On the Overconvergence of Sequences of Rational Functions

On the Overconvergence of Sequences of Rational FunctionsAuthor(s): J. L. WalshSource: American Journal of Mathematics, Vol. 54, No. 3 (Jul., 1932), pp. 559-570Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2370901 .

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ON THE OVERCONVERGENCE OF SEQUENCES OF RATIONAL FUNCTIONS.*

By J. L. WALSH.

If a sequence of rational functions of the complex variable z converges to a given function f (z) on a given point set C with a certain degree of approximation, then that sequence may necessarily converge to the given function or its analytic extension on a larger point set containing C in its interior. For instance, it was shown by S. Bernstein that convergence of the sequence of polynomials p4z(z) of respective degrees n = 1, 2, ill such a way that we have for some R > 1

I f (Z) pn(Z) I I MlRn,( 1_

implies uniform convergence of the sequence en(z) in some ellipse whose foci are the points 1 and - 1. Various generalizations of this result have been studied by the present writer, where the degree of approximatioln of polynomials is given in a certain way not on the poinlt set - 1< z < 1, but on a muore general poinlt set, f and also where the approximating functions are rationial functions more general than polynomials and the poinlt set is a region or rectifiable curve.. It,is the object of the present paper to set forth a result which inieludes all of those meiltioned, and which indeed seems to be the most general result of its, type. In the previous study, we considered prinmarily approximation in a circular region by rational functions whose poles lie in another circular region; in the present paper more general poilnt sets instead of circular regions are contemplated in these two roles. We shall first prove aln iilteresting special case, and then indicate the modifications for the more general theorem.

I. Let C be an arbitrary closed Jordan region of the extended ? z-plane

aild let D be an arbitrary closed point set having no poinit in common with C.

7 Presented to the Society, March 25, 1932. t Miinchner Berichte (1926), pp. 223-229. : Transactions of the American Mathematical Society, Vol. 30 (1928), pp. 838-

847 and Vol. 34 (1932), pp. 22-74. ? That is to say, we adjoin the point at infinity to the usual (finite) z-plane;

consequently the notions closure (of a poilnt set), Jor-dan curve, and Jordan region, are most conveniently interpreted on the sphere (stereographic projection of the plane) instead of in the Dlane itself.

559



560 J. L. WALSH.

Let CR (Z) denote the curve in the z-plane which is the image of the circle

I w I R > 1 when the complement of C is mapped onto I w I > 1 in such a way that the point z = z' of D corresponds to w = co. The closed regions containing C bounded by the various curves CR (z') have obviously the region C in common when z' varies over D., and conceivably have a still larger point set in common. We denote by CR this common point set whatever it may be. It will appear later that the point set common to the regions considered is itself closed, and it will also appear that C' actually lies interior to CR.

LEMMA I. If r"(z) is a rational function of z of degree n * whose poles lie in D and if we have

(1) I r (z) ?3M, for z on C,

then we have also

(2) I r,(z) ? MRn, for z on CR.

Let zl, Z2, * zn be the poles of rn (z); these lie in D, and need not be distinct. The function rn(z) need not have nf poles; if there are fewer than n poles, only an obvious modification is to be made in our reasoning. Let v = pj (z) be a function which maps the exterior of C onto the exterior of y: I w = 1 so that the point z = zi corresponds to w = co. The function 4i (z) is analytic exterior to C and (if suitably defined on C itself) continuous in the corresponding closed region except for a pole of the first order at z = zi. The modulus of ci (z) on C is unity. The function

rn (Z) /01 (Z) 02 (Z) . . .

O'n (Z)

is analytic exterior to C, continuous in the corresponding closed region, and for z on C we have by (1)

(3) I rn(Z)/01(Z)02(Z) * * (Z) M _1.

Since inequality (3) holds for z on C it holds also for z exterior to C. Throughout CR exterior to C we have

I + (z) < 1 R?

so that (2) now follows from (3) for z in CR exterior to C. But CR contains C in its interior, and rn(z) has no poles in CR, SO (2) is valid for z anywhere in CR.

* That is to say, rn (z) can be written in the form

(aoz?n + alzn-? + * * ? +an)/(bozn + b1zn-T + * * * + where the denominator does not vanish identically.



