THEORY OF SERIES AND SUMMABILITY ON THE EXISTENCE OF

THEORY OF SERIES AND SUMMABILITY
ON THE EXISTENCE OF SUMMATION FUNCTIONS
FOR A CLASS OF DIRICHLET SERIES
N
SHMUEL AGMON
Let f(s) be analytic in the half-plane 9l(s) è 00 in part of which it is represented by the Dirichlet series f(s) = X) ane~x"\ A function œ(x) continuous in
the interval [0, 1] is said to be a summation function of f(s) in the half-plane
3L(s) è (To if in each bounded set of the half-plane we have uniformly:
(1)
a»« ( V ) e~x»s = f(s).
lim E
The well known method of summation by the typical means due to M. Riesz
which is applicable whenever f(s) is of a finite order in the half-plane (that is
to say we have there uniformly: /(cr + it) = 0( 111 *)) is of the type (1) with
a>(x) = (1 - x)k'
(k' > k).
The object of this paper is to point out that summation functions satisfying
(1) exist for a much wider class of functions whose order may be infinite. We
prove:
THEOREM. Let f(s) be defined as above. Suppose that for 9l(s) ^ ao we have uniformly :f(<r + it) — 0(eß{ t )) where ß(u) is an increasing function of u such that:
/°° ß(u)u~2 du < oo. Then there exists a summation function w(x) depending only
on ß(u) such that (1) is satisfied.
Another type of theorem is obtained (also generalizing a result of M. Riesz)
when we suppose that f(s) is holomorphic only in the open half-plane 9t(s) > <r0
and continuous in the strip 9i(s) ^ cr0, 11 — U | < 6. If, furthermore, we have:
\f(a + it) | < A exp [<p(l/(T — do) + ß( 111 )] (o- > (To) where <p(u) and ß(u) are
increasing functions of u such that J00 ((p(u) + ß(u))u~2 du < oo, then a summation function œ(x) depending only on <p(u) and ß(u) exists such that (1) is satisfied
uniformly in any segment <r = <r0, 11 — to \ ^ ô' < ò. (co(x) is also independent
of t0.)
R I C E INSTITUTE,
HOUSTON, TEX., U. S.
A.
408
THEORY OF SERIES AND SUMMABILITY
409
GENERATING FUNCTIONS FOR TOTALLY POSITIVE
SEQUENCES. PRELIMINARY REPORT
M. AISSEN, I. J. SCHOENBERG, AND A. WHITNEY
A sequence of reals, {an}, (n = • • • , — 1, 0,1, 2, • • • ) is called totally positive
(t.p.) if the four-way infinite matrix, A = || at-_fc ||, is totally positive; i.e. all the
minors of A are non-negative.
I. J. Schoenberg (Courant Anniversary Volume, 1948) made the following
n
CONJECTURE. If a_fc = 0 (k > 0), oo ^ 0, and [an\ is t.p., then ^^_=o anx
is the Taylor series of a function of the form
l t d + «»*)
(1)
fix) - Ce"x -1
11(1
-ßmx)
i
where C, w, an, ßm are non-negative constants (C ?£ 0) and ]C°° «n, X)°° ßm
îonverge. That such functions generate t.p. sequences can be proved without
difficulty.
We prove the following weaker form of the conjecture.
n
THEOREM. If a-k = 0 (k > 0), Oo ?± 0, and {an} is t.p. \\ then 2 ? Q>nX is the
Taylor series of a function of the form
n d + «.*)
fix) = 6ff(°> -i
I I U - ßfnX)
1
öftere cew , ßm satisfy the same conditions as in the conjecture, and g(x) is an entire
function. Furthermore eo{x) generates a t.p. sequence.
UNIVERSITY OF PENNSYLVANIA,
PHILADELPHIA, PA., U. S.
A.
ON T H E SUMMATION OF MULTIPLE FOURIER SERIES
K. CHANDRASEKHARAN
Let f(x) = f(xi, • • • , xk) be a function of Lebesgue class Li and of period 2w
n each variable. Let fQ(x, t) be the spherical mean of/, of order zero, at x, and
[et S&(x, R) be the Riesz mean of order ô ^ 0 of the multiple Fourier series of /
when summed spherically (cf. S. Bochner, Trans. Amer. Math. Soc. vol. 40
(1936) pp. 175-207). Then it is a fundamental result of Bochner that (i) fQ(x, t) =
)(1) as t -» 0 implies SS(R) = o(l) as R -> oo for ô > (k - l)/2, and (ii) that
?
or a fixed x, no local hypothesis on/will yield the conclusion for ò g (k — l ) / 2
.he value (k — l ) / 2 being the critical exponent. Now, spherical means of order
410
SECTION II. ANALYSIS
p ^ 0 have been defined (cf. K. Chandrasekharan, Proc. London Math. Soc.
vol. 50(1948) pp. 210-229), and it seems plausible, in analogy with (ii), that for
p > 0, no hypothesis of the form fv(t) = o(l) as t —» 0 will yield the conclusion
S\R) = o(l) as R -> oo for Ò ^ p + (k - l ) / 2 . We show that this is not the
case by proving a number of results with varying local hypotheses on fp(t)
leading to conclusions about SS(R) for h + (k - l ) / 2 < 8 ^ p + (k - l)/2
where h is the greatest integer less than p. We also consider the converse problem
where we postulate the behaviour of SS(R) as R —» oo and deduce that of fp(t)
as t —> 0.
TATA INSTITUTE OF FUNDAMENTAL RESEARCH,
BOMBAY, INDIA.
