Degree of Polynomial Approximation to an Analytic Function as Measured by a Surface Integral

Degree of Polynomial Approximation to an Analytic Function as Measured by a SurfaceIntegralAuthor(s): J. L. WalshSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 48, No. 1 (Jan. 15, 1962), pp. 26-32Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/71095 .

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26 MATHEMATICS: J. L. WALSH PROC. N. A. S.

Since we are interested in connected groups, we can replace the hypothesis of Theorem I by the following. Let M1 be a compact Riemannian manifold which admits an infinitesimal non-isometric conformal transformation. We may also assume that M is orientable; if M is not orientable, we have only to take an orientable twofold covering space of M.

Remark: A compact symmetric space which is a rational homology sphere is isometric to a sphere, except for SU(3)/S0(3) [Kostant, B., "On invariant skew

tensors," these PROCEEDINGS, 42, 148-151 (1956)].

* This author was supported by the U.S. Air Force through the Office of Scientific Research of the Air Research and Development Command, under contract AF 49(638)-967.

I Goldberg, S. I., "Groups of automorphisms of almost Kaehler manifolds," Bull. Amer. Math. Soc., 66, 180-183 (1960).

2 Goldberg, S. I., "Groups of transformations of Kaehler and almost Kaehler manifolds," Comm. Math. Helv., 35, 35-46 (1961).

3 Lichnerowicz, A., Geometrie des Groupes de Transformations (Paris: Dunod, 1958). 4 Nagano, T., "The conformal transformations on a space with parallel Ricci tensor," J. Math.

Soc. Japan, 11, 10-14 (1959). 5 Nagano, T., and K. Yano, "Einstein spaces admitting a one-parameter group of conformal

transformations," Ann. Math., 69, 451-461 (1959). 6 Tachibana, S., "On almost analytic vectors in almost Kaehlerian manifolds," Tohoku Math J.,

11, 247-265 (1959).

DEGREE OF POLYNOMIAL APPROXIMATION TO AN ANALYTIC FUNCTION AS MEASURED BY A SURFACE INTEGRAL*

BY J. L. WALSH

DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY

Communicated November 16, 1961

The object of this note is briefly to indicate the proof of Theorems 1 and 2 below, which in the plane of the complex variable relate (i) degree of polynomial approximation in a region as measured by a surface integral to (ii) integrated Lipschitz conditions on the boundary; topic (ii) has previously been relatedl to (iii) degree of polynomial approximation as measured by a line integral over the boundary, so we have thus relations connecting (i), (ii), and (iii). Theorems 1 and 2 have re-

cently been stated without proof by Alper,2 who therefore has priority; the present proofs were obtained quite independently before the publication of reference 2; their publication here seems warranted by differences in treatment, and also be- cause of additional results (Theorems 3 and 4 below) concerning approximation by bounded analytic functions not mentioned in reference 2. Theorems I and 2 for the case p = 2 were proved previously,3 as Alper remarks; new methods are needed to treat the case p I 2.

Following Hardy and Littlewood, degree of trigonometric approximation in the

pth power mean to a real function f(O) was treated by Quade,4 who proved that

degree of approximation of order l/n(k+a)P, measured as in (1) on Iz I = 1 and where k is integral, 0 < a < 1, is necessary and sufficient that f(eti) have a kth derivative which satisfies a pth power integrated Lipschitz condition of order a.


Since we are interested in connected groups, we can replace the hypothesis of Theorem I by the following. Let M1 be a compact Riemannian manifold which admits an infinitesimal non-isometric conformal transformation. We may also assume that M is orientable; if M is not orientable, we have only to take an orientable twofold covering space of M.

Remark: A compact symmetric space which is a rational homology sphere is isometric to a sphere, except for SU(3)/S0(3) [Kostant, B., "On invariant skew

tensors," these PROCEEDINGS, 42, 148-151 (1956)].

* This author was supported by the U.S. Air Force through the Office of Scientific Research of the Air Research and Development Command, under contract AF 49(638)-967.

I Goldberg, S. I., "Groups of automorphisms of almost Kaehler manifolds," Bull. Amer. Math. Soc., 66, 180-183 (1960).

2 Goldberg, S. I., "Groups of transformations of Kaehler and almost Kaehler manifolds," Comm. Math. Helv., 35, 35-46 (1961).

3 Lichnerowicz, A., Geometrie des Groupes de Transformations (Paris: Dunod, 1958). 4 Nagano, T., "The conformal transformations on a space with parallel Ricci tensor," J. Math.