THE OVERCONVERGENCE OF SEQUENCES. 561

Lemma I is analogous to, and a generalization of, a lemma used by Bernstein in the proof of his result already mentioned. The present method is closely related to that used by M. Riesz * in his proof of Bernstein's lemma.

Lemma I can obviously be extended to apply to point sets C more general than Jordan regions; we shall discuss this fact in more detail later. Let us return to the original case considered and use our original notation.

LEMMA II. If C is a Jordan region andd if the point set D (having no point in commoon witk C), is a Jordan region bounded by a curve which! can be denoted by Cp(g) [where g is some point of D], then the point set CR is the Jordan region containing C bournded by the curve C'v(g), where

v (Rp +1)2/ ( + P)

Let w = (z) be a function which maps the exterior of C onto the exterior of y: w 1 so that the point z = corresponds to w = o. Then the function

w= (1 - 1w))/ (w -w) [1 - (zl ) ((z) ]/[(z) - (zi)]

maps the exterior of C onto the exterior of -y': |i = 1 so that z = z1 corresponds to w' = co. The restrictions Iw ?_, p, I ((z1) I I p, z1 in the closed region not containing C bounded by Cp (g), z1 in D, are all equivalent. All points z not in CR can be expressed by the conditions

(4) 1((i(() >? 1 Pzi I _p(z1) - (2( ) | = , I+ 5)1P

or by the equivalent conditions

(5) |- I. I> Wi >R w|

where w (z), w= 4(z1). Conditions (5), considered as restrictions on w, are not difficult to

transform. We substitute

w'= (1 -ww)/(w -'w1),

and write (5) in the form

(6) w =(iW'+1)/(,I+WV ) Iw'1?R>1, >wiI?p>1. Here w' and w1 are arbitrary, subject to the conditions indicated, and we seek the locus of w. The transformation in (6) transforms the unit circle

*Acta Mathematica, Vol. 40 (1916), pp. 337-347.



562 J. L. WALSH.

! 1 and its exterior into I w 1 and its exterior, so we obviously have w >1. Let us set I w' w> iR, I =piop, so we have by (6)

WW = (R12p12 + W1W' + ii1'iU! + 1)/(R'1 + w,w' + ii wV' + p12).

This last expression can be written as

(7) R,~12 Pi2 + 2X + 1 R12 + 2X + p,12

a fraction whose numerator is greater than its denominlator, which is positive. We are considering the fraction for -pj?R1 ? x ? piR1. The denominator vanishes for x = - (R12 + pi2)/2 _ Rp, so, since the function (7) is represented by an equilateral hyperbola whose asymptotes are the axes, its minimum occurs for x = p1Rj, and this minimum value is

[(Rlpl + 1)/(R, + pl)]2. The relation

(Rip, + 1)/(Ri + pi) (Rp + 1 )/(R + p)

is a consequence of the conditions R, _ R, pi _p, so we have showvn that the complement of CR lies in the Jordan& region not contairning C bo'uaded by the curve Cv(g), v= (Rp + 1)/(R( + p).

Reciprocally, let z be given in the Jordan region not containing C boulided by this curve Cv (g) ; we shall prove that z belongs to the complement of CR. When this situation is transformed onto the w-plane, it reduces to the following. Given w, in absolute value not less than (Rp + 1)/ (R + p); to find w-t and w1 such that we have (6) satisfied.

If I w _R, it suffices to set ' ==w, w, =oo. If we have

R > |w I >~! (RP + IV (R+ p),

we define the quantity pi by the equation

I w |-(Rpi + 1)/(R + pi)

so we have pi _ p. Then (6) is satisfied if we merely set

w=--pw(R +pi)/(Rp, + 1), w'==R(Rp, + 1)/w(R + pl);

we have I w1 I = pi, I w' I = R, and Lemma II is completely proved. It follows from Lemma II that the complement of CR never has a point

ill common with C, no matter how D (having no point in common with C) may be chosen. For such an arbitrary D lies in some Jordan regioll Dt bounded by a Jordan curve Cp(a). Einlargement of the original D to the new D' merely adds points to the complement of Cm or leaves that comiplement



THEC OVERCONV'ERGENCE OF SEQUENCES. 563

unchanged, and the complement of the CR corresponding to D' has no point in common with C. Thus C itself always lies interior to CR.