ON THE PARTIAL SUMS OF A TAYLOR SERIES
V. F. COWLING
The geometric technique of value region considerations as introduced by
Leighton and Thron (Duke Math. J. vol. 9 (1942) pp. 763-772) for continued
fractions is adapted to Taylor series to prove the following typical result: Let
sn(z) = a0 + a[z + • • • + anzn. If | aft/aft_i | ^ 1, n = 1, 2, • • • , and if z lies
within the elliptical region Ek: (x - (k - 2)/2k)2 + (Ä.2/4(/c - l))y2 = 1/4
(z = x + iy) for k ^ 2 then | sn(z) — a0k/2 \ S | a0 \ k, n = 0,1, 2, • • • .
UNIVERSITY OF KENTUCKY,
LEXINGTON, KY., U. S. A.
SUR LES THÉORÈMES TAUBÉRIENS POUR LES
SÉRIES DE DIRICHLET
HUBERT
DELANGE
Ks
Soit la série de Dirichlet YJX°° anè~ , où 0 < Ai < X2 < • • • < \n < • • • et
lim Xn = + oo. Supposons cette série convergente pour 9l[s] > 0 et désignons
par f(s) la fonction qu'elle représente.
Le résultat suivant est bien connu:
p étant un nombre positif ou nul, si, quand s tend vers zéro par valeurs réelles
positives, f(s) = As~p + o[s~p], on peut en conclure que, pour n infini,
ai + a2+ - - + an = [A/Y(p + 1)] \pn + o[\y, pourvu que Tune des deux
conditions suivantes soit satisfaite:
(1) \an\S
M(\n - \n-i)\pnl pour n â 2,
(2) an réel, an è —M(\n — X^-OXT1 pour n è 2, 'et lim inf„_*+<» \Hpan à 0.
Notre but est d'indiquer comment, si Ton renforce l'hypothèse sur f(s) en
faisant intervenir son comportement pour les valeurs complexes de s, soit simple-
THEORY OF SERIES AND SUMM ABILITY
411
ment au voisinage du point s = 0, soit au voisinage de tous les points de la
droite 9i[s] = 0, on peut remplacer les conditions (1) et (2) par de plus larges.
Dans nos énoncés, il est entendu que seules interviennent les valeurs de s
satisfaisant à 9L[s] > 0, et que s~p = f~pe~pl° si s = re%e avec | 0 | < TT/2.
1. Si l'on suppose que, quand s tend vers zéro par valeurs réelles ou complexes, f(s) — As~p + 0[rl~~p<p(r)], où <p(r) est une fonction positive non-croissante pour r > 0 et telle que l'intégrale J0 <p(r) log (1/r) dr dans le cas p > 0,
ou /o <p(r) dr dans le cas p = 0, soit convergente, on peut remplacer (1) et (2)
par
{1') an = o[(Xn - Xn_x)XpJ
(n - * + oo),
(2') an réel, Min (an , 0) = o[(Xn — Xn-i)Xn], et
lim infn_>+00 \npan â 0.
2. Â l'hypothèse indiquée ci-dessus, ajoutons l'une des suivantes suivant
que p = 0 ou p > 0 :
a. Pour le cas p = 0: f(s) reste bornée au voisinage de chaque point de la
droite 9l[s] = 0,
b. Pour le cas p > 0: Pour chaque y 5* 0, quand s tend vers zéro,
on a f(iy + s) = 0[r~pyl/(r)], où \J/(r) est une fonction positive non-croissante
pour r > 0 et telle que l'intégrale J0 \//(r) dr soit convergente (cette fonction
pouvant dépendre de y).
Alors (1) et (2) peuvent être remplacées par:
<1") | fln | ^ M(\n - \n-i)K
pour
(2/;) an réel, a« à — M(\n — ^n-ii^n
n ^ 2,
pour n ^ 2,
et
lim infn^+00 X^pan è 0.
En fait, ces résultats sont établis comme conséquences d'énoncés plus généraux relatifs à l'intégrale de Laplace.
N.B. Depuis que cette communication a été présentée au Congrès, l'auteur a
reconnu que les hypothèses sur f(s) dans (2) peuvent être élargies. Cf. C. R.
Acad. Sci. Paris t. 232, pp. 589-591.
UNIVERSITY OF CLERMONT-FERRAND,
CLERMONT-FERRAND, FRANCE.
LES PERMUTATIONS CLIVÉES
A R N A U D
D E N J O Y
/ désignant l'ensemble des entiers positifs, (N) son ordination dans le sens
de la croissance, toute ordination différente étant appelée une permutation de
(N), la somme (imbriquée) d'une suite de permutations Pk (k = 1, 2, • • • ) est
la permutation P des entiers n décrivant 2":
n=f1(k\j)
=
ik + j){ +j 1)
K - +j
0 ^ 0 , / . + ^ 2)
412
SECTION II. ANALYSIS
définie par
fi(k | 0) < fi(k\j)
< fx(k + 1 | 0) s i i è 1,
fi(k I j) < fiik | f) si j < f selon Pk .
Posons
fpiS | j) = /p(fci, • - •, K I j) = /*-i(&i, • • • , V i I ip-i) si jp-i = fi(kp | ; ) .
Une permutation P est dite cKvée si elle est soit la permutation-unité (N non
permutée) soit une somme de permutations clivées (c'est une définition antirécurrente).
G étant une famille de suites finies d'entiers positifs S = (ki, • • • , kv), une
suite S de G est dite ouverte ou close dans G selon qu'elle commence ou non une
autre suite de G; G est dite progressive si toute suite commençant une suite de
G est dans G; douée du caractère (A) si, pour toute suite ouverte S de G, toutes
les suites (S, K) (K à 1) sont dans G.