Soc. Japan, 11, 10-14 (1959). 5 Nagano, T., and K. Yano, "Einstein spaces admitting a one-parameter group of conformal

transformations," Ann. Math., 69, 451-461 (1959). 6 Tachibana, S., "On almost analytic vectors in almost Kaehlerian manifolds," Tohoku Math J.,

11, 247-265 (1959).

DEGREE OF POLYNOMIAL APPROXIMATION TO AN ANALYTIC FUNCTION AS MEASURED BY A SURFACE INTEGRAL*

BY J. L. WALSH

DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY


The object of this note is briefly to indicate the proof of Theorems 1 and 2 below, which in the plane of the complex variable relate (i) degree of polynomial approximation in a region as measured by a surface integral to (ii) integrated Lipschitz conditions on the boundary; topic (ii) has previously been relatedl to (iii) degree of polynomial approximation as measured by a line integral over the boundary, so we have thus relations connecting (i), (ii), and (iii). Theorems 1 and 2 have re-

cently been stated without proof by Alper,2 who therefore has priority; the present proofs were obtained quite independently before the publication of reference 2; their publication here seems warranted by differences in treatment, and also be- cause of additional results (Theorems 3 and 4 below) concerning approximation by bounded analytic functions not mentioned in reference 2. Theorems I and 2 for the case p = 2 were proved previously,3 as Alper remarks; new methods are needed to treat the case p I 2.

Following Hardy and Littlewood, degree of trigonometric approximation in the

pth power mean to a real function f(O) was treated by Quade,4 who proved that

degree of approximation of order l/n(k+a)P, measured as in (1) on Iz I = 1 and where k is integral, 0 < a < 1, is necessary and sufficient that f(eti) have a kth derivative which satisfies a pth power integrated Lipschitz condition of order a.



V()L. 'f+, I U(L lVAinllp l-ILVI All d. 1J. VV.ilJIl .i

Results analogous to these but in approximation by polynomials in the complex variable exist,l where approximation is measured by a line integral over a Jordan curve C:

1 If(z)

- Pn(Z) I 1dz 1, p > 1. (1) c

Approximation may also be measured by the surface integral over the interior 1) of C:

ff If(z) - pn(Z) IP dS. (2) D

A function f(w) analytic in Iw I < 1 is said to be of class H,(p > 1) provided f2 If(re'0) Ipdo is uniformly bounded for 0 < r < 1; under these conditions, boundary values f(eie) for approach "in angle" exist for almost all 0, and f If(Ve") IpdO exists. A function f(w) is said to be of class Lp(k,a) on y: Iw I 1 if f(z) is of class Hp and if we have

f7r If(k)(eio)

- f(k)ei(?

+ d) PdOe < A j I Pa; (3)

here and below, unless otherwise indicated, we suppose p > 1, k a non-negative integer, and 0 < a < 1. Numbers A and B with or without subscripts represent constants independent of n, w, and z, which may have different meanings in different formulas.

Here (in contrast to Alper) we suppose f(w) rather than f(k)(w) to be of class Hp; in the direct theorems on approximation, it then follows that f(w) on y is of power-series type, and the results follow frorn the continuity properties of f(k)(w) on y itself; in the indirect theorems, the hypothesis of approximation to f(w) in the mean on -y by polynomials or other functions analytic on and within -y (with uniformly bounded norms on Iw I = r < 1) implies at once that f(w) is of class Hp.

If C is an analytic Jordan curve, we say that f(z) is of class Hp or Lp(k, a) on C if the transform of f(z) is of class Hp or Lp(k, a) on y when the interior of C is mapped conformally onto Iw I < 1; it followsl' 5 that (3) is equivalent to

f IF(s) - F(s + h) "ds ? A1 Ih I", (4) c

where s is arc length on C and F(s) is the kth derivative of f(z) with respect to s on C. Let llf(z) - Pn () II and l!f(z) - pn(z) 11' denote the real non-negative pth roots of

(1) and (2). We prove THEOREM 1. Let C be an analytic Jordan curve whose interior is D. If the func-

tion f(z) is analytic in D, a necessary and sufficient condition that f(z) be of class Lp(c, a) on C is that there exist polynomials pn(z) in z of respective degrees n = 1, 2,.., such that

llf(z) - P(z) 11' ? k+a+i/p> p > 1, k > 0, 0 < a < 1. (5)