It is also true under our present hypothesis, namely that C is a Jordani region and D an arbitrary closed set having no point in common with C, that the point set CR is closed. In fact, CR is the point set common to a finite or infijnite number of closed regions (each bounded by a curve CR?(z') and containing C) and therefore closed.

It is not true under our present hypothesis that CR is necessarily connected, as we proceed to illustrate by an example. Ljet us choose C as the unit circle, and D as the point set 2, 2w, 2W2, where () is an imagilnary cube r oot of unity. The curve CR (z') is a, circle of the coaxial family determined by z' (considered as a null circle) and C. For large values of R, the three circles CR (2), CR(2ow), CR (2w2) are small circles which do not intersect each other and which contain the points 2, 2w, 2o2 respectively in their interiors. As R decreases, those circles vary as follows: they increase in size, intersect each other, become straight liles, become circles not containing the respective points 2 2W, 2,i2 but containing C, decreases in size, and finally approach (C. At the stage where these circles CR(Z') are still circles colntaining the respee- tive points 2, 2W, 2W2 but inter secting each other, the point at infinity is exterior to the circles CR(Z') anid henice a point of CO, yet cannot be connected with C by a broken line consisting wholly of point of CR.

By means of Ljemma I we shall be able to prove

THEOREMA I. Let rn (z) b'e a sequence of rational functions of r-espective degr?ees nm such that we have for z in aC ar-bitr-ar-y closed Jordanm regionm C

(8) f f (z) --n(Z) I _ ?I/p?n, p > 1.

If the poles of the functions 'n (Z) lie in the closed set D, where D has no point in co?n?non wivth C, then thte sequence rn(z) contverges unifor?nly for z i?n CR, where R < p"/2.

If (8) is valid for z in C and if the functio- r, L+l(z) - rn(z) is of degr ee n + 1, with its poles in D, then the sequence rn (z) converges unifor?nly for z in CR, wher-e R < p.

If (8) is valid for z in C and if there exists a sequence r'n(z) of functionls such that r'n(z) - rn(z) is a rational fun?tction of degree n with its poles in D, if we have (9) f (Z) - I'. (Z) ] 3

M/pn, z in (,t

and if the sequence r/%(z) contverges uniformly for z in C.,,. where R < p (ft may or may not depend on the- sequence r'n(z) ), then, the sequence rn(z) also converges uniformly for z in CUR.



564 J. L. WALSH.

We prove the various parts of Theorem I in order. From the two inequalities

f(z) -rn(Z) I _ M/pn, I f(z) -rn+1(Z) I M/pn+l

satisfied for z on C, we have

r rn+1(Z) -rP(Z) l ?VI( +p),

also for z on C. The rational function rn+i(z) - rn(z) is of degree 2n + 1, so Lemma I yields for z on CR,

| rn+l(z) -rn(z) |- n )R2n+l -MR (1 + )R)n.

Hence the sequence rn (z) converges as we have asserted. It follows also from the proof just given that if r,+1 (z) - r% (z) is of

degree n + 1, with its poles in D, we have for z on CR

| rn+l(Z) -rn(Z) | + Rn+?

so the sequence rn (z) converges uniformly for z on CR, R < p. The assump- tion that rn+i1(z) - r. (z) is a rational function of degree n -- 1 is satisfied for instance by the sequence of functions corresponding to a series of the type*

_ pi_ + (Z - p+ ((Z- 2) (Z -Pl) (Z- 2) (Z-/3)

z (X1 (Z - (z -X c2) (Z - cl) (z - 2) (z (3)

If we have both (8) and (9) valid, we obtain similarly for z on C,

I rn(z) -r',n(z) I 2M/p8,

and Lemma I gives the inequality

I rn(z) -r/n(Z) I _ (2M/pn)Rn

for z on CR. Thus the sequence rr (z) - r(z) converges uniformly for z on CB, R < p. The sequence qr' (z) is known to converge uniformly for z on CR, so it follows that the sequence rn(z) converges also in this manner, and the proof of Theorem I is complete. If (9) is used, we do not need to know that rn(z) is rational or of degree n.

In each of the parts of Theorem I it is naturally true that the sequence rn(z) converges to the function f (z) (or its analytic extension) for z in CR if CR is connected and otherwise for z in that connected part of CR containing C; for by (8) itself the limit of the sequence is f(z) on C, and the limit of the sequence is analytic for z on CR.