A toute permutation clivée P correspond une famille G progressive dont les
suites S sont ainsi caractérisées que les nombres n = fp(S \ j) où j décrit I,
forment une section ordinale de P, immédiatement précédée par fp(S | 0) et
suivie par fP(Sp-i,
kp + 1 | 0). La suite S est close ou bien ouverte dans G
selon que la permutation des j semblable à l'ordination des n = fp(S \ j) par P
est ou non identique à (N).
Réciproquement, soit G une famille de suites S = (ki, • • • 7 kp) et $((?) l'ensemble des formes <j>, savoir fp(S \ 0) si kp ^ 2 et fp(S \ j)(j è 1) si S est close.
1° Si G est progressive et vérifie la condition (A) : L'ensemble ^ des valeurs n des
formes <j> est identique à I; 2° 3>((?) étant ordonné alphabétiquement selon la
croissance des arguments des 0 et ^ semblablement à $(G), la permutation
obtenue est clivée. Pour que la permutation clivée P soit bien ordonnée, il faut
et il suffit que G vérifie la condition (B) : Toute suite indéfinie KI,K2, • • • d'entiers
positifs commence par une suite close de G.
UNIVERSITY OP PARIS,
PARIS, FRANCE.
A CLASS OF NONHARMONIC FOURIER SERIES
R. J. DUFFIN AND A. C. SCHAEFFER
A sequence {Xn}, n = 0, ± 1 , ± 2 , • • • of real or complex numbers is said to
have uniform density 1 if there exist constants L and 8 such that | Xn — n | ^ L
and | Xn — \m | ^ 8 > 0 for n ?£ m. It was shown by the authors (Amer. J.
Math. vol. 67 pp. 141-154) that if an entire function f(z) of exponential type
y, y < ir, is uniformly bounded at the points {X„}, then it is uniformly bounded
on the real axis. This result was applied to problems concerning the coefficients
of power series. The central result of this paper is that if $3 I /(^«) 12 < °°, then
THEORY OF SERIES AND SUMMABILITY
413
fix) £ L2(— oo. oo). An essentially equivalent statement of this result is that
if g(x) £ L2i~y, y), then there are positive constants A and B which depend
exclusively on 7, L, and 8 such that
y
/
y
M
I
»y
I gix) |2 dx g X)
/
—00 j J— y
~y
|2
0(B) exp (iXn&) <fo
I
_§ B /
J— y
| g(x) \2 dx.
This is referred to as the frame condition. If Xn = n, then A = B = 2ir is Parseval's theorem. The proof that a constant B exists is quite direct. The proof of
the existence of the constant A is made to depend on a closure theorem for
functions analytic in a circle. The existence of A implies that the set
{exp ii\nx)} is closed on the interval (—7, 7); however, closure on this interval
does not imply the existence of A or of B.
Abstract considerations in Hilbert space show that the frame condition gives
nonharmonic Fourier series of the form X) ^ exp ii\nx) quite similar properties
to ordinary Fourier series. However, the situation is more complicated because
the set {exp ii\nx)} is highly dependent on an interval of length less than 2r.
Most of the previous studies of nonharmonic Fourier series have been for the
independent case. In view of this fact, an L2 theory of these series is developed
which considers properties of conjugate frames, expansion coefficients, mean
convergence, and pointwise convergence. Some of these results overlap those
of Paley and Wiener, Levinson, and Boas.
CARNEGIE INSTITUTE OF TECHNOLOGY,
PITTSBURGH, PA., U. S. A.
UNIVERSITY OF WISCONSIN,
MADISON, W I S . , U. S. A.
THE GENERAL FORM OF HYPERGEOMETRIC SERIES
OF TWO VARIABLES
A. ERDéLYI
Sponsored by the ONR
A formal double power series X) Amnxmyn is called a hypergeometric series
if Am+itVl/Amn = fim,n) and Amtn+i/Amn
= gi^n) are rational functions of m
and n. The most general form of such series has been subject to controversy
and as far as the author knows the explicit result is given here for the first time.
Appell, Birkeland, Horn, Kampé de Fériet, Mellin, and others have investigated series in which Amn is a gamma product, i.e. of the form
(1)
ymn = u (rfe + nun, + Vin)/T(*)}
where the c. are arbitrary (possibly complex) constants and the Ui and Vi are
(positive, negative, or zero) integers. The question has been asked whether
this is the most general form of the coefficients of a hypergeometric series.
414
SECTION II. ANALYSIS
Clearly,
(2)
f(m,n)g(m
+ 1, n) = firn, n +
l)gim,n)
for all non-negative integers m, n and hence identically in m and n, and conversely, it is easily seen that every rational solution of (2) generates a hypergeometric series. Birkeland (C. R. Acad. Sci. Paris vol. 185 (1927) p. 923) stated
that every rational solution of (2) can be decomposed into linear factors, and
this leads essentially to (1) as the most general form. However, 0 . Ore (C. R.
Acad. Sci. Paris vol. 189 (1929) p. 1238) noted that Birkeland's result is not
entirely general and gave (Journal de Mathématiques (9) vol. 9 (1930) p. 311)
a thorough analysis of the rational solutions of (2).
