute the first derivative of S +,+ f . We have ( d dx S +,+ f ) (x) = 1 √ 2π d dx ∫ ∞ 0 (cos(xy) + sin(xy)) f (y)dy = 1 √ 2π ∫ ∞ 0 f (y) d dx (cos(xy) + sin(xy)) dy = 1 √ 2π ∫ ∞ 0 y f (y) (cos(xy) − sin(xy)) dy = S +,−(y f )(x), where the interchange of the derivative and the integral is valid since f ∈ S(R+). Applying the same procedure, we obtain that d2 dx2 (S +,+ f )(x) = −[S +,+(y f )](x), d3 dx3 (S +,+ f )(x) = −[S +,−(y f )](x), d4 dx4 (S +,+ f )(x) = [S +,+(y f )](x), and the result follows by induction. (iii) Integrating by parts and having into consideration that f ∈ S(R+), we obtain S +,+ ( d dy (y f ) ) (x) = 1 √ 2π ∫ ∞ 0 d dy (y f (y)) (cos(xy) + sin(xy))dy = 1 √ 2π y f (y)(cos(xy) + sin(xy))|0 − 1 √ 2π ∫ ∞ 0 y f (y)(−x sin(xy) + x cos(xy))dy = − 1 √ 2π x ∫ ∞ 0 y f (y)(cos(xy) − sin(xy))dy = −xS +,− (y f ) (x). With same procedure, we deduce that S +,+ ( d2 dy2 (y f ) ) (x) = −x2S +,+ (y f ) (x) S +,+ ( d3 dy3 (y f ) ) (x) = x3S +,− (y f ) (x) S +,+ ( d4 dy4 (y f ) ) (x) = −x4S +,+ (y f ) (x), and the result follows by induction. (iv) From a direct computation, it follows ||S +,+ f ||∞ = sup x∈R+ ∣∣∣∣∣ 1 2π ∫ +∞ 0 (cos(xy) + sin(xy)) f (y)dy ∣∣∣∣∣ ≤ 1 √ 2π sup x∈R+ ∫ +∞ 0 |(cos(xy) + sin(xy))| | f (y)|dy ≤ 1 √ 2π sup x∈R+ ∫ +∞ 0 √ 2| f (y)|dy


INTRODUCTION
Convolutions are very important operations in Functional Analysis, Integral Operators and Integral Transforms and in their applications. In fact, probably the most important property of a convolution -besides generating new functions upon previously given two or more functions -is to satisfy a factorization property which is typically intrinsically associated with one or more than one integral operators. In most of the cases, such factorization property is fundamental to solve consequent integral equations which can be characterized by those convolutions. The properties of the convolutions may also be very diverse up to the concrete definition of each convolution (cf. [1,2,7,10,15,23,24,25]). The integral operators which are associated with each one of the know convolutions may have also a very distinct nature (see, e.g., [3,4,5,6,8,9,11,12,13,14,21,22]). Moreover, the range of possible applications of convolutions to other sciences is also wide (cf. [16,17,18,19,20,26]). Here, we will introduce some new convolutions associated with oscillatory integral operators on the positive half-line. We start by defining those operators.
Let L 1 (R + ) denote the Lebesgue space of all absolutely integrable complex-valued functions on R + . For any function f ∈ L 1 (R + ), we define the integral operator S ±,± by respectively.
The main aim of this work is to introduce new convolutions for the operators just presented. That is to say, in the present case, to introduce new functions, generated by two previously given functions, which will exhibit certain factorization identities when considering the above operators applied to all those functions. This will achieved in the final section of this work.

FUNDAMENTAL PROPERTIES OF THE INTEGRAL OPERATOR S +,+
In the present section, we will be concentrated in studying some fundamental properties of the operator S +,+ : this will be the case of some mapping properties.
The Integral Operator S +,+ in the Framework of the Schwartz Space Let us denote by S(R + ) the Schwartz space on R + , and by C ∞ 0 (R + ) the Banach space of all continuous and infinitely differentiable functions on R + that vanish at infinity.
In this subsection, the operator S +,+ is studied in the framework of the Schwartz space. We prove that if f ∈ S(R + ) then, S +,+ f ∈ S(R + ). The following lemma shows some basic properties of the integral operator S +,+ in the Schwartz space S(R + ).
(i) For any x ∈ R + , we have (ii) Let us compute the first derivative of S +,+ f . We have (cos(xy) + sin(xy)) f (y)dy where the interchange of the derivative and the integral is valid since f ∈ S(R + ). Applying the same procedure, we obtain that (iii) Integrating by parts and having into consideration that f ∈ S(R + ), we obtain With same procedure, we deduce that and the result follows by induction. (iv) From a direct computation, it follows Using the previous results, we conclude, in the following theorem, that S +,+ f ∈ S(R + ) for f ∈ S(R + ).

Proof.
From Lemma 1, S +,+ f is continuous and for any pairs of integers k and α, we have that Therefore, and since f ∈ S(R + ), we have Thus, S +,+ f ∈ S(R + ).
The Integral Operator S +,+ in the Framework of the L 1 (R + ) Space We will now consider the integral operator S +,+ within the framework of the L 1 (R + ) space. With this aim, let us first recall two known results which will be useful in this study.
Theorem 3 (Lebesgue dominated convergence) Let ( f n ) n∈N be a sequence of Lebesgue measurable functions defined on a Lebesgue measurable set E such that the pointwise limit f (x) = lim n→∞ f n (x) exists. Assume that there is an integrable g : for each x ∈ R + . Then, f is integrable as is f n for each n and where a and b are real numbers, we have Theorem 5 Let f ∈ L 1 (R + ). Then, S +,+ f is a uniformly continuous function on R + and

NEW CONVOLUTIONS
In this section, we introduce new convolutions (i) (i = 1, 2, 3) for the operators S +,+ and S − +,+ defined in the domain L 1 (R + ) and we prove that these convolutions have a corresponding multiplicative factorization property.
In what follows, let us consider the Heaviside and the signal functions defined by respectively. For f ∈ L 1 (R), let us consider the Hartley transform (cos(xy) + sin(xy)) f (y)dy.

Definition 1
For any f and g in L 1 (R + ), we define three new convolutions (i) (with i = 1, 2, 3) by where u ∈ R and 0 are the zero extension operators from R ± into R (depending on the correspondig extension region which is clear in each application of the extension operator).
From the last identities, we obtain the following results, just for x ∈ R + : In the same way, we obtain Finally, +H(−u) 0 ( f (| − u − t|) + f (−u + t)))g(t)dt du.
The last theorem exhibits intricate factorization properties, for the new convolutions, which also relate by that identities three different integral operators: S +,+ , S − +,+ and H 1 . The authors intent to continue the present study in a forthcoming paper where some of the potentialities of the three new convolutions here introduced will be applied in different contexts.