Variable convolution operators

We consider first operators on 1.2 (Rft) with kernels of the form i x \

^R3

^-At the end of the section we shall indicate what can be done with the more general form I x y \

^R^j-The setting will be the algebra, which we call 38 {^), of G1 functions from £1 to the algebra S8 of the last section. More exactly, ceg8(Sl) if c is a function from 0. to S8 for which there is a continuous function, denoted by

grad c,

229

230 H A R O L D W I D O M from Q. to ^-vectors with components in 3S^ satisfying

( 6 . 1 ) \W-c{y)-{x-y).gv^c{y)\\^=o[\x-y\)

for \x—y\->o. The subscript indicates that the norm taken in (6. i) is the norm in SS.

In the set of nonnegative elements of S8 with its natural partial ordering, bounded subsets have least upper bounds. We define, for ce^(fi.)

(6.2) c=f.u^b.\c{x)^ ^.^.|grad^)|.

Here |grad c{x) \ is taken to mean the sum of the norms of the components of grad c{x).

The norm in SS(fl) is given by

I k I I^(Q)

⁼

I ^ll^^11^ 1 1 ^

-The set ^(Q) is a Banach algebra under convolution defined by ( q * ^ ) ( x ) = q ( ^ ) * ^ ) .

It is convenient to associate with each CE^(Q) a function of two variables c(x, t) defined everywhere on QxE"; given xeO. let c{x, t) be an everywhere defined function of t (more exactly a constant times 8[t) plus an everywhere defined function of t) rep-resenting the element c(x) of S8. Clearly we have for all x

( 6 . 3 ) \c{x, t) |^), |grad^, t) \^{t)

for almost every t (and for t == o which corresponds to the S summands of the distri-butions c(x)). The gradient, of course, is taken with respect to the variable x.

For any ce^(Q.) we denote by W^c] the integral operator on L^RQ) with kernel ( x \

^R⁵ ^)-Note that with the notation of the last section

^,.)=^)^,.)

where 7 is an ordinary function of two variables. Correspondingly WR^] is multipli-cation by a(^/R) plus the integral operator with kernel

Jx \

^R^J in the classical sense. It follows that

l|WR[c]||^||a[|,+J^^)^||c||^).

In the special case c{x]=f{x}c with ceSS the operator on L^E") with kernel Ax)c(x—_y)

230

has inverse exactly equal to the operator with kernel Ayr'h^-y)

where h is the inverse of c in St. This suggests that in denning the approximate inverse UR^] ofWp[/'] in general, one should replace the first term of (5.14) by

*(t-)

where h is the inverse of c in S9{^1).

In analogy with the last section our basic assumption will be that (6.4) c==Exps, se^{0.).

As with 3§^ this is equivalent to ceSS{^)

? bounded away from o on ^ixE'1

(the circumflex denotes Fourier Transform with respect to the second variable) plus, in dimension one

A arg ?(x, ^) == o for each x.

— oo < S < °o

This follows from the Arens-Calderon form of the Wiener-Levy theorem and the fact that the most general homomorphism of^(d) onto the complex numbers is of the form

c-^^x, S)

for some {x, S^e^xE^ The proof of this is not hard and is left as an exercise for the interested reader.

If (6.4) holds we define the elements h and h^ of ^[0.) in analogy with the last section. The mapping y^y, just as before, is to satisfy (5.1) and (5.2). Then we define V^[c] to be the operator on L^RQ) with kernel

l y \ r i y \ i y - \

(6.5) h [-.x-y - h^ { ^ x - y + t \h^\--,-y+y-t \dt.

\K. / Jt.v{y)>0 V< / \K /

When carrying out the procedure of the last section we encounter new operators which arise because c{x, t) is not independent of x. Given q, c^eS§(^l) we write

VR^I? ^2]

for the integral operator on L^RQ) with kernel

Ki - ^ - ^ -^i - ^ - ^ K2 - ^ - J ^ JRQ\ \K. ; \K /; \K ;

and VR^, c^\ for the operator with kernel given by a similar formula but with absolute value signs around the integrand. Although one such operator will make a significant contribution to the asymptotic formula for the determinant, these operators are all reasonably small in certain senses.

