Pacific Journal ofMathematics

APPROXIMATION BY HOLOMORPHIC FUNCTIONS ONCERTAIN PRODUCT SETS IN Cn

BARNET MORDECAI WEINSTOCK

Vol. 43, No. 3 May 1972

PACIFIC JOURNAL OF MATHEMATICSVol. 43, No. 3, 1972

APPROXIMATION BY HOLOMORPHIC FUNCTIONSON CERTAIN PRODUCT SETS IN C*

BARNET M WEINSTOCK

In this paper we prove several theorems concerning appro-ximation by holomorphic functions on product sets in Cn whereeach factor is either a compact plane set or the closure of astrongly pseudoconvex domain. In particular we show thatevery continuous function which is locally approximable byholomorphic functions on such a set is globally approximable.Our results depend on a generalization of a theorem ofAndreotti and Stoll on bounded solutions of the inhomogeneousCauchy-Riemann equations on certain product domains.

1* Statement of results* If i f is a compact set in Cn let C(K)denote the Banach space of continuous complex-valued functions onK with the uniform norm, and let H(K) denote the closure in C(K)of the space of functions which are holomorphic in some neighborhoodof K. When n = 1, each function in H{K) is the uniform limit of asequence of rational functions which are finite on K and the spacesH(K) (usually denoted R{K) in this instance) have been extensively-studied. In particular, the following properties of H(K) are well-known in the case n = 1 (cf. Chapter 3 of [2]):

(1) If U is a neighborhood of K, feC\U), and df/dz = 0 onK, then f\KeH(K).

(2) If f e C(K) and if for each x e K there is a neighborhoodUx of x in C such that f e H(K f] ϋβ), then feH(K).

(3) If μ is a complex Borel measure on K, then μ = dμ/dz where

β(z) = —π

is locally integrable on C. A measure μ is an annihilating measure

for H(K)(i.e., \fdμ = 0 for all f e H(K)J if and only if μ is sup-

ported on K.

Properties (l)-(3) are not valid for arbitrary compact sets of C*9

even if one restricts one's attention to holomorphically convex, oreven polynomially convex sets. A celebrated example of Kallin [6]shows that (2) fails in general for polynomially convex compact sets.Also, Chirka [3], by modifying her example, has shown that for eachpositive integer s there is a compact holomorphically convex set K8

in C3 and a function / , e C°°{K8) such that / . $ H(KS) although dfs

811

812 B. M. WEINSTOCK

vanishes on Ks to order s.1

In this paper we consider compact sets K c Cn of the form K =KL x x Kr where each K{ is either a compact set in C or theclosure of a strongly pseudo-convex domain in Cni. For such K weprove the following theorems:

THEOREM 1.1. If U is a neighborhood of K, feCr+ί(U), anddaf/dza = 0 on Kfor all a = {au , O with Σ a* ^ r, then f e H{K).

THEOREM 1.2. If f e C(K) and if for each xe K there is a neigh-borhood Ux such that f e H(K Π Ux), then feH(K).

THEOREM 1.3. A measure μ on K is an annihilating measurefor H(K) if and only if there exist distributions Xl9 , Xn of order<^ r — 1 with support in K such that μ = Σ dXj/dz3 .

It is possible that Theorem 1.1 remains valid if / is merelyrequired to satisfy df/dzj = 0 on K, 1 j ^ n. We know of no counter-example.

Theorem 1.2 implies an approximation theorem of the Keldysh-Mergelyan type if, in addition to the above hypotheses, K has the"segment property", i.e., if there is an open cover {Z7J of dK andcorresponding vectors {wj such that for 0 < t < 1 z + twζ lies in theinterior of K whenever z e K Π [/<. In this case every function whichis continuous on K and holomorphic in the interior of K satisfies thehypotheses of Theorem 1.2 so lies in H(K). In particular, if K is aproduct of smoothly bounded domains, then K has the segmentproperty. The case r = 1 when K is the closure of a strongly pseudo-convex domain in Cn has been treated by Lieb [8] and Kerzman [7].We use their method to prove Theorem 1.2.