Ore's results should enable one to construct the most general hypergeometric
series, but it seems that this construction has not been carried out. In the present
paper Ore's theorems on the rational solutions of (2) are analyzed and their
bearing on the structure of Amn is investigated. It turns out that the only factors
disregarded by Birkeland are rational functions of m and n. More precisely,
the conclusion is reached that the coefficients of the most general hypergeometric series of two variables are of the form Amn = R(m,n)ymnambn where R is
a fixed rational function of its two variables, a and b are constants, and ymn is a
gamma product. This is equivalent to saying that the most general hypergeometric series of two variables results from i h e application of a rational differential operator R(x(d/dx), y(d/dy)) to a hypergeometric series of the Horn-Birkeland type. This answers a question left open by Horn (Math. Ann. vol. 105
(1931) p. 381) and justifies the general practice of restricting attention to series
of the Appell-Horn-Birkeland type.
CALIFORNIA INSTITUTE OF TECHNOLOGY,
PASADENA, CALIF., U. S. A.
ON THE ASYMPTOTIC DISTRIBUTION OF
CERTAIN SUMS
N . J.
FINE
Let ((t)) = t — [t] — 1/2, the square brackets denoting the greatest integer
function. It has been shown by Kac [J. London Math. Soc. vol. 13 (1938) pp.
131-134] that the distribution of the sums N~ll2^n<N((2nt))
is asymptotically
normal with mean 0 and variance 1/4. The author, in §8 of his dissertation
[Trans. Amer. Math. Soc. vol. 65 (1949) pp. 372-414], considered the sums
X^n<_v((2w£ ~ 1/2)) and proved that they are bounded uniformly in N and t.
The question was therefore raised as to the behavior of iV"1/2]Cn<jv((2n_; — ß)),
0 _ë ß < 1. It is proved here that the distribution of these sums is asymptotically normal with mean 0 and variance a2 = ^2n^i2~n((ß — ßn))2, where ßn is the
fractional part of 2nß. Thus ß = 1/2 is the only case with zero variance. Ex-
THEORY OF SERIES AND SUMMABILITY
415
)licit results are obtained for the joint distribution of pairs of sums of the above
ype.
Similar results are obtained for the relative frequency with which the fracional part of 2nt, 0 S n < N, falls in a given interval. Here, again, the distri>ution is asymptotically normal, If the interval is (0, ß), then the mean is ß
nà the variance is r2/N, where r2 = ß - ß2 + 2X^i2~" n {min (ß, ßn) - ßßn\.
UNIVERSITY OF PENNSYLVANIA,
PHILADELPHIA, PA., U. S. A.
A CLOSURE CRITERION FOR ORTHOGONAL FUNCTIONS
Ross E.
GRAVES
In this paper the author gives a simple necessary and sufficient condition for
. sequence of orthogonal functions to be closed in L2. In theory, the question
f closure is reduced to the evaluation of certain integrals and the summation
f an infinite series whose terms depend only upon the index n. The simplest
3rm of the final result reads
THEOREM la. Let {<pn) be a set of orthonormal functions on a finite interval
a, b) and let c be any number such that a ^ c g b. Then
2
/
7 i = l Ja
2
2
/ <Pn(t) dt dx £ J[(a - c) + (6 -* c) ],
| Jc
)here equality holds if and only if {<pn} is closed in L2 on (a, b).
Actually, the above theorem is a special case of a more general result.
THEOREM I. Let pit) be a function whose zeros and discontinuities have Jordan
intent zero such that for x £ (a, b), p(t) £ L2 on min (c, x) < t < max (c, x),
Hier e a S c ^ b (a and b may be infinite; c is fixed). Let w(x) be a measurable
mction almost everywherefiniteand positive such that w(x) j * \ p(t) \2 dt £ Li on
%, b). Then for any family of functions {<pn} orthogonal and normal on (a, b),
OO
E
nU I
/
n = l Ja
nX
/ viÏÏPnit) dt
22
pu \
wix) dx g /
Je
pX
/ I pit) |2 dt wix) dx,
Ja I Jo
here equality holds if and only if {<pn} is closed in L2 on (a, b).
The insertion of the functions pit) and wix) serves two purposes. First, it
nables us to extend the results of Theorem la to the case where the interval
i, b) is infinite; and, second, proper choice of these functions greatly facilitates
le calculation of the integrals involved and the summation of the resulting
îries.
Applications of Theorem l a are made to the trigonometric and Legendre
motions, while the closure of the Hermite and Laguerre functions is established
y means of the more general Theorem I.
UNIVERSITY OF MINNESOTA,
MINNEAPOLIS, MINN., U. S. A.
416
SECTION IL ANALYSIS
SUR UNE NOTION D E CONTINUITÉ RÉGULIÈRE AVEC
APPLICATION AUX SÉRIES D E FOURIER
J. KARAMATA
Soit \(t) une fonction logarithmico-exponentielle définie pour t > 0 et telle
que
\(t) —» 0 lorsque t —» 0.
Nous dirons que f(x) est régulière d'ordre X(t) au voisinage du point x, si en
posant
*(0 = ffr + *>)+ f(x - t) - 2/(_r),
on a
*>(0 = 0{X(OJ pour t £ t ' £ t + t>\(t), t -> 0.
Plus \(t) tend rapidement vers zéro, plus la fonction f(x) est régulière.
Lorsque la fonction satisfait à la condition de Lipschitz
|*(0I £M'\®,
elle est certainement régulière d'ordre \(t), mais elle peut l'être d'un ordre plus
élevé.
T H é O R è M E . Lorsque la fonction f(x), continue au point x, y est régulière d'ordre
\(t), sa série de Fourier convergera en ce point si
L+o x(0t dt
converge.
UNIVERSITY OF GENEVA,
GENEVA, SWITZERLAND.
