f t(y)dy f h(x)g(xy) dx fk 4 a. «..

CERTAIN OPERATORS AND FOURIER TRANSFORMS ON L2 RICHARD R. GOLDBERG 1. Intrductin. A well knwn therem f Titchmarsh [2] states that if fel2(0, ) and if g is the Furier csine transfrm f/, then G(x)=x~1Jx0g(y)dy is the csine transfrm f F(y)=J (f(x)/x)dx (bth Fand G being in L2). The same result applies t sine transfrms. In this paper we prve the fllwing result fr a wide class f functins \p: If g is the csine transfrm f fel2 then /. t(y/x)g(y)dy 0 is the csine transfrm f F(y) =J^x~1ip(y/x)f(x)dx. (The same result again applies t sine transfrms.) The therem f Titchmarsh stated abve is the special case f ur result in which i/' is the characteristic functin f (0, 1). We shall prve the abve result by develping prperties f a certain class f bunded peratrs n L2. Finally we shall cnstruct a class f self-adjint bunded peratrs which cmmute with the Furier csine (r sine) transfrm. 2. Preliminaries. We shall dente Lp(0, ) by Lp, (p = l, 2) with the Lp nrm / p defined as usual as (J\f(x)\ "dx)llp. If T is a linear transfrmatin n L2 int itself then 7 is defined as iub rg M g 8. eel1 We shall make use f the Schwarz Inequality: if/, gel2 then fgel' and /g i^ / j g *, and its Cnverse: if fr each hel2, \\Gh\\iSA\\h\\2 then GEL2 and \\G\\2fkA. 3. A Class f bunded peratrs n L2. Lemma. If\p(y)^0 Prf. Fr y > 0 and J^(y)y~1,2dy = A < then fr any g, hel2 f t(y)dy f h(x)g(xy) dx fk 4 a. «.. J 0 J 0 Received by the editrs Octber 30, 1958. 385

386 R. R. GOLDBERG [June /, 1 /* I gixy) \2dx = I y J Therefre, by the Schwarz Inequality, I g(x) \2dx. Hence I h(x)g(xy) dx S \\h\w \\g\\i. /i I, I, y112 1 I, /, a r* 00.» 00 t(y)dy I /z(z)g(ry) dx S \\h\\i\\g\\2 J ^(y)y-1/2(fy= 4 /s 2y 2. The first part f the next therem was prved in a much different frm by Schur [l]. Therem 1. Let \p be nn-negative with J^(y)y~ll2dy = A <. Let \p define the linear transfrmatin T n L2 as fllws: Tg = G means G(x) = I if/[ ) g(y)dy x J \ x / (g E L2). Then T is a bunded peratr n L2 and T\\ SA. Furthermre if we define T* as J,= 1 / x\ 4>[ )f(y)dy (}EL2), y \y / then T* is the adjint f T and s T*\\ SA. Prf. We shall first shw that GEL2 and that G[ 2g^ g 2. Fr any hel2 we have /' i I G(x)h(x) i dx S I C -L L \ Kx) I dx j r + I / ) y \, g(y),\ dy J x J \xi /» 00 (% CO /» 00 /» I A(#) rfa; I ^(y) g(xy) \ dy = j ^(y)dy I A(a;)g(xy) dz. The last iterated integral cnverges (abslutely) by the lemma justifying the change in rder f integratin. Thus by the lemma \\Gh\\iS A\\g\\2\\h\\2. The cnverse f the Schwarz Inequality thus implies that GEL2 and c7 2 S A\\g\\2. Since G=Tg this shws that 7g 2^.4 g [2 fr all g A2 and s T is a

1959I CERTAIN OPERATORS AND FOURIER TRANSFORMS ON U 387 bunded linear transfrmatin n L2 int itself (bunded peratr) and \\t\\ SA. The first part f the therem is thus established. Nw chse any/, gel2. Then with (a, b) defined as f^a(x)b(x)dx, the usual inner prduct in L2, we have (1) (Tg,/) = rf dx["+ (-) g(y)dy, J X J \ x / and (2) (g, T*f) = f g(y)dy [ - <P (~)f(x)dx. J J X \ X / The integrals in (1) and (2) cnverge abslutely by the lemma and hence are equal. Thus (Tg,f) = (g,t*f) which, by definitin f adjint, shws that T* is the adjint f T. Finally, since [F* = 7], we have F* -fka and the prf is cmplete. In passing we remark that the integrals defining F and G in the statement f Therem 1 exist nly almst everywhere. 4. Relatin t Furier transfrms. We shall write Uf = g if g is the Furier csine transfrm f/. Thus if Uf=g then /2\i/2 r-r g(y) = l.i.m. ( /(/) cs ytdt fel2, B->» \ir / J where l.i.m. stands fr limit in the L2 mean. Furthermre g(y) = ( ) J f(t) cs ytdt if/ E L' C\ L2, the abve hlding fr almst all y. It is well knwn that ii fel2 and Uf = g then gel2 and Ug=f. Mrever U is a self-ad jint peratr (U= U*). It will be readily verified that everything we prve abut the Furier csine transfrm U will als hld fr the Furier sine transfrm. Therem 2. Iftyis nn-negative,ipel',and f$(y)y~ll2dy< then where T, T* are as in Therem 1. Prf. It is sufficient t prve TV = UT*