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232 H A R O L D W I D O M

Lemma (6.1). — If c^c^3S(fX) then as R->oo

VR[q, ^-^(i), VH^, ^^(R^2).

If moreover c^eSS then the estimated trace of

WR^IVJ^]

is O^R1-1).

Proof. — It follows from (6.1) and (6.3) that

^ — , x—z

' ( ^ ^ " / ^ i ' ^ " ^ ) ^AR"

( x

¹ !^-^!^-^)

for some constant A. Hence the kernel ofV^q, ^] is at most (6.6) R-^x-^c^x-y)

where ^===Aq*^. (Note that the 8 summand of ^4 plays no role here.) We may write this as

R - -l| ^ — ^ | ^ ( ^ — ^ ) ^ ^ ^ ; ^ _ y ) < , R ^ 4 - R- l| ^ — J;| ^ ( ^ — J;) X { ^ ^ / ) : | x - ^ | > e R } •

The first of these summands is the kernel of convolution with a function whose L^ norm is at most c || 6*4 [[i. Since \x—y\ is at most a constant times R for x,yeRQ,^ the second summand is at most a constant times the kernel of convolution with a function whose L^ norm is

^\^c^dx=o^ as R-^oo. Hence VR^, ^2]= = ooo(I)•

We use the same decomposition to estimate the Hilbert-Schmidt norm. The square of the first summand is

R -2| ^ - ^ |2^ 4 ( ^ - ^ )2^ ^ ^ . . 1 , _ y ^ , R ^ < £ R -l| ^ - ^ | ^ ( x - ^ )2.

This must be integrated over R^ X Ri2. Integration with respect to x gives at most eR.-l(\x\c^x)2dx.

Subsequent integration with respect toy gives at most a constant times sR^¹""¹. Similarly the square of the second summand is at most a constant times

R-^x-^c^x-y)2^^^^^^^

and the integral of this over RQ x R^2 is at most

Rn ~ ^lvo ^L^ ^X \ ^c ^ ^dx

which is ^(R^1) as R—»-oo. Thus we have shown that the square of the Hilbert-Schmidt norm of V^[^, ^] is ^(R⁷¹"¹).

To prove the last part of the lemma, note first that the estimate (6.6) implies 232

that VR[CI, eg] has estimated trace zero. Hence we may assume €3 has no 8 summand.

The estimated trace in question is at most the estimated trace of the operator with kernel LR-' \c^-z)\\z-y\ c,{z-y)dz.

This is the resultant of two nonnegative Hilbert-Schmidt kernels. The estimated trace is therefore equal to the absolute value of the trace, which is

Uw^-^^-y^y-^c^x-y^dxdy

^LL^i^Ki^-^+^-jo ² )^

and this is 0(R"-1).

Theorem (6.1). —Assume c='Exps with seg§{^), and define V^[c] by (6.5). Then Wa[c] is invertible for sufficiently large R and we have, as R-^oo

WRH-^UR^] +o»(i), WRM-^URH+^Rt"-1'/2).

Moreover for any c^eSSifi) the operator

WR^] (WBM^-URM+UBMV^, h]) has estimated trace o(R"~¹).

Proof. — If Uy.(x,y) denotes the kernel of V^c] then WRMUaM

has kernel

L ^C ^ ^X ~ ^{^} ) ^VR{z ~~^ ^z= =S^ ^C (i' ^x - ^z ) ^u ^-^ ^{d^}

+ L( ^c (^ ^x - ^z )- ^c (i' ^x - ^z )) ^v ^ ^dz '

By Theorem (5.1), or more exactly its proof, the first term in the right is the kernel of an operator

I+ER where

E^=^(i), ER^R^-¹)/²)

and for each ^eB the product WR^ER has estimated trace ^(R^-1).

The second integral on the right side is the kernel of VRM]+ER.

Here Ep has kernel of the form

^^^'L^^'- ² )-^^^-^) ¹¹ ^^)^

233

234 H A R O L D W I D O M

where HR has the following properties. First

|HR(^)|^->O for some keg§ without a 8 summand. Second

(6.7) LLlH^^^12^^= = o(R n~l)•

These things may be checked by referring to the proof of Lemma (5.7).