It is sufficient5 to prove Theorem I for the case that C is the unit circle, which we proceed to do. Let f(z) be given of class Lp(k,a) on C. Then (ref. 4, Lemma C, Theorems 5 and 8), we have

_1 2,7r if(ei) - sn(ei) PdO < n Al (6)




where s,(z) is the sum of the first n + 1 terms of the Maclaurin development of f(z). However, f(z) - s,(z) has a zero of order at least n + 1 at the origin, whence for 0 < r < I O_<r_

I. Air(n+l)p Jr If(re'i) - s,(re'0) PdO <- (

271' --

n(k?a)p

and (5) follows. Indeed, this method shows that for an arbitrary function f(z) of class Hp an in-

equality Ilf(z) - pn(Z) 1 -_ Ane, where the p,(z) are polynomials in z of respective degrees n and E,- 0, implies Ilf(z) - s0(z) 1 ' < Aien,nl/P.

LEMMA 1. If P(z) = ao + alz + . . . + az", we have

ff 1P(z) IPdS -- + I P(e:?) dO. (7) lz1<1 np +- 2

We shall prove (7) by establishing (r < 1)

Ml(r) = f fI P(r,O) P dO > - f z ' I P dO = rT a (8)

under the condition 1M(1() = 1; without the latter assumption, (8) is equivalent to

Ml(r) > r'lm(l). (8')

Denote by m the largest integer such that a.m , 0; the case P(z) 0 is trivial. As r -> o , we have

lM(r) am I rmv; (9)

the symbol i indicates that the ratio of the two members approaches unity, whence the difference of their logarithms approaches zero. If k denotes the smallest integer such that ak I 0, we have as r -' 0

Af(r) la, lPr"". (10)

By the generically monotonic character of M1(r) as a function of r for an arbitrary

analytic function, we may write

P() 111) = , a (z) ) = 1 a, - -T ^(I) ki a,1^1' - n ^0) - i- Z = 0 = O

We continue to assume M(1) = 1 and are now in a position to use Hardy's theorem6 that log Ml(r) is a convex function of log r. When log r becomes positively infinite, the curve F: y = log 1M(r), x = log r, in the (x,y)-plane is by (9) asymptotic to the line Li: y = qi + mpx, qi = p log la.m I 0; when log r becomes negatively infinite, P is by (10) asymptotic to the line L2: y = q2 + kpx, q2 = p log lak I 0. The line L1 either coincides with Lo: y = npx, or to the right of the y-axis lies below Lo; the line L2 either coincides with Lo or cuts Lo in the origin 0 or in the third quadrant. Since M1(1) = 1, the convex curve F if not coincident with Lo in x > 0 cuts Lo at 0 with forward and backward slopes less than that of Lo (as a matter of fact6 the slope is continuous); hence, r either coincides with Lo in the third quadrant or lies above Lo there. Then (8) is established, and (7) follows by multiplying (8')



VOL. 48, 1962 MATHEMATICS: J. L. WALSH 29

through by r dr and integrating with respect to r from 0 to 1. Inequality (7) becomes an equality if and only if k = m = n.

The monotonic character of M(r) yields M(r) < M(1) for 0 < r < ., and integration yields an inequality in the sense opposite to that of (7). Together we have the sharp inequalities

7f IP(e)) Pdo < S P(z) (pdS < - o P(e`) pd0. np + 2 Ilz<i 2

A classical method due to de la Vallee Poussin now enables us to complete the proof of Theorem 1. Let (5) be satisfied, where f(z) and p,(Z) are given, and we have k + a > 0. If n is given, we choose m to satisfy 2m-1 < n < 2m; inequality (5) implies (ref. 7, ?5.3) uniform convergence of the sequence p,(Z) on every lz I r(< 1) to some limit function f(z) necessarily analytic in lz I < 1, so for \z I < 1 we

may set

f(z) -

pn(Z) [p2m() - pN(Z)] + [P2m+l(Z) - 2m(Z)] + .

By (5), we have

jlf(z) - pM(Z) +' -< A/2m(ka+ +/p) IIf(z) - p2+i(zl) lI' < A/2(m+;)(+a+./p)

jP2m(Z) - Pl(z) il' < 2A/nk+a+I/, Jp2+l(Z) - p2mn(z) l 2A/2m(k+ . ...

By Lemma 1, we now write

ljP2m(z) - Pn(Z) 11 A12mPv/nk+?+l/P A12m/P/2(m-1)(k+a+l/p)

1jp2m+1(z) - p2m() ?| A12(m+-)/P/2m(k+af 1/), etc.