388 R. R. GOLDBERG [June TUf = UT*f fr felt\l2 since L'C\L2 is dense in L2 and T, T*, U are cntinuus n L2. Accrdingly, chse any felt\l2 and let g = Uf, G=Tg, F = T*f. We need nly shw that G= UF. With c= (2/tt)112 we have G(*) =- - f *( )«(y)<*y = f *( )dy f f(t) cs ytdl x J \x / x J \x / J c rx rx /y\ = I f(t)dl rp I ) cs ytdy x J J \xj f(t)dt I 4/(y) cs xyfcfy /I CO /» CO * = c/ t*/ Kt")C0S^ //* cs xydy I 1 ^ I /y\ Jf(t)dt A(y) cs xydy. The integral in (3) cnverges abslutely since \f/,fel'. This justifies the changes in rder f integratin and als shws that FEL'. Thus G = UF which is what we wished t shw. Remark. If we set then if G = Tg, F= T*f we have My) = l, ^ysu My) =, y> l, i rx rxfix) Gix) = - giy)dy, Fiy) = J-^dx. X J 0 J y X Frm Therem 2 we see that if g= Uf then G= UF. This is the therem f Titchmarsh mentined in the intrductin. 5. A mre general result. We may drp the hypthesis that^ga' in Therem 2. T see this chse any nn-negative \p such that f"ipiy)y-1,2dy = A<<x> (but nt necessarily such that ypel'). Fr» = 1, 2, define

19591 CERTAIN OPERATORS AND FOURIER TRANSFORMS ON L> 389 tn(y) = $(y), l/nfkyfk n; $n(y) =0, 0 fk y < 1/n; n < y <. Then fg\(/n(y)y~ll2dy = An<'x and, by the Lebesgue cnvergence therem, lim An = A. n >«Mrever if T, Tn are defined by \[/, \p as in Therem 1 then T Tn is defined by ip \p» and thus, by Therem 1, (4) \\T - Tn\\ fk A - An^0 as «-+«>. But \pn beys the hyptheses f Therem 2. Hence Letting TnU = VT*. n > and using (4) we have TV = VT*. We have thus shwn that TU= UT* even fr T defined by a nnnegative xp fr which we assume nly f^p(y)y~ll2dy<. We nw state this in detail. Therem 3. Let \p be nn-negative with fi/(y)y~llidy<. Define the linear transfrmatin T n L2 as fllws: 1 r /y\ Tg = G means G(x) = I ^ I I g(y)dy. x J a \ x I Then T is a bunded peratr n L2. Mrever if T* is the adjint f T and U is the Furier csine transfrm then TV = VT*. Remark. This therem, translated back int classical terminlgy, is the generalizatin f the therem f Titchmarsh stated in the intrductin. 6. Operatrs that cmmute with the csine transfrm. In rder fr T t be self-adjint (T= T*) we see frm the definitin f T, T* in Therem 1 that it is sufficient t have r ~ *( ) = *( )» 0<X,y<c; x \x/ y \y/

390 R. R. GOLDBERG (5) +(y) = im V 0 <y < c. y \y / Suppse then that we have a nn-negative functin xp defined n (0, 1 ] such that (6) J <P(y)y-1l2dy < J and define xp(y) fr y>l Then if yi < 1 we have by *(y) = * ( ), 1 < 3- < c. y \y/ i> ( ) = yna(yi) s that^(y) = (l/y)^(l/y) fr all y>0 (i.e. (5) hlds). Frm (5) and (6) we have f Hy)y~ll2dy = f yp ( j y~3l2dy = f \P(y)y-V2dy <. This and (6) imply \p(y)y-ll2dy < c s that the hyptheses f Therem 3 hld. Frm (5) we cnclude that the T defined by xp is self-adjint s that we have the fllwing cnsequence f Therem 3. Therem 4. Let xp be nn-negative n (0, 1 ] with flxp(y)y~ll2dy < c. Define xp(y) = (l/y)^(l/y) fr y > 1. Then if T is as in Therem 1 TU = UT. In ther wrds T cmmutes with the Furier csine transfrm. References 1. I. Schur, Bemerkung zur Therie der beschrdnkten Bilinear fr men mil unendlich vielen Verdnderlichen, J. Reine Angew. Math. vl. 140 (1911) p. 23. 2. E. C. Titchmarsh, Intrductin t the thery f Furier integrals, Oxfrd, 1937, p. 93. Nrthwestern University