Thus

W R H U R H ^ I + V ^ A J + E ^ + E ^ . By Lemma (6.1)

VRM]=^(I), E^^(i).

(Note that the kernel of E|^ is at most that ofV^^].) Hence in particular Vf^[c]

is right invertible.

To see why this implies WR^] is also left invertible, write c{x,t)==^x)8{t)+?{x,t)

where 7 has no 8 summand. Our hypothesis implies that ai{x) is nonzero on 0; in fact a(;v) is the exponential of the coefficient of the 8 summand ofj. Therefore WR^] consists of an invertible operator, multiplication by a (A:), right multiplied by the operator

Wnp+a-^]

which differs from I by a compact operator. Hence, by the Fredholm alternative, left and right invertibility for WR^] are completely equivalent.

Hence WR^] is invertible for sufficiently large R and

WKM-^UR^KI-VR^Aj-E^-E^+^^R-1).

Since

|UR(^)I^O(^)

for some ^B, what remains to be verified is that for each 6-3 eB WR[^, W^]Ej,

have estimated traces equal to ©(I^-¹). That the first does has already been mentioned.

As for the second, it follows from (6.7) that the kernel ofEJ^ has absolute value at most P-^-^l

times the kernel of an operator which is ^(R^-^2). It follows that E|^ itself has estimated trace zero. One may assume therefore that ^ has no 8 summand, and Schwarz's inequality shows that the estimated trace of

WR^K is at most ©(R^-^²) times

^(LL^-^^-^ ² ^^ ²⁹

234

If 0. is contained in the ball B(o, A) then this is at most a constant times

^"Ui^HM

2

!^)!

2

^

2

-^'

1

-^).

Theorem (6.2). — Assume that s ==Log ce38{0^ and that Vf^[s] is nuclear/or each R>o.

Then with determinant defined by (4.3), and with 9 any function satisfying the conditions (4.6) and (4.7), we have as R—^oo

log det WR^^R^ lim f dx fe-^s-1^, t)dt

£ —^" 0 */ &2 J

+* ^RW-l L<^>o^)^ ^ -^

-iR'"¹^^^ -^.grad^.^+^R'-¹).

Proof. — The first term on the right side is, by (5. i8) trW^M.

As in the proof of Theorem (5.2) we first embed c in the family

<;(X)=ExpXj-and show that

tr WR[V(X)]W^(X)]-¹ == tr W^M + .(R-¹)

uniformly for ^ in any compact set. The only difference with the corresponding point in the proof of Theorem (5.2) is that now we must show

trW^<)U^]V^ h]=0(K¹-¹).

This follows, however, from the last part of Lemma (6.1).

Thus once again it suffices to consider the family c{\)=c+\S and to compute, for this family

^detlogWRM-trW^]-1. By the preceding theorem this is

tr U^M-tr UKMV^, Aj+^R-¹).

The first term, the trace of U^], is

(6.8) trW^]-f iy\ ^J-^-J+^J-.-j.+J^

JRQ Jt.v{y)>0 \^ f \K /

=trW^h]-Rⁿ-^l( dz[ t.v{z)h^ t)h^-t) dt+o{Rⁿ-^l).

JaQ Jt.v{z)>0

235

236 H A R O L D W I D O M

The proof of this is like the corresponding part of the proof of Theorem (5.2). One uses an argument based on (5.34) to show that if in the integral on the left side of (6.8) the quantity R~¹^ is replaced by R^J the error incurred is ^(R^"¹). We omit the details.

We are left with

trURMVaM].

Now UR^] is the sum ofW^^] and another operator whose contribution to the trace of the product can only be o{'Rn~l), by (6.7) and Lemma (6.1). Therefore what we must compute is

(6.9) trW,[A]V,[(,*]

°J J *(^-^f (^•'-^-^•'-Mi-'-^

JRQJRO \K. j JRQ\ \K J ^K ff ^K f

²

-First, observe that we may replace

^,.-.

in this integral by

^-with error ^(R⁷¹""¹). The reason is that the difference between these functions is the kernel of the operator V^A, 8] and

trV^SJVR^A]^^--1) by Lemma (6.1).