* Compare Angelescu, Bulletin de l'Acadermie Roumaine, Vol. 9 (1925), pp. 164-168; Walsh, Proceedings of the National Academrny of Sciences, Vol. 18 (1932), pp. 165-171.




II. It is clear that the reasoning used in proving Lemmas I and II can be

extended to include point sets C much more general than Jordan regions; the reasoning holds with only minor changes if C is any region or other closed point set whose complement is connected. But if the complement of C is not connected, ouT previous discussion requires to be supplemented, and we proceed to indicate how that can be accomplished.

Let C and DI be arbitrary closed point sets with no common point, such that the Dirichlet problem (for arbitrary boundary values) has a solution for every region S(z') defined as the totality of points which can be joined to a given point z' of DI by broken lines lines not meeting C. Such a totality of points actually forms an open region and its boundary consists entirely of points of C, as the reader will easily prove.* As before, let CR(Z') denote the curve or curves in the z-plane which are the image of the circle I w = R < 1 when the region S(z') is mapped onto I w I > 1 in such a way that the point Z = z' corresponds to w, so. The mapping need not be smooth and the mapping function w = +(z) is not uniquely defined, but the mapping function axists by the hypothesis on the region S(z'), and the point set C'R(z') is uniquely defined. Let SR(z') denote the complement in the z-plane of the set I +(z) I > R > 1, so that SR(z') is closed and contains C, and is bounded by CR(z'). Let CR denote the point set common to all the point sets SR(z'), where z' takes on all possible positions in D1; the given set C surely belongs to CR. It can be proved as in connection with Lemma II that C lies interior- to OR, and that CR is closed.

LEMMA III. If r (z) is a rational function of z of degree n whose poles lie in D and if we have

(1) I rn(Z) I CM, z on C, then we have also (2) I rn(z) I< MRn, z on CR.

The proof of Lemma III is slightly more complicated than that of Lemma I because of the fact that in the present case the point set C may separate the plane. Let z1, Z2, , * * Zn be the poles of r, (z), not necessarily all distinct; if there are fewer than m poles, obvious modifications are to be made in our discussion. Let, z1, Z2,* , * zm be the poles of rn(z) which lie in S(z1). Denote by 1(z), 402(Z), . .

4.n(Z) functions.which map the region S(zi) onto the exterior of y: I w | 1 so that the respective points zl, Z2, . . . Zin

correspond to w = oo; these functions need not be single-valued, but that does not affect our reasoning. The function

* Compare for instance Walsh, Cretle's Jourrnal, Vol. 159 (1928), pp. 197-209.



566 J. L. WALSH.

r'n (Z) /C1p (Z) P2 (Z) . . .

Om (Z)

is anialytic in S(z,) except possibly for branch points, anld its modulus is single valued in S(z ). For z oni C, and hence for z on the boundary of S(zi) inequality (1) is valid, so for z on the boundary of S(z1) we have

(10) I r (z)/P1(z)02(z) . . .kM(Z) ? M.

To be sure, +$(z) is not properly defined on the boundary of S(z1), but as z approaches that boundary, l i (z) I approaches unity and r" (z) is continnuous; iniequality (10) is to be interpreted in that sense. It follows that (10) is valid throughout the region S(z1), a region within which the function on the left is analytic except.possibly for branch points. Throughout the region common to S (z1) and CR we have

I (z) I|R, so from (10) we derive (2) for z in the region common to S(z1) and CB. In a similar manner it is shown that (2) is valid for z in the region common to CR and any S(z1) containing poles of r.(z).

Inequality (2) is valid on the boundary of each region S(z1) containing poles of rn(z); denote by E the set complementary to the totality of such regions S(zj). Then inequality (2) is valid at each boundary point of E; the function frn(z) is analytic throughout E, so inequality (2) is valid at each point of Y, and is valid in particular at each point of CR belonging to .

We have already proved the validity of (2) at each point of CR not in , that is,at each point of CR lying in a region S(zj) containing poles of r (z), so the proof of Lemma III is complete.

We add without proof the remarks * that if RI is less than R, the set CR' is interior to the set CRB if E is an arbitrary closed set interior to CR, there exists an R' < R such that CR' contains E in its interior; if P represents an arbitrary point interior to CR, there exists a value of R" less than R such that P lies on the boundary of CR'-

Lemma III can now be applied to establish a result analogous to Theorem I; here C is no longer a Jordan region. In the present case C and D are arbitrary closed point sets with no conimmon point, such that the Dirichlet problera (for arbitrary continuous boundary values) &M a solution for every region S(z/) defited as the totality of points which can be joined to a given point z' of D by broken lines not meeting C.

