A Duality in Interpolation to Analytic Functions by Rational FunctionsAuthor(s): J. L. WalshSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 19, No. 12 (Dec. 15, 1933), pp. 1049-1053Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/85885 .

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VOL. 19, 1933 MA THEMA TICS: J. L. WALSH 1049

omission and to say whether its correction will lead only to a new inter- pretation of the constants of our equations or to an actual change of their form. Another simplification is the neglect of polar and excited states: While there is reason to assume that their influence is small, its exact estimate is still lacking. Further inaccuracies were discussed in our pre- ceding paper: The use of Bloch's periodicity condition in place of the actual border conditions, the restriction to interactions between adjacent atoms and to first order perturbations. With these omissions the mathe- matical side of the theory becomes simple and elegant and, as the pre- ceding sections show, it is sufficient to represent the actual conditions at low temperatures. However, the prospects of carrying this treatment to a higher degree of approximation are not very favorable. The mathe- matical simplicity is lost and the theory becomes rather cumbersome because one has to increase the accuracy in so many different directions, For determining the Curie point one should, perhaps, try a different approach to the problem, for instance, that recently outlined by L. Bril- louin" although it seems that this method, too, is difficult to be carried through without considerable idealizations.

1 R. I. Allen and F. W. Constant, Phys. Rev., 44, 228 (1933). 2 P. S. Epstein, Phys. Rev., 41, 91 (1932). 3 P. Weiss, Phys. Zeits., 9, 361 (1908). 4 W. Heisenberg, Zeits. Physik, 49, 619 (1928). - 5 F. Bloch, Ibid, 61, 206 (1930). 6 R. Forrer, Journ. Phys. et le Rad., 1, 49 (1930). 7 F. Bitter, Phys. Rev., 37, 91 (1931). 8 A. Preuss, Thesis of Zurich (cited by Forrer). 9 We infer this from the fact that our formula (8) with a suitably chosen constant

represents well the saturation curves obtained by Weiss and his pupils for hetero- crystalline materials.

10 The formula given by Bloch (note 5) leads to a still much slower decline than ours. u L. Brillouin, Journ. Phys. et le Rad., 3, 373, 565 (1932); 4, 1 (1933).

A DUALITY IN INTERPOLATION TO ANALYTIC FUNCTIONS BY RATIONAL FUNCTIONS

BY J. L. WALSH

DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY

Communicated October 24, 1933

The following examples make clear a certain duality in interpolation to analytic functions by rational functions; reversal of the r61les of pre- scribed poles and points of interpolation reverses the region or regions of convergence, provided the bounding curves remain unchanged.

Ia. If the function f(z) is analytic for I z < R, then the sequence of

VOL. 19, 1933 MA THEMA TICS: J. L. WALSH 1049

omission and to say whether its correction will lead only to a new inter- pretation of the constants of our equations or to an actual change of their form. Another simplification is the neglect of polar and excited states: While there is reason to assume that their influence is small, its exact estimate is still lacking. Further inaccuracies were discussed in our pre- ceding paper: The use of Bloch's periodicity condition in place of the actual border conditions, the restriction to interactions between adjacent atoms and to first order perturbations. With these omissions the mathe- matical side of the theory becomes simple and elegant and, as the pre- ceding sections show, it is sufficient to represent the actual conditions at low temperatures. However, the prospects of carrying this treatment to a higher degree of approximation are not very favorable. The mathe- matical simplicity is lost and the theory becomes rather cumbersome because one has to increase the accuracy in so many different directions, For determining the Curie point one should, perhaps, try a different approach to the problem, for instance, that recently outlined by L. Bril- louin" although it seems that this method, too, is difficult to be carried through without considerable idealizations.

1 R. I. Allen and F. W. Constant, Phys. Rev., 44, 228 (1933). 2 P. S. Epstein, Phys. Rev., 41, 91 (1932). 3 P. Weiss, Phys. Zeits., 9, 361 (1908). 4 W. Heisenberg, Zeits. Physik, 49, 619 (1928). - 5 F. Bloch, Ibid, 61, 206 (1930). 6 R. Forrer, Journ. Phys. et le Rad., 1, 49 (1930). 7 F. Bitter, Phys. Rev., 37, 91 (1931). 8 A. Preuss, Thesis of Zurich (cited by Forrer). 9 We infer this from the fact that our formula (8) with a suitably chosen constant

represents well the saturation curves obtained by Weiss and his pupils for hetero- crystalline materials.

10 The formula given by Bloch (note 5) leads to a still much slower decline than ours. u L. Brillouin, Journ. Phys. et le Rad., 3, 373, 565 (1932); 4, 1 (1933).

A DUALITY IN INTERPOLATION TO ANALYTIC FUNCTIONS BY RATIONAL FUNCTIONS

BY J. L. WALSH

DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY

Communicated October 24, 1933

The following examples make clear a certain duality in interpolation to analytic functions by rational functions; reversal of the r61les of pre- scribed poles and points of interpolation reverses the region or regions of convergence, provided the bounding curves remain unchanged.