Next, let us see what error is incurred in the integral (6. g) if the z integration is taken over all E^ If 7i, c are the least upper bounds for h and grad c as in (6.2) the error has absolute value at most

f f R-^x-y^y-x^dxdyf ^(x-zWz-y) dz.

JRQJRQ • -/ 1 ^ } -^(R^ v ) \ J )

The proof of Lemma (5.6) shows that the inner integral, as a function on RQ X RQ, has Lg norm ©(R""1). Therefore the triple integral is at most ©(R^"1^) times

^'(LjRJ^-^l2^-^^^)1'2-0^"1'2)

just as at the end of the proof of Theorem (6.1).

Thus we have shown that

trURHV^A]

is equal to

f f *(^-W (^'-^-'{i-'-'Mi.'-^

JROJRQ \K / JE^ \-><- / \K. j ] \K j 226

plus o{'Rn~l). We leave it to the reader to check that the error incurred upon replacing I x \ [ y \

^R^-'j-^R5'-^

by

R-\x-y).g^dcl^x-z\

is also o^K¹-¹). Since

grad s == h * grad c

(this is just the old formula (log/)'==/'// suitably interpreted) we conclude that trUKMVRM]=f f R-^l-y-x\{x-^.g^dsl-x-^dxdy+o{R^^

JR^JRO V¹"" / V^ /

Again the error incurred upon integration with respect to x over Eⁿ rather than RQ is o^'SJ¹"¹) and so

trURMV^, ^R^J^JAO;, -x)x.gr^ds(^ x) dx+o{K1-1).

Recall that we are dealing with the family (6.10) ^ ( X ) = 6 - + ^ S

and are trying to verify that the trace of Wa[^]~1 differs by o(R!^~l) from the sum of the derivatives with respect to X of the three terms appearing on the right side of the formula given in the statement of the theorem. The trace ofU^] is given by (6.8).

That the two terms on the right side of (6.8) are the derivatives of the first two terms of the formula is no different from the corresponding fact in the last section.

There remains the verification of the identity

2 h[y, —x)x.gro.ds{jy, x) dt= . s{y, —x)x.gr2.ds(y, x) dx

for eachjy. The right side is (prime denotes differentiation with respect to X) J/(7, —x)x.gr2ids{jy, x) dx+Js{y, —x)x. grids'{jy, x) dx

=J^'(j/, —x)x.grads{^ x) d x — j s { j y , x)x.gr3.ds\^ —x) dx

=2p'(7, —x)x.gr2ids{jy, x) dx—j x. grades {jy, x)s\y,—x)} dx.

The first term is exactly

2 \h{y, —x)x.gro.ds{y, x) dx

since A'==J for the family (6.10). We shall shall show that the second term is zero.

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238 H A R O L D W I D O M

It suffices to show that for any SESS without a 8 summand, and with A = = E x p ( — s ) ,

one has

^xs{x)h{—x)dx==o.

It is simplest to assume first that

(6.11) J(i+H)|^)l^<oo.

Then also

( 6 . 1 2 ) J^(I+M)|^)|^<0)

since L^ with weight function i + M is a Banach algebra under convolution. If h = h + 8 then taking Fourier transforms gives

i +J^)^ dx = exp i-\s[x)e^ dx).

Because of (6.11) we can differentiate (that is, take the gradient of) both sides with respect to ^ to deduce

fx^e^ dx == -(xs^e^dx exp (-fs^e^ dx\ = - f ^ dx\ys[y\h[x -y}dy.

Therefore

xh[x) ==^ys[y\h{x —y}dy

almost everywhere. If s is a continuous function with compact support then both sides are continuous, the equality holds everywhere, and at x==o it gives

o^fjys(y)7i{-jy)dy=fjys(y)h(-j)dy.

For general seSS we use the facts that the continuous functions with compact support are dense in SS and that

{c^c^)-^fxc^x)c^x)dx

is [|| IH-continuous. This completes the proof of the theorem.