Then, the sequence p5(z) converges in the mean on C, to a function of class Hp that we may denote byfi(z), and we have

Ilf(z) - p(z) II ' A121/P[2--(m)(k+a) + 2-m +a (k + .' ' ] < A2/n,k+

where A2 is independent of n and z. It follovws that the function fl(z) is of class Lp(k,a) on C and that the sequence p,(z) converges uniformly on every IzI <

r(<l) to the function

1 f:L(t)dt fi(X) = -. - 7? 2ri c t - z

sof(z) is this latter function throughout z I < 1, and Theorem 1 is established. Theorem 1 fails to treat the case a = 1. Here, a new category Z(k,p) of func-

tions, introduced by Zygmund in the real domain and by Walsh and Russelll in the complex domain, is involved. An arbitrary function given on C can be approxi- mated1 in the pth power mean as in (1) by polynomials of respective d.egrees n with degree of approximation l/n(k+l)P when and only when it is of class Z(k,p) on C. We have the

COROLLARY. Theorem 1 remains valid if L.p(k,a) is replaced by Z(k,p) and if a = 1 in (5), k > 0.

Theorem 1 fails also to characterize the functions for which polynomials p,(z) exist satisfying (5) with k + a + lip > 0 even though k = --1, that is for which



30 MATHEMATICS: J. L. WALSH PRoC. N. A. S.

1 - Ilp < a < 1. Here, we use the categories L,(k,a) and Z(kc,p) for c = --1;

compare reference 1. THEOREM 2. If C and D are as in Theorem 1, and if f(z) is analytic in D, a neces-

sary and sufficient condition that f(z) be on C of class Lp(- 1,a), 1 - l/p < a < 1, or of class Z(- l,p), a = 1, is the existence of polynomials pn(z) satisfying (5) with k - -1.

It is sufficient5 to establish Theorem 2 when C is the unit circle. LEMMA 2. Let f(z) be analytic in D: 'z I < 1, and let polynomials pn(z) in z of

respective degrees n exist such that we have

l)f(z) < pn(z )ll' < d > 0. n

If F(z) is the indefinite integral f J f(z)dz in D, and if S,(z) is the sum of the first n + 1 terms of the Maclaurin development of F(z), then we have

A2 IJF(z) - n(Z) 11' <

On the circle z I = r(<l), we have dF(re')/ddO = [dF(rei?)/dz]irei?. Then (ref. 4, Theorem 8), we have (r < 1.)

I F Ar" -2- 0 F IF(rei) - Sn(re'i) \PdQ <-2 f l\f(re5i) - s^(re'5) "Pdl,

2 nr kr - 2wnP

where s,(x) denotes the sum of the first n + I terms of the Maclaurin development of f(z) and A is independent of r and indeed of f(z). Also we may write (ref. 4, Lemma C)

r 2 f(re) -(re ) P I(re) - p,n(rei) IPdO, 2 np - 27rnp

since rP < 1. Integration of these two inequalities from r = 0 to r = 1 after multi- plication by r dr then yields

IIF(z) - S(z) 11' < B2 lf(z) - p(z) n

whenever this last member has a meaning. Lemma 2 follows. We return to Theorem 2. If (5) is satisfied with k = -1, k + a + 1/p > 0, it

follows from Lemma 2 and then from Theorem 1 or the Corollary that the indefinite integral F(z) off(z) is of class .Lp(0,a) if k = -1, I - 1/p < a < 1, or of class Z(O,p) if k --1-, a =- 1. Consequently, f(z) is of class Lp (- ,a), 1 - l/p < a < 1 or

Z(- 1,p), a = 1, respectively. Conversely, let f(z) be given on C: z I = 1 of class Lp(- l,a), 1 - lip < a < 1,

or of class Z(O,p), a = 1. We assume f(O) = O, which involves no loss of generality, for as in (6) the greatest degree of mean approximation to f(z) on z I < 1 is given by the partial sums s (z) of the Maclaurin development, and this degree is the same forf(z) and forf(z) -f(O):

Ilf(z) - sn(z) ' = II[f(z) - f(0)] - [s(z) - f(0) I

'



VoIL. 48, 1962 MATHEMATICS: J. L. WALSH 31

With f(O) = 0, the function f(z)/z is also of class Lp(-l,a) or Z(-l,p) on C. We study in some detail the function

_z) f( ) F(z) - of( dz, 2

of class Lp(O,a) or Z(O,p) on C. LEMMA 3. If Pn(z) is a polynomial of degree n, we have

zP,P(z) j' 2n IPn()ll'. (II)