If we consider Theorem 1.3 in the case r = 1 we conclude thateach annihilating measure for H(K) where K is the closure of astrongly pseudo-convex domain is the ^-divergence of an w-tuple ofmeasures supported on K (distributions of order 0). This implies thefollowing localization theorem for annihilating measures which is well-known in case n — 1 [2, Lemma 3.2.11] and which is a sort of dualversion of Theorem 1.2, which it clearly implies.

THEOREM 1.4. Let K be the closure of a strongly pseudoconvexdomain in Cn. Let μ be an annihilating measure for H(K). If

is a finite open covering of K there exist annihilating measures1 The referee has pointed out that Kallin, in an unpublished remark and without

knowledge of Chirka's paper, observed that her counterexample could be made to yieldthis additional property.

APPROXIMATION BY HOLOMORPHIC FUNCTIONS 813

fo for H(K Π Ui) (in particular each μt is supported in Ϊ7f) such that

All of our results are derived from an estimate (Theorem 2.2below) for Cauchy-Riemann operator in certain product domains. Thistheorem is a generalization of a theorem of Andreotti and Stoll [1] forthe case of a polycylinder. Our proof, like theirs, follows the inductionprocedure used in the proof of the familiar Dolbeaut-Grothendiecklemma, but we make essential use of the representation theorem ofGrauert and Lieb [5] for bounded solutions of the Cauchy-Riemannequations in strongly pseudo-convex domains.

2* The basic estimate*

DEFINITION 2.1. An open set G in Cn is called admissible if (a)n = 1 and G is a bounded open set or (b) n > 1 and G is a stronglypseudo-convex domain with C°° boundary.

The following theorem is due to Grauert and Lieb [5] in the casen > 1, and is simply a restatement of known properties of the Cauchykernel when n — 1.

THEOREM 2.1. Let G be an admissible open set in Cn. Thenthere exists a differential form Ω(ζ, z) of type (n, n — 1) in ζ, of type(0, 0) in z, defined in a neighborhood of G x G such that

( i ) Ω is of class C°° off the diagonal of G x G:(ii) there is a neighborhood of dG x G in G x G in which dzΩ —

0;(iii) if g is a bounded (0, 1) form of class C°° such that dg = 0

on G, and if

f{z) = - f g(ζ) A Ω(ζ, z)JG

then f e C^iG) and df = g in G;(iv) there is a constant A(G), independent of z such that

JG n

where α(ζ, z) is any coefficient of Ω and dm is Lebesgue measure onG;

(v) G has a sequence {(?„} of admissible neighborhoods whoseintersection is G such that {Δ(GV)} is a constant sequence.

Also, G is the union of an increasing sequence of admissible opensets {Gv} for which {Δ(GV)} is a constant sequence.

814 B. M. WEINSTOCK

If G is an open set in Cn we denote by BC~(G) the space of func-tions on G whose derivatives of all orders are bounded and continuouson G. We will need the following corollary of Theorem 2.1.

COROLLARY. Let G be an admissible open set in C%. Let Uk beopen in C%k, k = 1, 2. Suppose that

g,(z, x,w), , gn(z, x, w) e BC°°(G x U, x U2)

and that g — Σ gjdZj satisfies dzg — 0 in G x Uι x U2. Then thereexists f e BC°°(G x V1 x U2) such that

( i ) dj = g in G x E x ί72;(ii) H/il^ J ^ m a x ^ J I ^ H;(iii) if D is a differential operator on C%1 x C 2 with constant

coefficients, then

\\Df\\£J(G)m&x\\Dgj\\.j

(In particular, if each gό is holomorphic in U2 for fixed (z, x) 6G x U1} then f has the same property.)

Here | . | / | | = s u p σ x c Γ l X E 7 2 1 / | .

Proof. Let f(z, x, w) = — \ g(ζ, x, w) A Ω(ζ, z), where Ω is as inJG

Theorem 2.1. Since all the derivatives of g are bounded we maydifferentiate under the integral sign as often as we wish. The corol-lary then follows immediately from Theorem 2.1.

Let G — 6?! x x Gr be an open set in Cn where each G{ is anopen set in Cnκ If / is a function on G and g — Σ QAZJ is a (0, 1)form on G we will use the following notation:

r » = max \\d«fldz«\\ΰa e A k , t

(where a = (au •••, an) is an ^-tuple of nonnegative integers, \a\

d"f/dz° = d^

and

Akft = {a — (al9 , an): \a\ ^ k and ^ = 0

if j > nx + + wj)

\\g\W = \\g\\(ok'r)

APPROXIMATION BY HOLOMORPHIC FUNCTIONS 815

We can now state our basic result.