Ia. If the function f(z) is analytic for I z < R, then the sequence of

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1050 MATHEMATICS: J. L. WALSH PROC. N. A. S.

rational functions of respective degrees n with poles at infinity (i. e., sequence of polynomials in z of degree n) and which interpolate to f(z) in the origin, converges uniformly to f(z) for I z | < R.

lb. If the function f(z) is analytic for I z | > R, then the sequence of rational functions of respective degrees n with poles at the origin (i. e., se-

quence of polynomials in 1/z of degree n) and which interpolate to f(z) at

infinity, converges uniformly to f(z) for I z I _ R. IIa. If the function f(z) is analytic for I p(z) I < A > 0, p(z) = (z - Zi)

(z - z2)... (z - zr), then the sequence of rational functions of respective degrees vn - 1 with poles at infinity (i. e., sequence of polynomials in z of degree vn - 1) and which interpolate to f(z) in the points Zk each considered

of multiplicity n, converges uniformly to f(z) for | p(z) | < s.

lib. If the function f(z) is analytic for I p(z) I >- t > 0, p(z) = (z - zi)

(z - z2). . (z - z,), then the sequence of rational functions of respective degrees .vn with poles in the points Zk each of multiplicity n and which inter-

polate to f(z) at infinity, converges uniformly to f(z) for I p(z) I > p. IIIa. if the function f(z) is analytic for I z < R, A < R < B, then the

sequence of rational functions of respective degrees n with poles in the n

points (B')l/n and which interpolate to f(z) in the n + 1 points (A n1+ )) i/(n+ 1), converges uniformly to f(z) for z I < R.

IIIb. If the function f(z) is analytic for I z ? R, A < R < B, then the

sequence of rational functions of respective degrees n with poles in the n points (A')l'n and which interpolate to f(z) in the n + 1 points (B'+N)1/('+1),

converges uniformly to f(z) for I z I -

R. IVa. If the function f(z) is analytic for 1/R < | z ? < R > 1, then the

sequence of rational functions of respective degrees 2n whose poles lie in the

origin and the point at infinity each of multiplicity n and which interpolate to f(z) in the (2n + l)st roots of unity, converges to f(z) for 1/R ? I z ? < R.

IVb. If the function f(z) is analytic for I z I < 1/R and for I z j > R > 1, then the sequence of rational functions of respective degrees 2n - 1 whose

poles lie in the (2n - l)st roots of unity and which interpolate to f(z) in the

origin and point at infinity each of multiplicity n, converges to f(z) for I z I <

I/R and for I z R. In each of these theorems we are dealing with the extended plane, i.e.,

with the plane closed by the adjunction of the point at infinity. A rational function of degree n is a function of the form

aoz' + azl' + . + a.

boz' + blznl' + ... + bn

where the denominator does not vanish identically. All of these theorems are included in the general theorem proved below.

Theorems Ia and Ib are classical. Theorem IIa is well known and due to

Jacobi. Theorem IIb is new, and is remarkable in that the region of con-

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VOL. 19, 1933 MA THEMATICS: J. L. WALSH 1051

vergence of the sequence of rational functions need not be simply connected, whereas the sequence is defined by interpolation in a single point. Theorem IIb extends readily to the case of an arbitrary infinite region bounded by a finite number of Jordan curves, and the rational functions of interpola- tion can be defined for all degrees. Theorems IIIa and IIIb were recently established by the present writer,' as was also Theorem IVa.2 Theorem IVb is new, but is similar to a theorem recently proved by Ketchum.3 The theorem of Ketchum, like Theorem IVb, iLnvolves rational functions whose poles lie in roots of unity and which converge for Iz ? 1/R and

z j > R > 1 to an arbitrary function analytic for those values of z; the sequence of Ketchum does not seem, however, to be defined by inter- polation.

Our general theorem is the following:

THEOREM. Suppose the relation

nI

lim (z - - fln)( - 2n)..(Z

- n+n) = (z) g O, Fin aj, (1)

n-coo (z - aln)(z - O2n). . .(z - a,n)

to be valid uniformly in some region D. Suppose C: ((z) = R and r: :(z) =

R1 > R are two curves or finite sets of curves in D such that f(z) is analytic in a closed region or set of closed regions D, bounded by r and containing none of the points ain all the points in, and containing a closed region or set of closed regions D2 bounded by C. Then the sequence of rational functions r,(z) of respective degrees n with poles in the points ain which interpolate to f(z) in the points fin converges to f(z) uniformly in D2.

In (1), the fraction on the left is intended to represent an analytic func- tion of degree n + 1 whose zeros are the points j3in and whose poles are the points ai,, plus the point at infinity. In case a point ati or fin is at infinity, this notation requires some formal modification, and the same is true of the equations below.

f(z) - rn (z) =

(Z- lin)(z--2,n)) ... (z -

~n,n) (t -

an) (t - a2n). . (t -

n,n)f(t)dt, (2)

(z - ln) (z - 2n) . . . ( - Otn,n) (t- ln) (t

- 2n) . . (t- F+nl,n)(t- )

for z in D2. The integrand (for z on C and t on r) may be split into three factors, one of which behaves in modulus like R.', one of which behaves in modulus like l/R' and the third of which is uniformly bounded. Conse- quently the left-hand member approaches zero uniformly for z on C, hence uniformly for z in D2, and the theorem is proved.