On the circumference [zj = r(<1), Zygmund's Lemma asserts

o Pn(re?) dO ? 2Pn I P,(re'i) PdO,

which by suitable integration yields (11). By Theorem 1, or the Corollary if a = 1, there exist polynomials p,n() of respec-

tive degrees n satisfying

A jiF(z)

- p(Z) l' ;-Azp_ (12)

Again we use the method of de la Vallee Poussin, by setting for lz I < I and 2m- 1 n < 2m

F(z) - Pn(z) = [p2m(Z) - n(z)] +- [p2m+1(Z) - p2m(Z)] + ....

Inequality (12) implies

jiF(z) - p2m(z) l' A/2m(a+l/P), IIF(z) - p2m+l() 11' A/2(m+l)(a+l/P) . ;

jjP2m(z) - pn(Z) II' 2A/na+l/p, lp2m+l(z) - p2m(z) l 2A/2m(a+l/P) ....

By Lemma 3, we now have

j1zp2m(z) - ZPn(z) I' _ 4A2m/na+l/p <

4A2m/2(m- 1)(a+]/p)

lIzplm+i(z) - zp2'n(z) 1' < 4A2m?l/2m(a+l/P), etc.

Thus, the sequence Zp'n(Z) converges in the mean on \z I < 1, by (12) necessarily to the function zF'(z) -f(z), and we have

IIf(z) - pn(Z) l'

< 4A2m[2-(m-1)(a+l/P) + 2-m(a+l/) + . . . ] An-a-/

which implies (5) with k = -1 and completes the proof of Theorem 2. Theorems 1 and 2 apply5 to the study of approximation by bounded analytic

functions: THEOREM 3. Let the function f(z) be analytic interior to the analytic Jordan curve

C whose interior is D, and let A denote a bounded region containing C + D. A

necessary and sufficient condition that f(z) be of class Lp(k,a), 0 < a < 1, k + a + I/p > O, or of class Z(k,p), k > -1, a = 1, is that there exist functions fn(z), n = 1, 2, .. ., analytic in A satisfying

Ifn(z) I < AR", z in A;

llf(z) -

fn(z) j' < A/nk+a+l/P



32 MICRIOBIOLOGY: JORDAN, YARMOLINSKY, AND KALCKAR PROC. N. A. S.

THFEOREM 4. Let C be an analytic Jordan curve whose interior is denoted by D, and let A be a bounded region containing C + D. Let f(z) be analytic in D, and for each positive M11 let FM(z) be the (or a) function analytic and of modulus not greater than 11 in A for which the norm JIF(z) - f(z) 11' in D is least; let yM denote the least value of this norm. Suppose k + a -+ 1/p > O, where k is an integer and O < a < 1. Then, a necessary and sufficient condition that f(z) be on C of class Lp(k,a) if 0 < a < 1 and of class Z(k,p) if a = I is

lim sup [log M.l/](k+a+l/p) < + o. (13) M-co

It follows, that (13) is valid if (13) holds merely for the numbers 11 = 1Mn of a monotonic sequence such that MIn -- oo, log Ml+it /log Mn < oo.

* Research supported in part by the Air Force Office of Scientific Research, Air Research and D)evelopment Command.

i Walsh, J. L., a.nd H. G. Russell, Trans. Amer. Math. Soc., 92, 355-370 (1959). 2 Alper, S. J., Doklady Akademii Nauk. S.S.S.R., 136, 265-268 (1961). 3 Walsh, J. L., Proc. Amer. Math. Soc., 10, 273-279 (1959). 4 Quade, E. S., Duke Math. Jour., 3, 529-543 (1937).

Walsh, J. L., these PROCEEDINGS, 45, 1528-1533 (1959). 6 Hardy, G. H., Proc. London Math. Soc., 14, 269-277 (1915). 7Walsh, J. L., Interpolation and Approximation, Colloquium Publications of the Alnerican

Mathematical Society, vol. 20 (New York: 1935). 8 Walsh, J. L., these PROCEEDINGS, 37, 821-826 (1951).

CON7TROL OF INDUCIBILITY OF ENZYM11ES OF THE GCALACTOSE SEQUENCE IN ESCHERICHIA COLI*

BY ELKE JORDAN, t MICHAEL B. YARMOLINSKY, AND HERMAN M. KALCKAR,t

MCCOLLUM-PRATT INSTITUTE AND DEPARTMENT OF BIOLOGY, THE JOHNS HOPKINS UNIVERSITY?


