f t(y)dy f h(x)g(xy) dx fk 4 a. «..

Transcription

1 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,

2 386 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

3 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*

4 388 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

5 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/

6 390 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 E. C. Titchmarsh, Intrductin t the thery f Furier integrals, Oxfrd, 1937, p. 93. Nrthwestern University