ON IRREGULAR POINTS OF NORMAL CONVERGENCE
AND M-CONVERGENCE FOR SERIES
OF ANALYTIC FUNCTIONS
BENJAMIN LEPSON
A series of complex-valued functions is said to be M-convergent on a set S
if the sum of the least upper bounds on S of the terms is finite, while, following
Laurent Schwartz, the series is said to be normally convergent on S if the series
of absolute values is uniformly convergent on S. Given a series of functions
each analytic in a domain D of the complex plane which converges absolutely
at every point of D, we define the sets of irregular points in D of normal convergence and of M -convergence in the expected manner, and denote these sets
by N and M respectively. Let U be the set of irregular points in D in the sense
of Montel for the sequence of partial sums of the given series. It was shown
THEORY OF SERIES AND SUMMABILITY
417
previously by the author (Bull. Amer, Math. Soc, Abstract 56-3-247) that
N = M and that M is nowhere dense. It is clear that M contains U. The following example shows that M may actually be larger than U.
Let D be the interior of the square bounded by the axes and the lines x = 1
and y = 1. Let Pn(z) be a polynomial such that | Pn(z) \ < 1/n in that portion
of D with y ^ 1/2 + 1/n and | Pn(z) — 1 | < 1/n2 in the portion of D with
y t 1/2 + 2/n. Let Qn(z) = Pn+i(z) - Pn(z). Let {Nn} be a sequence of
positive integers such that | Qn(z) \ < Nn for all z in D, and put Rn(z) =
Qn(z)/(nNv). Let Sn(z) be the ?ith term of the series whose first Ni terms
are each Ri(z), whose next 2N2 terms are each R2(z), etc. Then the series
Si(z) — Si(z) + S2(z) — S2(z) + • • • is the desired example, since U is empty
while M is that portion of the line y = 1/2 contained in D.
If the domain D is simply connected, the set M can be characterized geometrically following the method of Hartogs and Rosenthal. It is found that the
necessary and sufficient conditions that a set be the set of irregular points in
D of M -convergence of an absolutely convergent series of analytic functions
and that it be the set of irregular points in D in the sense of Montel for a convergent sequence of analytic functions are the same.
INSTITUTE FOR ADVANCED STUDY,
PRINCETON, N. J., U. S. A.
DIRECT THEOREMS ON METHODS OF SUMMABILITY
G. G. LORENTZ
By a direct theorem we understand an assertion that under some simple
conditions which restrict the structure of sn , this sequence is summable by a
given method A = (amn), that is lim.m-+x22namnSn exists. Thus, regularity of
A is a direct theorem. But besides convergent sequences, all useful methods
possess wide classes of summable divergent sequences, which can be easily
described. Thus if (and only if) the variation of the rath row of (amn) tends to
D, all "almost convergent" sequences sn described by
F:
lim - (sn+i + . . . - ) - sn+p) = s
uniformly in n
_9-*00 P
are A summable. We further call an increasing function ti(n) —> 00 a summability function (si.) of the first or second kind, according to which of the following conditions implies A summability of sn : sn = 0 except for a subsequence
sn„, whose counting function œ(n) (i.e. the number of nv ^ n) is ^ ß ( n ) ; or
Si + • • • + sn = 0(Q(n)). The following conditions characterize method A
chicli have ü(n) as a s i . of the first or second kind: ììmm-,0OA(m, ti) = 0, A (m, ti)
3eing the least upper bound of X..* I amnP | for all sequences nv with œ(n) _g
ì(n), or lim^ooXlnfifa) | amn — am,n+i | = 0. And in order that A have some
418
SECTION II. ANALYSIS
s i . of the first or the second kind it is necessary and sufficient that amn—*0
uniformly for m —> oo or that X!n | <W — «m.n+i | —» 0. Since F has no si., it
follows that any method A 3 F is strictly stronger than F even for bounded
sequences. Replacing summability by absolute A summability in the above
definitions, absolute s i . are obtained. A method A has absolute s i . of the first
kind if and only if the variation of the nth. column converges to 0. The statement
that a condition on sn is not a Tauberian condition for a given method A is a
direct theorem. Therefore, theorems and methods described above are a good
means of showing that a certain Tauberian condition is the best possible of
its kind. We have: (a) If ti(n) is a s i . for A, then sn — Sn~i = o(fì(n)_1) is not
a Tauberian condition f or A ; (b) If nx < n2 < • • • , cn ^ 0 and
n
(
*+i
^
lim <Max X) | a»n 1 > = 0,
m-*oo ^
v
n„+l
')
n
_»+i
X) cn à 8 > 0,
n„+l
then sn — Sn-i = 0(cn) is not a Tauberian condition. To apply (a) and as a
problem interesting in itself we find all s i . of the Abel, Riesz, Euler, Borei,
Riemann, Hausdorff methods. The determination of the absolute s i . is more
difficult.
UNIVERSITY OF TORONTO,
TORONTO, ONT., CANADA.
T H E DETERMINATION OF CONVERGING FACTORS FOR THE
ASYMPTOTIC EXPANSIONS FOR T H E WEBER
PARABOLIC CYLINDER FUNCTIONS
J. C. P. MILLER
When determining the numerical value of a function from an expansion in
series, the method commonly used is to evaluate the series in the form
S = Sn + Rn
in which Sn is a partial sum, and Rn the corresponding remainder term. For
many series, n may be readily chosen so that Rn is negligible and S may be taken
as approximately equal to Sn . When, however, convergence is slow, or, as in
the case of asymptotic expansions, the series is divergent, it may be necessary
to evaluate Rn closely, rather than to reduce its value below a prescribed, negligible upper limit. J. R. Airey introduced the idea of a converging factor; i.e., he
expressed Rn in the f o r m i Cnun , where ±un is the nth. term in the series, and
concentrated on the expansion of Cn in the form of a series suitable for computation. The methods by which he obtains expansions for Cn are empirical in
nature, but numerical tests have justified his results.