THEOREM 2.2. Let G — Gx x x Gr be an open set in Cn whereeach Gι is an admissible open set in Cnκ Let g be a C°° (0, 1) formin G such that dg — 0 in G and \\g\\{G~ι) < °° Then there exists f eC°°(G) such that

(i) df = g in G

(ϋ) WfWσ^iWWgWZ-"-Here A is any number ^ 1 such that A A(Gi) as defined in Theorem2.1.

Proof. We first prove the theorem in the case when each coeffi-cient of g is in BC°°(G).

If g is a (0,1) form on G, then g — gl where each gι is a (0,1)form on Gι with coefficients depending only on the other variables asparameters. For each k, 1 k ^ r we consider the following Asser-tion k:

Let G = G1 x x Gr be as above. Let g be a (0,1) form on Gwhose coefficients lie in BC°°(G) and such that dg — 0 in G. Supposethat

where each gι is a (0,1) form on Gi9 (with coefficients depending alsoon the other variables). Then there exists f e BC°°(G) such that

(i) df = g(ii) 11/11, :g (34*||firr-ι *-1J.We shall prove Assertion 1 and then show that for k = 1, 2, ,

r — 1, Assertion k implies Assertion k + 1. (Of course Assertion rimplies Theorem 2.2 in the case g is of class BC°°.)

If g satisfies the hypotheses of Assertion 1, then g is a 5-closed(0,1) form in Gx whose coefficients are holomorphic functions in G2 x• x Gr for fixed z in Gx. Assertion 1 thus follows directly from theCorollary to Theorem 2.1.

Suppose now that Assertion k is true and that g satisfies thehypotheses of Assertion (k + 1). Then

Notice that dg = 0 implies dk+1gk+1 = 0 (where dk+ι differentiates only

with respect to the variables from Gk+1) and that the coefficients ofg are holomorphic in Gk+2 x x G>. Applying the corollary to Theo-rem 2.1 we conclude the existence of u e BC°°(G) such that dk+ίu = gk+ι

and such that u is holomorphic in Gk+2 x x Gr. Let s = g — du.

816 B. M. WEINSTOCK

Then 3s = dg — 0 and s is clearly a sum of (0, l)-forms involving onlydifferentials in the variables from Gu •••, Gk. By Assertion k thereexists t e BC°°(G) such that dt = s.

Let / = u + t. Then 9/ = g. Also,

But

I k \\(k~l,k~D

Σ/|

max

Hence

This concludes the proof in the case when g is of class BC°°.Suppose now that g satisfies the hypotheses of Theorem 2.2. By

Theorem 2.1 and what has been proved so far, we can find a sequenceof open sets {Gv} and a constant C independent of v such that

( i ) Gu c Gu+1 c G

(ii) G= UGV

(iii) there exists /„ e J5C°°(G,)such that dfu = # on Gy and

For each / let Sμ — {fv\Gμ:v ^ μ). Since Λ — fμ is holomorphicon Gμ for each v, and {/„ — fμ \ Gμ} is uniformly bounded, Sμ is rela-tively compact by MonteΓs theorem. Thus we may choose, for eachμ, a subsequence {fv,μ} of Sμ which converges uniformly on Gμ, suchthat {fv,μ+1} is a subsequence of {fv,μ}. Then the diagonal sequence{fμ,μ} converges uniformly on each Gμ to a continuous function /defined on all of G. But / is in fact in C°°(G) since, if v > μ9 fvv — fμμ

is holomorphic on Gμ and /„„ — fμμ—>f — fμμ on G^. Thus f — fμμ isholomorphic, hence in C^iGμ) so / e C~{Gμ) for each μ. This alsoshows that df = g on G. Finally, if z e G then there exists μ suchthat ze Gv for v ^> μ. This means that

APPROXIMATION BY HOLOMORPHIC FUNCTIONS 817

but f(z) = \\mfμμ{z) so

3* Proofs of Theorems 1.1 and 1.2.