THEOREM II. Let r. (z) be a sequence of rational functions of respective degrees n such that we have for z on C

(8) 1 f~ M(z -- \ rn (Z <

M7Pn /An 1

* In the proof of these remarks it may be found convenient to use the results of Lebesgue, Palermo Rendiconti, Vol. 24 (1907), pp. 371-402.




If the poles of the functions r4(z) lie in D, then the, sequence rn(Z) conver-ges uniformly for z on CR, where R < p1/2.

If (8) is valid for z in C and if the function r.+i (z) -r.(z) is of degree n + 1, with its poles in D, then the sequence rn(z) converges uniformly for z in CR, where R < p.

If (8) is valid for z in C and if there exists a sequence /'n(z) of functions such thLat r', (z) - rn (z) is a rational function of degree n with its poles in D, if we have (9) f(z) - r'n(Z) M/pn, z in C,

and if the sequence r/,(z) converges uniformly for z in CR, where R < p, [R may or may not depend on the sequence r'4(z)], then the sequence rn(z) also converges uniformly for z in CR.

The proof of Theorem II follows directly that of Theorem I.

III. If the point set C of Lemma III actually separates the plane, and if the

various regions into which C separates the plane are known to contain respectively specific numbers of poles of r, (z) less than n, then it may occur that a sharper result than Lemma III can be established, with a corresponding application more general than Theorem II. We give only a very simple illustration of this remark:

LEMMA IV. Let C be an arbitrary Jordan curve in whose interior the origin lies. If r2n(Z) is a rational function of z of the form

r2n(z) -,anZ n + a n+z. n+ + + aO + a1z + a2z2 + + a,fZn

and if we have for z on C (11) I r2n (z) M_

then we have also I r2n(z) I _ MRn

for z on CR, where D consists of the origin and the p'oint at infinity. The proof is essentially contained in the discussion already given in

connection with Lemma III. Lemma IV will be used in proving a result related to Theorem I:

THEOREM 1II. Let C be an arbitrary Jordan curve in whose interior the origin lies. Let r24(z) be a sequence of rational functions of the form

r2'n (z) Hn,A(z) + rni (z) p

/n (Z) = ao0 + a1.nZ + a,,Z2 +-** + annZ"p rn" (z) = a 2 z-z-1 -2,z2 -1 - + a-nn



568 J. L. WALSH.

such that we have for z on C

(12) t(z)- r2, (Z) MIP /n, p > 1

Then the sequence r21f(z) converges for z in CR, uniformnly for z in CR,, R, < K , where R = p. The point set D is to be taken as consisting of the origin and the poir&t at ifinnity.

The region CR is bounded by tw6 curves, C'R = CR( oo ) exterior to C an d C = COR (0) interior to C; the sequence /, (z) converges for z interior to COp, uniformly for z interior to C0, - < p IR, and the sequence rm"/(z) converges for z exterior to Cp", uniformly for z exterioil to C'",J a,< p = R.

The first part of Theorem III follows directly, for we have by (12) for z on C

ff(Z)- r2,(Z) ?M/pn, r2f(z) r2n+2(z) ! M/pn+L,

I r2n+2(z) - r2n (z) I _ (M/,pn) (1 + l/p); and this last inequality yields by Lemma IV, since the function

1 2n+2(z) - r2n (z)

is of the form used in Lemma IV and of degree 2n + 2,

I r2n+2(z) - r2n(z) I C (M]1/pfn) (1 + 1/p) R 1n+1

for z on CRi. Thus the sequence r2.1. (z) converges uniformly for z on CRB if

R, <p. The limit in. Cp (p = R) of the sequence r2n (z) is in Cp an analytic

function of z, and coincides on C with f (z), so this limit is the analytic extension in Cp of f(z) and will be denoted by f (z). The function f (z), defined in the ring bounded by C'p and Cp", may be written in the form

f (Z) =f (Z) + f2 (Z) ,

where f, (z) is analytic interior to C'p and f2 (z) is analytic exterior to Cp"

and vanishes at inffirity: we have

f( (z) 1r f ( t) dt < P, z interior to C'u,

(13) 27r1iCc t-z I < p,

f2 (Z) f7ri- ]t _dt z exterior to Ca";

these integrals are independent of the precise value of a'. In a similar way we have

(14) ( ~ - ,) r2, t dt z interior to CJT,

r"'(14) ~ r2 n(t) dt rn"z I z exterior to Ca".




