A Note on Integral Transformations

Let k be a non‐negative integer, and S k the class of all real‐valued functions s defined on [1,∞) such that every sɛS k satisfies the conditions: s k (t) is absolutely continuous in every finite interval of [1,∞), and∫ 1 ∞| s ( t ) | d t 6 ∞Let BV [1, ∞) denote the class of all real functions of bounded variation on[1, ∞). The following problem ( P ) arises in Absolute Convergence Factor problems:What conditions are necessary and sufficient in order thatA ( x ) = ∫ 1 ∞ a ( x , t ) s ( t ) d tshould converge for every x ⩾1 and belong to BV [1,∞), whenever sɛS k ? In [1], we gave conditions forB ( x ) = ∫ 1 ∞ a ( x , t ) s ( t ) d tto exist for every x ⩾1 and satisfy∫ 1 ∞| B ( x ) | d x < ∞ ,whenever sɛS k , under some restrictions on b(x, t) ; [1; Theorem 4]. Under somewhat severe restrictions on a(x, t) , it is possible to write∫ 1 ∞| d A ( x ) | = ∫ 1 ∞| A ' ( x ) | d x ,A ' ( x ) = ∫ 1 ∞a x ( x , t ) s ( t ) d t ,and find the conditions required in (P) by using (5). But in the theorem that we give here, the restrictions placed on a(x, t) are mild, and are easily satisfied in the applications. The technique used hinges on the following easily proved lemma, which can be usefully applied in problems of this kind. LEMMA. A necessary and sufficient condition for AɛBV[1,∞) is that there exist aconstant M, such that | B(z, y) | ⩽ M for all z ɛ X and all y ɛ Y where X is the class of all increasing sequences z, with z 0 = 1, Y is the class of all finite sets of non‐negative integers, andB ( z , y ) = ∑ r ɛ y{ A ( Z r + 1 ) ‐ A ( Z r )} .When (6) is satisfied∫ 1 ∞| d A ( x ) | ⩽ 2 M .THEOREM. Suppose that ( I ) a(x, t) is measurable in [1, ∞], for every x ⩾ 1, and (II)V ( t ) = ∫ 1 ∞| d x a ( x , t ) | ɛ L ( 1 , T ) f o r e v e r y T = 1.Then necessary and sufficient conditions for (2)to converge for every x ⩾ 1 and A ɛ BV, whenever s ɛ S k , are that (i) for every x ⩾ 1, T = 1∫ 1 T| a ( x , t ) | d t < ∞ ,(ii) for every x ⩾ 1, there exist K 0 , T 0 such that |a(x, t)| ⩽ K 0 for almost all t ⩾T 0 . (11) (iii) there exist K, T' such that v(t) ⩽ K for almost all t ⩾ T′. (12) Proof. It is trivial that (i) and (ii) are sufficient for the convergence of (2) whenever s ɛ S k If (iii) holds, andA 1 ( x ) = ∫ 1 T a ( x , t ) s ( t ) d t ,A 2 ( x ) = ∫ T ∞ a ( x , t ) s ( t ) d t ,then| ∑ r ɛ y{ A 2 ( Z r + 1 ) ‐ A 2 ( Z r ) }| ⩽ K ∫ T ∞| s ( t ) | d t ,1 which is independent of z ɛ X and yeY. Also| ∑ r ɛ y{ A 2 ( z r + 1 ) ‐ A 2 ( Z r ) }| ⩽ K ∫ T ∞| s ( t ) | d t ,which is independent of z ɛ X and y ɛ Y. Hence, by the lemma, A ɛ BV[1,α). Conversely, (i) and (ii) are necessary for the convergence of (2) whenever s ɛ S k , see [ 1 ; Lemma 1]. Next we have |B(z, y)| ⩽ M for all zɛX, yɛY, whenever sɛS k , (16) where M is indepedent of z and y, but may depend on s, i.e.| ∫ 1 ∞ b ( z , y , t ) s ( t ) d t | ⩽ M forallz ɛ Y , whenevers ɛ S k ,(17) whereb ( z , y , t ) = ∑ r ɛ y{ a ( z r + 1 , t ) ‐ a ( Z r , t ) } .Now, for every z ɛ X, y ɛ Y,∫ 1 T| b ( z , y , t ) | d t ⩽ ∫ 1 T V ( t ) d t = M T ,which is independent of z and y, and hence † there exist K, T' such that, for all z ɛ X and all y ɛ Y, |b(z, y, t)|⩽½K for almost all t ⩾ T'. Since X is independent of z and y, it follows from the lemma that V(t) ⩽ K for almost all t ⩾ T. Thus (iii) is necessary.