Proof of Theorem 1.1. We may suppose that / has compact

support in U. Choose φ e C°°(Cn) such that φ ^ 0, U = 1 and φ - 0

outside the closed unit ball. For each d > 0 define fδ by

Λ(s) = \ /(« — Sw)φ(w)dw .

Then fδeC°°(Cn),fδ->f uniformly as <5->0 and for each a,

(dafδ/dza)(z) - ί(3α/δ/3^)(^ - δw)φ(w)dw

so if G is an open set in Cn,

w h e r e G 5 = {z — dw: zeG,\w\^l).

For each ί, 1 ^ i ^ r we can find a sequence {&£} of admissibleneighborhoods such that Kt — Π G< and such that {Δ{Gt)\ is a constantsequence. Let us denote the constant by Δi9 Choose A ^ 1 such thatA ^ At for 1 ^ i ^ r.

Let ε > 0 be given. Choose δ0 such that | | / - fδ\\κ < ε/2 if δ <δ0. Choose v such that if G = Gί x x G>, then

Then there exists d < §0 such that

By Theorem 2.2 we can choose u e C°°(G) such that du = 3/δ and

Then h = fδ — u is holomorphic in a neighborhood of i£ and

l l / - Λ | | i C ^ | | / - / , | U + H Λ - A I U

< e / 2 + | | « | U

<e/2+(3/f)' | |Λ| |5

<s/2

< ε .

818 B. M. WEINSTOCK

Proof of Theorem 1.2. (Here we follow Lieb [8].) Since K iscompact we can choose finitely many neighborhoods UXl, , UXm whichcover K. Let U = UXί U U UXm. Choose sequences {G-} of admis-sible neighborhoods of Ki9 1 ^ i ^ r as in the proof of Theorem 1.1and let Δ be as above.

Let ε > 0 be given. Choose v such that Gv = (?ί x x Gv

r liesin Z7, and such that there exist holomorphic functions h3- on Ux. Π Gv

with | | / - Λyll^.n* < e. Notice that \ht - hs\ ^\f - ht\ + \f -%\ <2ε on Ux. Γ\UX. Γ\ K. By choosing v larger if necessary we may supposet h a t \hi%- hj\< 4ε on UH Π ^ Π G\

Choose a C°° partition of unity φl9 , φm subordinate to the cover{Ux.}. Letg^ΣjiΦiih-hi). Notice that ||flrfc||^fcnJ5:<_2e. A l s o ^ - ^ -hs - hk and dg, - dgk = 0 on UXj Π ϊ7βA; Π G\ Thus {3 -} defines a (0, 1)form g on Gv which satisfies dg — 0 so by Theorem 2.2 we can findu e C°°(GV) that du = # on Gv and

But

and

3^ f c/3^ - Σ (d'Φi/S^KK - hi) .i

Hence \\g\\{G»~ι) ^ 4εC where C is a constant depending only on thepartition of unity {^}.

Let bj = #j — M. Then 3&y = 0 on UXj Π Gv and δy — bk = gά — gk —hj — hk. Thus {^ — bj} defines a holomorphic function h on Gv and

- h\\κ rg max ( | | / - hό\\κ + | | ^ |U) + \\u\

^ ε + 2ε

where C and J are independent of ε. Since ε was arbitrary, thiscompletes the proof.

4* Annihilating Measures for H(K). If G is an open set in Cn

we denote by A(0>1)(G) the space of differential forms of type (0,1) onG of class C°°. We topologize A(0>1)(G) as the direct sum of n copiesof the Frechet space C°°(G) with the topology of uniform convergenceon compact subsets of G of derivatives of all orders. The dual spaceof C°°(G) is the space of distributions on Cn whose support is a compactsubset of G. We will identify the dual space of A{0Λ](G) with thespace of distributions with compact support in G.

APPROXIMATION BY HOLOMORPHIC FUNCTIONS 819

LEMMA 4.1. Let G be a domain of holomorphy in Cn, K a compactsubset of G, and μ an annihilating measure for H(K). Then thereexist distributions \, , λΛ with compact support in G such thatμ = —Y.dXjjdZj.