The purpose of this note is to present expressions for the converging factors
THEORY OF SERIES AND SUMMABILITY
419
appropriate to the asymptotic expansions for the parabolic cylinder functions.
These are solutions of Weber's differential equation
(A)
g - (ia* + a)y.
There are two independent expansions, both satisfying the equation (A),
and each such that individual terms of the expansions satisfy a two term recurrence relation. It follows that in each case Cn satisfies a second order differential equation and a first order difference equation. If un is the term of least modulus in either series it is found that h(= f.r2 ± a + 1 — n) is small; the sign
depends on which of the two series is considered. From either, or both, of the
equations, differential or difference, it satisfies, Cn can be determined in the
form of a series of the form
in which n does not occur explicitly. Results have been determined up to and
including 04(A) in each case, apart from the term independent of h in a* in the
case of that asymptotic expansion which ultimately has all its terms of similar
sign. The evaluation of this constant would require the preliminary evaluation
da&/dh.
NATIONAL BUREAU OF STANDARDS,
WASHINGTON, D. C , U. S. A.
MODULAR TRANSFORMATION OF CERTAIN SERIES
SAUL ROSEN
The Hardy-Littlewood method as modified by Rademacher can be used to
obtain an infinite series expansion for modular functions. These series themselves will then exhibit an absolute or relative invariance under modular transformations. The problem of applying the transformations:
,
T
=
ar + b
CT + d'
a b
c d
= 1
directly to the series has previously been considered in the case of the absolute
modular invariant J(r) by Rademacher. This is now extended to a considéraLion of functions belonging to modular subgroups, in which the invariance is
not absolute, where certain roots of unity appear in the transformation equations. The function
fix) = U (1 + *") = e - " " " ^ ,
x = e2"v
420
SECTION II. ANALYSIS
is considered in detail. This function belongs to r 0 (2), the modular subgroup for
which c = 0(mod2). The Hardy-Littlewood method leads to the expansion
/(#) = Y^.=*QnXn where
o = — _ë
2
1/2
2rihn,k
X! ti
ft^l.fcodd AmodA;l(A,fc)=l
e'
^T
Y)
( 7T ,Y
^24/r2/
JI=1 MK/* ~" 1) ' V
[-(-à)rThe fì^.A are rather complicated roots of unity related to those which appear
in the transformation theory for the r\ function.
In this paper the series above is shown to belong to T0(2) by direct application of the transformations which are the generators of T0(2) : r' = r + 1
and r' = r/(2r + 1). In the latter case it was found convenient to consider
r ' = r/(2r + 1) as the resultant of the transformations r' = — 1/r, T' =
r — 2, and / = — 1/r. r' = — 1/r is not in T0(2). It is thus necessary to introduce series corresponding to the cosets of T0(2), which is of index 3 in the
full modular group. These series are the expansions, by the Hardy-Littlewood
method of the associated functions fi(x) and f2(x), related to fix) by modular
transformation. A reciprocity law for the roots of unity tih,k is derived and used
in the transformation of the series.
DREXEL INSTITUTE OF TECHNOLOGY,
PHILADELPHIA, P A . , U. S. A.
UNIQUENESS THEORY FOR HERMITE SERIES
WALTER RUDIN
We consider series of the form S = ^JSan<j>nix), where 4>n(x) is the normalized Hermite function c?f%l%Dn{e*\
cn = (-l) n (2^!7r 1 / 2 ) 1 / 2 .
DEFINITIONS. (A) Let F(t) be defined for | x — t \ < 8. For 0 < A < 8, let
yit) = yit; F, h) be the solution of y" - (t2 + l)y = 0, such that y(x + A) =
F(x + A), y(x - A) = F(x - A). Put AkF(x) = y(x; F, h) - F(x), andAFfa) =
Hmh-*o2AhF(x)/h2. A*F(x), A*F(x) are defined likewise with lim sup, lim inf in
place of lim. If F"(x) exists, then AF(x) = F"(x) - (x2 + l)F(x). In particular, A<j>n(x) = — (2n + 2)(/)n(x). (B) We write / Ç Hp if / Ç L on every finite
interval, and if /ü* | xvf(x) \ e~x2/2dx < °o. If / £ Hp for every p ^ 0, we write
feH.HfeH,
and an = \!„f(x)<t>n(x) dx, we write S = S(f). (C) If / £ Ho,
put tif(x) = — f1OQf(t)K(x, t) dt, where K(x, t) ìé the Green's function of the
system y" - (x + l)y = 0,z/(±°o) = 0. (D) We write S <E K if
-T/oan<t>n(x)/(2n + 2) = S(F),
and F is continuous. (E) f*(x), f*(x) denote the upper and lower Poisson sums
of S.
THEORY OF SERIES AND SUMMABILITY
421
RESULTS. I. Let p è 0 be given. Suppose (1) F(x) is continuous and F G H.p J
(2) A*F(x) > — oo and A*F(x) < +°o_ except possibly on countable sets Z?j
and E2 ; (3) lim s\iph->QAhF(x)/h è 0 on Ei, lim inîh^0AhF(x)/h g 0 on E2 ;
(4) there exists j/ G # p such that y (a) g A*JP(îB). Then AF G Hp and _F(a) =
fìAF(a;) for all x.