In Escherichia and in Salmonella there exists a sequence of gratuitously inducible

enzymes of galactose metabolism largely determined by a sequence of closely linked genes collectively called the gal region. Genetic studies in E. coli have been facili- tated by the discovery, due to Morse, Lederberg, and Lederberg,l that coliphage lambda transduces specifically genes of the gal region and that, apart from infre-

quent acts of recombination, the transductants tend to persist as partial hyperploids termed syngenotes. Where the hyperploid region is heterozygous they are termed

heterogenotes. Transduction of genes of the gal region in S. typhimurium has been achieved with the aid of phage PLT-22 by Z. Hartman2 and by Nikaido and Fukasawa.3 This phage, however, is not specific for the gal region and does not form stable syngenotes. A number of galactose mutants have been characterized genetically, largely through transductional analysis by E. M. Lederberg working with E. coli4 and by Z. IIartman working with S. typhimurium.2 Of these, several have been analyzed biochemically, 5-7 with regard to the enzymes of galactose

32 MICRIOBIOLOGY: JORDAN, YARMOLINSKY, AND KALCKAR PROC. N. A. S.

THFEOREM 4. Let C be an analytic Jordan curve whose interior is denoted by D, and let A be a bounded region containing C + D. Let f(z) be analytic in D, and for each positive M11 let FM(z) be the (or a) function analytic and of modulus not greater than 11 in A for which the norm JIF(z) - f(z) 11' in D is least; let yM denote the least value of this norm. Suppose k + a -+ 1/p > O, where k is an integer and O < a < 1. Then, a necessary and sufficient condition that f(z) be on C of class Lp(k,a) if 0 < a < 1 and of class Z(k,p) if a = I is

lim sup [log M.l/](k+a+l/p) < + o. (13) M-co

It follows, that (13) is valid if (13) holds merely for the numbers 11 = 1Mn of a monotonic sequence such that MIn -- oo, log Ml+it /log Mn < oo.

* Research supported in part by the Air Force Office of Scientific Research, Air Research and D)evelopment Command.

i Walsh, J. L., a.nd H. G. Russell, Trans. Amer. Math. Soc., 92, 355-370 (1959). 2 Alper, S. J., Doklady Akademii Nauk. S.S.S.R., 136, 265-268 (1961). 3 Walsh, J. L., Proc. Amer. Math. Soc., 10, 273-279 (1959). 4 Quade, E. S., Duke Math. Jour., 3, 529-543 (1937).

Walsh, J. L., these PROCEEDINGS, 45, 1528-1533 (1959). 6 Hardy, G. H., Proc. London Math. Soc., 14, 269-277 (1915). 7Walsh, J. L., Interpolation and Approximation, Colloquium Publications of the Alnerican

Mathematical Society, vol. 20 (New York: 1935). 8 Walsh, J. L., these PROCEEDINGS, 37, 821-826 (1951).

CON7TROL OF INDUCIBILITY OF ENZYM11ES OF THE GCALACTOSE SEQUENCE IN ESCHERICHIA COLI*

BY ELKE JORDAN, t MICHAEL B. YARMOLINSKY, AND HERMAN M. KALCKAR,t

MCCOLLUM-PRATT INSTITUTE AND DEPARTMENT OF BIOLOGY, THE JOHNS HOPKINS UNIVERSITY?


In Escherichia and in Salmonella there exists a sequence of gratuitously inducible

enzymes of galactose metabolism largely determined by a sequence of closely linked genes collectively called the gal region. Genetic studies in E. coli have been facili- tated by the discovery, due to Morse, Lederberg, and Lederberg,l that coliphage lambda transduces specifically genes of the gal region and that, apart from infre-

quent acts of recombination, the transductants tend to persist as partial hyperploids termed syngenotes. Where the hyperploid region is heterozygous they are termed

heterogenotes. Transduction of genes of the gal region in S. typhimurium has been achieved with the aid of phage PLT-22 by Z. Hartman2 and by Nikaido and Fukasawa.3 This phage, however, is not specific for the gal region and does not form stable syngenotes. A number of galactose mutants have been characterized genetically, largely through transductional analysis by E. M. Lederberg working with E. coli4 and by Z. IIartman working with S. typhimurium.2 Of these, several have been analyzed biochemically, 5-7 with regard to the enzymes of galactose



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Degree of Polynomial Approximation to an Analytic Function as Measured by a Surface Integral