Proof. C°°{G) and A(0A)(G) are Frechet spaces. Since G is adomain of holomorphy, 3 maps C°°(G) onto the closed subspace of (0,1)forms g satisfying dg = 0. The measure μ, considered as a continuouslinear functional on C~(G), annihilates the kernel of 3, so by a theoremof Dieudonne and Schwartz [4], μ is in the image of the adjoint,9*, of 3. But if λ = (λi, •••, λn) is an w-tuple of distributions withcompact support in G and / e C°°(G), then

(3*λ)(/) = λ(3/) = Σ M3//32,)

Thus for some λ^ •••, λΛ we have μ = — Σ δλ /δ^ .We will also consider, for a bounded open set G in Cw, the Banach

space Cr(G) of continuous functions on G whose derivatives of order^ r are bounded and continuous on G, with the norm

| | / | | S = max

where a is a 2w-tuple of non-negative integers and

Da =

A continuous linear functional on Cr(G) is easily seen to define adistribution on <7% with support in G. In addition, we will denoteby B{

r°>])(G) the space of (0,1) forms on G with coefficients in Cr(G),

topologized as the direct sum of n copies of Cr(G) with the norm

If G x c G 2 are two bounded open sets in Cn let R: B{

r°>ι)(G2) -»

B{ryl){G^ be the operator which restricts forms in G2 to G1# Then JB*is a norm-decreasing embedding of the dual space of B{?Λ)(GΪ) intoB?>ι){G2)* since, if λ e B^iG,)*, and 0 = Σ Λ is in B{

r°>ι)(G2),

\(R*\)(g)\

so | |Λ*λ||With these preliminaries we can proceed to the proof of Theorem

1.3.

Proof of Theorem 1.3. Let {Gv} be a sequence of bounded open

820 B. M. WEINSTOCK

neighborhoods of K such that( i ) G . c G U(ϋ) ϊ=nft(iii) there exists a constant C independent of v such that if g e

A{0'ί)(G1)ydg_= 0 in Gv, and \\g\\&~ι) < «,, then there exists feC°°(Gv)

such that df — g and

If μ is an annihilating measure for H(K) we can apply Lemma4.1 to obtain, for each v, an w-tuple λ* = (λ£, •••, λ£) of distributionswith compact support in Gu such that μ = — dXj/dzj. Let TΓV bethe subspace of Cr~1(G^) consisting of restrictions to Gv of C°° func-tions on C*. If f e Wv we can find h e C°°(GV) such that / — h isholomorphic on (?„ and |[Λ||βv ^ C\\df\\{J~ι) where C is the constant in(iii) above. Thus

|λ>(5/)| -

where \\μ\\ is the total variation of μ. This means that Xv defines acontinuous linear functional on the subspace dWv of B^KGJ) of norm^C\\μ\\. By the Hahn-Banach theorem there is an %-tuple, whichwe will continue to denote by λ% of continuous linear functionals onCr~\Gv) such that

(a) λ*(3/) = \fdμ for all / e C~(Cn)

(b) \\X*\\ ^ C| |μ| | .Now, by composing with the adjoint of the appropriate restric-

tion operator we may consider each λv so obtained as a continuouslinear functional on B^-KGi). Then the sequence {λv} constitutes abounded sequence of elements in the dual space of a separable Banachspace. Consequently, there is a subsequence {λv'} and an π-tuple λ —(X, , λw) of continuous linear functionals on Cr~ί(Gι) such thatλ"'—>λ in the weak star topology. Since each X)' is supported, as adistribution, on Gv, and since K = Γ\GU, it follows that the support ofeach λy, as a distribution, lies in K. Moreover, if / is a C°° functionon Cn then

λ(3/) = \imXy(df) =

i.e., μ= —'ΣidXj/dZj. Finally, it is clear that each X3 is of order< r — 1.

Conversely, suppose μ is a measure on K, and μ — Σ dXj/dz, whereλx, , Xn are distributions with compact support on K. If / is holo-

APPROXIMATION BY HOLOMORPHIC FUNCTIONS 821

morphic in a neighborhood of K, then df/dzj, 1 :g j ^ n, are identi-cally 0 in a neighborhood of K so

\fdμ = Σ (dXj

since the λ, are supported on K.