II. Suppose £ G i£. If F satisfies (2), (3), and if there exists y G H such
that y(x) Û A*F(x), then S = S(AJF).
III. Suppose S G /£. If (i) U(x) > — oo and/*(a;) < + oo. except possibly on
countable sets Ex and E2 ; (ii) F satisfies (3) ; (iii) there exists y G H such that
y(x) S U(x), thenf*(x) = U(x) p.p. and S = £(/*) = £(/*).
IV. Suppose an = o(nlß). If — oo < y(x) S U(x) _ë /*(a) < + » , where
» G H, then S = £(/*) = «(/*)V. If S converges to a finite function/ G ff, then S = $(/).
DUKE UNIVERSITY,
DURHAM, N. C ,
U. S.
A.
GENERALIZED FOURIER INTEGRALS
WARD C. SANGREN
It is well known that the Sturm-Liouville expansion of an integrable function
fix) behaves as regards convergence in the same way as an ordinary Fourier
series. It is shown that there exists a parallel situation in the case of certain
generalized Fourier integrals and the ordinary Fourier integrals. The generalized
Fourier integrals which are dealt with are special cases of the more general
types of expansions obtained by E. C. Titchmarsh (Eigenfunction expansions
associated with second-order differential equations, Chapter III).
In a previous paper presented to the American Mathematical Society, expansions were obtained which are associated with second-order differential
equations with step-function coefficients. These integral expansions are slight
extensions of ordinary Fourier integrals. Upon considering coefficients which
are discontinuous at interfaces and satisfy certain restrictions, it is possible to
obtain integral expansions which extend these expansions in a fashion similar
to the way the generalized Fourier integrals of the previous paragraph extend
ordinary Fourier integrals. In particular, the equi-convergence properties continue to hold.
MIAMI UNIVERSITY,
OXFORD, OHIO, U. S.
A.
422
SECTION II. ANALYSIS
LES FONDEMENTS D'UNE THÉORIE GÉNÉRALE DE
SÉRIES DIVERGENTES
RICARDO SAN JUAN LLOSâ
L'indétermination d'une fonction par son développement asymptotique et
l'illégitimité de la dérivation qui, remarquées par Poincaré dans son mémoire
original, furent, peut-être, la cause de ne pas remplacer sa convergence asymptotique par l'ordinaire de Cauchy [G. H. Hardy, Divergent series, Oxford, 1949, p.
28], on peut les reparer moyennant Vapproximation asymptotique optime (a.a.O.),
c'est-à-dire, dont les bornes mn du développement asymptotique f(z) ~ X_ln=o UnZn
dans un domaine renfermant l'origine soient ordonnément plus petites, mn <mn,
que les homologues mn d'une de toute autre fonction holomorphe quelconque
dans le même domaine, qui remplissent la condition d'unicité de CarlemannOstrowski. L'existence de cette fonction est la condition nécessaire et suffisante
pour la coincidence de toutes les fonctions holomorphes dans le domaine où elles
vérifient cette condition, coincidence que certainement n'a pas lieu toujours.
[R. San Juan Llosâ, Acta Mathematica t. 75, pp. 247-254.] Approximations
optimes sont la prolongation analytique ordinaire, la somme de Stieltjes, celles
qui remplissent la condition de Watson-Nevanlinna, etc.
La dérivée et primitive de l'a.a.o. en | zlla — pel*o/a | < p, a > 0, pQ — 8 <
p < Po + 8 c'est l'a.a.o. de la série dérivée et primitive respectivement, si les
bornes ra^ sont croissantes: mn < mn+i. Elle a aussi d'autes propriétés formelles.
En généralisant la méthode de décomposition appliquée premièrement [R. San
Juan Llosâ, Ibid.] a e~l , il en résulte que toute fonction \l/(t) avec
fo tn d\p(t) < oo peut se décomposer en deux a(t) + â(t) = \p(t) avec p.n =
fîfdW)
< a>,tellesqueE«=il/(M,n + r-p n ) 1/M = « , E ï ^ i VÛ^-K-p.) 1/n = « ,
pn > 0. étant une suite positive quelconque telle que X_]*=i l/pî/ n = °°. Comme
corollaire immédiat il en résulte ime ample généralisation d'un théorème de M.
Mandelbrojt [S. Mandelbrojt, Acta Mathematica t. 72, p. 16]. L'application aux
coefficients de séries de fractions simples avec des dérivées nulles à l'origine conduit, par la transformation de Laplace, à des séries avec des approximations
optimes différentes en deux domaines Rf < R. L'existence de a.a.O. différentes
dans domaines R et R' emprétants (R 0 Rf ^ 0) ou R* < R nous met devant
un dilemme: ou bien on s'abstient d'approximations optimes aussi naturelles
que celles de Stieltjes, ou on étendre la notion de somme d'ime série potentielle
X_]n=o anzn dans un domaine, étant admis que celle-ci n'est pas un concept inherent à la série exclusivement, mais à celle-ci plus le domaine.
Ceci n'empêche pas d'adopter comme prolongation radaile f(z) sur (0, 1)
l'a.a.o. dans un domaine | zlla — 1 | < 1, a > 0, si celle-ci coincide avec une
approximation holomorphe quelconque qui réalise la condition d'unicité dans un
autre d'une ampleur plus petite a' < a. On définisse X!_-*=o o,n =
hmz^i-f(z).