We have already proved the uniform convergenice of r2n (z) to f (z) on C'r and Go!'. Term-by-term integration of the equations just written yields

lirn /n'(Z) = fl(z), uniformly for z interior to COT, < a < p, n-*oo lim rn" (Z) f2 (z), uniformly for z exterior to C0r". n->oo

Theorem III is now completely proved. Theorem III yields a new result which the reader can readily formulate,

by the use of a linear transformation z (z'- a)/(z'- b). Theorem III, together with much more general results, has been proved

by de la Vallee Poussin * for the case that C is a circle. Approximation oil the unit circle by rational functions of the kind indicated is equivalent to approximation by trigonometric sums. Theorem III was previously proved by the present writer (loc. cit.) for the case that C is rectifiable; the previous proof is not valid in the present case. We recall also the following complement to Theorem III, which holds whether C is rectifiable or, not:

If the function f (z) is analytic in the closed regiom COR, then there exist rational functions r2n(z) of the form indicated such thact we have (12) satisfied for p =R for z on C.

We have stated Theorem III for the case that C is a Jordan curve. The reader will notice that Lemma IV, Theorem III, and its complement are still true if C is an arbitrary closed set which separates the plane into two simply connected regions containing respectively the points z = 0 and z = oo. Moreover, if we have p > or > 1, there exists M' such that

I f(z) -r2n(Z) ?M'r/pn z on Cr.

Consequently we have by the use of equations (13) and (14)

I f,(z) - .n(Z) | _ M"c1/pn z interior to C',, < af < p, f2 (z) - rW" (z) Mf M"n/p,n z exterior to C,,".

In particular we have

Ifl(z)- rn'(Z) ?M"/p&, z on C, f2(Z) rn"' (Z) M ' M/.pln, z on C,

where pi is an arbitrary number less than p and where M" depends on pi.

IV. There are various generalizations of our resualts which can be obtained

with little difficulty. We merely mention these and leave the details to the reader.

*Approwimcation des fonctions (Paris, 1919), Ch. VIII. 10



J. L. WALSH.

(1) Functions f (z) which are meromorphic instead of analytic on C may be approximated, where some poles of the approximating functions r. (z) are allowed on C, but with the requirement that the limit points of poles of the functions r+, (z) - rn (z) in Theorems I and II and of r2+2 (z) - r2.(z) in Theorem III should lie in D. The results are entirely analogous to those already obtained.

(2) The properties of a sequence r (z) and its limit often depend not primarily on the position of the poles of r. (z) but, on the limiting position of the poles of r,+1(z) - rn(z), that is on the limit points of those poles. A correspondiingly more general statement of our results can be formulated.

(3) As has already been suggested, Lemma IV and Theorem III are merely special cases of more general results, where C separates the plane or not and D consists of several separated parts. Results exist more general than Lemma III and Theorem II, if the various parts of D are known to contain respectively a specific number (less than n) of poles of rn (z) or of r.,1 (z) -r, (zz). This is true, moreover, whether or not parts of D are separated by C.

(4) Throughout the present paper our measure of the approximationi of rn(z) to f(z) has been

max f(z) -rn(z) I z on C.

There exist other interesting measures of the approximation of rn (z) to f (z) on C, such as (i) the line integral

I f(z)- rn(z)IP r dzj, p>O,

extended over the boundary of C, if that boundary is rectifiable; (ii) the surface integral

f|J I ff(z) - rn(z) P dS, p > O,

extended over the area of C; (iii) the integral

f If(z) r (z) IP I dv I, p,> 0,

extended over the circle y: w j - 1 after mapping of C or of such a region as S(z') onto the interior or exterior of y. Any of these new measures of approximation may be used, even with the introduction of a positive weight cor norm function, and under suitable restrictions there can be proved results analogous to those we have here established in detail.

(5) The approximation of harmonic functions by harmonic rational functions may be studied by our present methods.



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On the Overconvergence of Sequences of Rational Functions