Proof of Theorem 1.4. Let {φ{} be a C°° partition of unity sub-ordinate to the open covering {?7J, i.e., suppose ΦieC°°(Ui), φi hascompact support, 0 ^ φt ^ 1, and Σ ^ = l on a neighborhood of JBΓ.1/ μ is an annihilating measure for H(K), then by Theorem 1.3there exist measures λ1? , Xn on K such that

μ - - Σ 3λ,Λ = - Σ

- ~ Σ Σ ΦiidXsfdzj) - Σ Σ

= Σ { i" - Σ

where each JM4 is a measure compactly supported on Ut Γi K. If h isholomorphic on E7i Π if> then

for 1 ^ i ^ ^. Thus

REFERENCES

1. A. Andreotti and W. Stoll, The extension of bounded holomorphic functions fromhyper surf aces in a poly cylinder, Rice University Studies, 56 No. 2 (1970), 199-222.2. A. Browder, Introduction to Function Algebras, W. A. Benjamin, Inc., New York,1969.3. E. M. Chirka, Approximation by holomorphic functions on smooth manifolds in Cn,Math. USSR Sbornik 7 (1969), 95-114 (translated from Mat. Sbornik, 78 (120) (1969) No. 1).4. J. Dieudonne and L. Schwartz, La dualite dans les espaces (F) et (LF), Annales Inst.Fourier, 1 (1949), 61-101.

822 B. M. WEINSTOCK

5. H. Grauert and I. Lieb, Das Ramίrezsche Integral und die L'όsung der Gleichungdf — a im Bereich der beschrάnkten Formen, Rice University Studies, 5G No. 2 (1970),29-50.6. E. Kallin, A non-local function algebra, Proc. Nat. Acad. Sciences, 49 (1963), 821-824.7. N. Kerzman, Holder and Lp estimaties for solutions of du = f in strongly pseudo-convex domains, Comm. Pure Appl. Math., 24 (1971), 301-379.8. I. Lieb, Ein Approximationssatz auf streng pseudoconvexen Gebieten, Math. Ann.,184 (1969), 56-60.

Received September 22, 1971 and in revised form February 3, 1972. This researchwas supported by NSF Grants GP-11969 and GP-21326 at Brown University.

BROWN UNIVERSITY

PACIFIC JOURNAL OF MATHEMATICS

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H. SAMELSON J. DUGUNDJI

Stanford University Department of MathematicsStanford, California 94305 University of Southern California

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Pacific Journal of MathematicsVol. 43, No. 3 May, 1972

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group matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633Kenneth Hardy and R. Grant Woods, On c-realcompact spaces and locally bounded

normal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647Manfred Knebusch, Alex I. Rosenberg and Roger P. Ware, Grothendieck and Witt

rings of hermitian forms over Dedekind rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657George M. Lewis, Cut loci of points at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675Jerome Irving Malitz and William Nelson Reinhardt, A complete countable L Q

ω1theory with maximal models of many cardinalities . . . . . . . . . . . . . . . . . . . . . . . . . . 691

Wilfred Dennis Pepe and William P. Ziemer, Slices, multiplicity, and Lebesguearea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701

Keith Pierce, Amalgamating abelian ordered groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711Stephen James Pride, Residual properties of free groups . . . . . . . . . . . . . . . . . . . . . . . . . . 725Roy Martin Rakestraw, The convex cone of n-monotone functions . . . . . . . . . . . . . . . . . 735T. Schwartzbauer, Entropy and approximation of measure preserving

transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753Peter F. Stebe, Invariant functions of an iterative process for maximization of a

polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765Kondagunta Sundaresan and Wojbor Woyczynski, L-orthogonally scattered

measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785Kyle David Wallace, Cλ-groups and λ-basic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 799Barnet Mordecai Weinstock, Approximation by holomorphic functions on certain

product sets in Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811Donald Steven Passman, Corrections to: “Isomorphic groups and group rings” . . . . 823Don David Porter, Correction to: “Symplectic bordism, Stiefel-Whitney numbers,

and a Novikov resolution” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825John Ben Butler, Jr., Correction to: “Almost smooth perturbations of self-adjoint

operators” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825Constantine G. Lascarides, Correction to: “A study of certain sequence spaces of

Maddox and a generalization of a theorem of Iyer” . . . . . . . . . . . . . . . . . . . . . . . . . 826George A. Elliott, Correction to: “An extension of some results of takesaki in the

reduction theory of von neumann algebras” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826James Daniel Halpern, Correction to: “On a question of Tarski and a maximal

theorem of Kurepa” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827

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