Quelques propriétés formelles subsistent ainsi et d'autres avec des restrictions,
suffisantes pour développer une théorie de séries divergentes, dans laquelle une
THEORY OF SERIES AND SUMMABILITY
423
autre quelconque soit contenue, à cause de la nécessité des conditions d'unicité
adoptées.
UNIVERSITY OF MADRID,
MADRID, SPAIN.
ON A CLASS OF INTEGRAL-VALUED DIRICHLET SERIES
ERNST G. STRAUS
It is shown that if the Dirichlet series f(s) = X}*^ 0 * is integral-valued for
s = 1, 2, • • • , then either ak = 0 for almost all k, or Xl"-! I 0* | 1_e diverges
for every e > 0. Similar theorems are obtained if fis) is known to be rational
or algebraic for s = 1, 2, • • • . A theorem of the same type is obtained for
fis) = X^=i a * "~ _ö£-iM y a n d hence (since we did not assume the ak or bu to
be distinct) for every fis) = X3"-iw*a£ , where nu is integral.
UNIVERSITY OF CALIFORNIA,
Los ANGELES, CALIF., U. S.
A.
TAUBERIAN THEOREMS FOR SUMMABILITY (fii)
OTTO Szlsz
Denote by sn the nth. partial sum of a seriesXiïW ; if the series
(2A) X] ft-1Sn sin nh = R(h)
converges for small positive h, and if R(h) —> s as h —> 0, then _X <*>n is called
summable (Ri). This summation method has particular significance for Fourier
series; it was recently investigated in a paper by Hardy and Rogosinski (Proc.
Cambridge Philos. Soc. (1949)). The method is not regular; thus additional
conditions are needed in order that convergence imply summability (Ri). We
establish the condition (*) _XnW(| civ I — «0 = 0(1). We also prove that condition (*) and Abel summability imply Ri summability. For Fourier series we
prove summability (Ri) at each point of continuity. Finally we compare (Ri)
with Cesàro summability.
NATIONAL BUREAU OF STANDARDS,
Los ANGELES, CALIF., U. S. A.
424
SECTION II. ANALYSIS
REPRESENTATION OF AN ANALYTIC FUNCTION BY GENERAL,
LAGUERRE SERIES. PRELIMINARY REPORT.
OTTO
Szisz
AND NELSON YEARDLEY
The authors prove the following theorem for the general Laguerre expansion
Siz) ~ X)n=o a»a)Z.»a)(s) of the analytic function f(z) in the complex xyplane, where a(na) are the Fourier-Laguerre coefficients and {Lna)(x)} is the
set of general Laguerre polynomials of order a and degree n, orthogonal over
the interval (0, oo ) with respect to the weight function e^x*. In order that f(z)
possess a general Laguerre expansion of order a (a > —1) which converges toit for every point z inside the parabola p(da) : y2 = 4d«(_r + d2a) (where da =
—lim sup (2n 1/2 ) _1 log | a" \ ) it is necessary and sufficient that f(z) be analytic
in p(da) and that to every ba , 0 _g ba < da , there correspond a positive number B(oc, b) such that \f(z) \ ^ B(a, b) exp {x/2 - \ x \1/2[b2a -(r - x)/2]1/2}
(where r = (x + y2)112) for every point z in and on the parabola p(ba). Theabove theorem is equivalent to a generalization of a theorem by Pollard (see
Ann. of Math. vol. 48 (1947) pp. 358-365) and the methods used by the*
authors are essentially generalizations of those employed by Pollard for the casea = 0 in his paper referred to above.
UNIVERSITY OF CINCINNATI,
CINCINNATI, OHIO, U. S. A.
NATIONAL BUREAU OF STANDARDS,
Los ANGELES, CALIF., U. S. A.
PURDUE UNIVERSITY,
LAFAYETTE, IND., U. S. A.
SUMMABILITY MATRICES COINCIDENT WITH REGULAR
MATRICES, BANACH SPACE METHODS
ALBERT WILANSKY
Let A be a conservative summability matrix (i.e. one summing all convergent
sequences). With a natural assumption (normality) we ask if there is a regular
matrix summing exactly the sequences of A. If so, call A admissible. Methods
of Mazur (Studia Mathematica vol. 2) and the author (Trans. Amer. Math.
Soc. vol. 67) bring known Banach space techniques to bear on summability.
(1) Admissibility of A is shown equivalent to continuity (where defined) of / ,
where f(x) = lim xn, by use of the Hahn-Banach extension theorem. This
reduces the problem to examination of a metric. Hence (2) A is admissible if
there is a regular matrix summing all the sequences of A. The author (loc. cit.)
has shown that if A is admissible, then (3) limw X . A ä ä ^ X)* hmnank . Assume
this in the following. Let B be the matrix inverse to A. Let ak = limnanA;. (4)
If X) I ak%k | converges whenever x is summable A, then (2) implies that A is
THEORY OF SERIES AND SUMMABILITY
425
admissible. (5) If X)_X I anbnk | converges, then A is admissible. A straightforward argument shows that denial of (1) denies (3). (6) If X.fc'io0 a>n{P),kXh is
bounded if x is summable A for each choice of sequences m(p), n(p), then A is
admissible. An obvious application of the principle of condensation of singularities (Banach's treatise, p. 180) gives a boundedness condition on the biorthogonal sequence {Ln} {dn} (where Ln(x) = xn, dn = sequence whose nth term is 1,
others zero) which implies that {dn) is a basis for the space. This contradicts
<3) if (1) is denied.
LEHIGH UNIVERSITY,
BETHLEHEM, PA., U. S.
A.