We shall now indicate how one might obtain asymptotic results for more general operators W^^] with kernels of the form

( x y \

^R'R''-7

If one defines (supposing for the moment that c is defined for all values of its second variable)

c ° { x , t ) = c [ x , x — — t \

\

^R

/

238

then

/A: \ x y

c [^^-y ⁼ ' R ' R

/•u | __ Y __ 11 | —- r | __ __ Y __ i) |

' R\K / \K K5 y ~ R ' R5 -T Therefore an asymptotic formula for

log dot W^]

should be obtainable form Theorem (6.2) applied to the function c° (which, admittedly, -depends on R).

To a first approximation ^{x^ t) equals

c(x, x, t)—R~^lt.gr^Ld^c{x, x, t)

where the subscript refeis to the fact that the gradient is taken with respect to the second variable. Substituting this into the formula given by the theorem gives, after a little manipulation

logdetWR^-RMimJ dx fe-^e-1^, x, t}dt

+* ^Rn-l L^.L,)>o^ ^y ^ ^J ^ ^ ^ ^ -^

+^"^0^^^-^ -^x)^x'<.g^r!id2^s{^^ ^)~grad^(^,^, x))dx +.(Rn-l).

Here of course c[x^y^ t) is, for fixed x andj/, supposed equal to Exps{x,jy, t).

Note the asymmetry in the first two variables. This should not be unexpected since the form of the kernel also displays this asymmetry. If c is symmetric in its first two variables though, then so is s and the third term of the formula vanishes.

Although what we just described is the simplest way of formally deriving the asymptotic formula from what has already been done, it may not be the simplest way to go about proving it. Perhaps better would be to prove an analogue of Theorem (6.1) for the general case, where instead of the term

/ v \

"(i-'-^

appearing in (6.5) one uses

^-4

and continuing as before. The computations would certainly be quite lengthy and the whole thing might not be worth the effort. There can be no doubt that under .appropriate conditions the last asymptotic formula is correct.

239

240 H A R O L D W I D O M

REFERENCES

[i] R. ARENS and A. CALDERON, Analytic functions of several Banach algebra elements, Ann. of Math., 62 (1955),.

204-216.

[2] G. BAXTER, A norm inequality for a finite section Wiener-Hopf equation, ///. /. Math., 7 (1963), 97-103.

[3] T. BONNESEN u. W. FENCHEL, Theorie der konvexen Korper, Berlin, Springer, 1934.

[4] A. DEVINATZ, The strong Szego limit theorem, I I I . J . Math., 11 (1967), 160-175.

[5] I. C. GOHBERG and M. G. KREIN, Introduction to the theory of linear non-self adjoint operators, Providence (Amer.

Math. Soc.), 1969.

[6] B. L. GOLINSKII and I. A. IBRAGIMOV, On Szego's limit theorem, Math. U.S.S.R., Izvestija, 5 (1971), 421-444.

[7] R. E. HARTWIG and M. E. FISHER, Asymptotic behavior of Toeplitz matrices and determinants. Arch. Rat.

Mech. Anal., 32 (1969), 190-225.

[8] I. I. HIRSCHMAN, Jr., On a theorem of Kac, Szego, and Baxter, J . d'Anal. Math., 14 (1965), 225-234.

[9] —, On a formula of Kac and Achiezer II, Arch. Rat. Mech. Anal., 38 (1970), 189-223.

[10] M. KAC, Toeplitz matrices, translation kernels, and a related problem in probability theory. Duke Math. J . y 21 (i954)» 501-509«

[n] L. MEJLBO and P. SCHMIDT, On the eigenvalues of generalized Toeplitz matrices. Math. Scand., 10 (1962), 5-i6.

[12] L. NIRENBERG, Pseudo-differential operators, Proc. Symp. Pure Math., 16, Amer. Math. Soc., Providence, 1970.

[13] G. SZEGO, On certain hermitian forms associated with the Fourier series of a positive function, Comm. seminaire math. Univ. Lund, tome supp. (1952), 228-237.

[14] H. WIDOM, A theorem on translation kernels in n dimensions, Trans. Amer. Math. Soc., 94 (1960), 170-180.

University of California, Santa Cruz.

Manuscrit refu Ie 20 mars 1973.

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