Under broad restrictions on the function I (z), it is true that iff( z)

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1052 MA THEMA TICS: J. L. WALSH PROC. N. A. S.

is analytic on the closed set Di, then the sequence rn (z) converges to f(z) uniformly on that closed set.

Equation (1), valid uniformly on the curves C and r, implies the con- dition

n I

lim n/ (- al)(z- a2.)...(! - - o (3) n- > X (z

- 1 )(Z- ~f2n) ... (z- n,n) - - (z)

on C and r provided the points a,,+,n and #/n+l,n satisfy certain simple restrictions. Equation (3) is then sufficient for the application of the theorem just established, with the r61les of C and r, and of the ain and /in interchanged.

The points ain and /in need not be defined for every n. Condition (1) is of interest in connection with the condition

n+i

lim \(Z -

1)(Z -

02.)... (Z- ~+ ,n)| =

,(Z) (4)

which has recently been emphasized by the present writer4 as a useful and natural condition in connection with interpolation to analytic functions by polynomials. Many well-known and frequently utilized sets of points /in satisfy condition (4) and hence can be used in connection with (1); compare Theorems Ia-IVb. A similar remark holds for the points ain.

Equation (1) may clearly be utilized to give results on the degree of convergence of the sequence rn(z) to the given function f(z).

The duality that we have pointed out in connection with our main theorem is by no means dependent on condition (1). Let us consider the formal expansion of the particular function f(z) = l/(t - z) as a sequence of rational functions r,(z) found by interpolation in the points ,in and with

poles in the points aO,. We have

1 (S- n) ... (Z -

+f ,n)(t-ln) ... (t- n,n) (5)

t- E: ) , (z--an) . . (z - an,n) (t-ln). . . ( n l,n)(t- Z)

and equation (2) is found formally by multiplying (5) through by f(t)dt/ (2iri) and integrating over r. The new formal expansion is justified if

the right-hand member of (5) approaches zero uniformly for t on r with suitable restrictions on z, f(z) and the Bin. Equation (5) corresponds, if z is considered variable, to interpolation to the function l/(t - z) in the

points /,in by a rational function of degree n with poles in the points ai,n; but if t is considered variable, equation (5) corresponds to interpolation to the function 1/(t - z) in the points ain (and infinity counted twice) by a rational function of degree n + 1 with poles in the points /in. It fre-

quently occurs in practice that this exceptional interpolation at infinity can be replaced by interpolation in two suitably chosen additional points

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VOL. 19, 1933 MA THEMA TICS: J. L. WALSH 1053

an+',,n and an+2,n. Under such circumstances, if the right-hand member of

(5) approaches zero uniformly for t on a curve or set of curves r and z on a curve or set of curves C, if r bounds a closed region or set of closed regions Di containing none of the points ain, all the points /in, and containing a set of closed regions D2 bounded by C, and if C bounds a closed region or set of closed regions D' containing none of the points /i,, all the points ain, and

containing a set of closed regions D' bounded by r, then the sequence of func- tions r,,(z) of respective degrees n with poles in the points ain found by interpo- lation in the points n,, to a function f(z) analytic in D1 converges to f(z) in D2; the sequence of functions rn+1(z) of respective degrees n + 1 with poles in

the points i,n found by interpolation in the points ain to a function f(z) analytic in Di converges tof(z) in D2. The proof may be given directly from (5) and

(2). The general theorem involving condition (1) is a special case, but

one which is in rather more manageable form.

It is not unusual for the development of an analytic function f(z) in a

series of polynomials pn(z) (for example, Legendre polynomials) to be

derived by contour integration from the development

1 - EPn(z)qn(t); t - z

it frequently occurs that a function f(z) analytic in a region or set of

regions can be developed in terms of the p,,(z), and that a function q5(t)

analytic on the complementary set can be developed in terms of the qn(t). Nevertheless, the remark we have made relative to interpolation seems to

be new, and subject to many applications other than those we have given here. In particular, our remark applies to such series as

E -- -- ---, --,an bn(z - /1)(Z - 32) ... (Z- a). (z - a,)(z - 2) .... (z - a3n)'

The former series corresponds to a sequence of rational functions of re-

spective degrees n with poles in the points a1, a2, ..., an found from

f(z) by interpolation at infinity; the latter series corresponds to a sequence of rational functions of respective degrees n with poles at infinity found

from f(z) by interpolation in the points /3, /2, ..., An+. 1 Walsh, J. L., Trans. Amer. Math. Soc., 34, 22-74, 65 (1932). 2 Walsh, J. L., Proc. Nat. Acad. Sci., 19, 203-207, Theorem III (1933). 8 Ketchum, P. W., Bull. Amer. Math. Soc., 39, p. 347 (Abstract) (1933). 4Walsh, J. L., Proc. Nat. A cad. Sci., 19, 959-963 (1933).

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