258

HIGHER AND I T E R A T E D H A U S D O R F F DERIVATIVES

by

R. F. Rinehart (~) (Durham, North Carolina)

and

Jack C. Wilson (Pella, Iowa)

w 1. Introduction. Let s~ be a finite dimensional linear associat ive algebra

with identity over F, the real or complex field. Let f be a function with domain

and range in ~ . The concept of H-differentiabil i ty [1, 2, 3] of f has been

generalized by Wagner [6], Volovel ' skaya [5], T rampus [4], et al. to higher orders

of H-differentiability, as follows. Let ~1, - . . , ~, be a basis for ~c~ with ~1 the

identity of .c~. Let n st

= x, F, = . . . ,

Definition 1.1. Let the component functions fi(x~ . . . . , x,) possess all partial

derivat ives of m-th order in an open set cg of A (2). Then the m-th differential

of f(~) at a fixed ~E~)~ is defined to be the multilinear function of m variables,

, m O, L (1.1) d " ( / ( ~ ) ; ~', . . . ~ ' " ) = y, Y, d'~d;~.. , d(m'e

' r = l S , , S ~ . . . . . S m = [ l C ~ X s m ~ X s r a - 1 " " " C ~ X s l s m r

where ~"' ~ ~ d'/'E. " ' i~,-" ,, d~ E F, and the partial derivat ives are evaluated at ~.

(l) Supported by ONR grant. (~) More properly, on the open set in E" induced by the open set c'/[ of ~ .

HIGHER AND ITERATED HAUSDORFF DERIVATIVES 2 5 9

If the m-th differential is expressible in the form

n �9 ~ 8 , m , ) _ _ v g . ~ . 8 ' ~ i ~ 8 " 8 .... d ~ ( f ( ~ ) ' ~ " ' " ' "" il...,~+~ ~ " '" ~i,,,+,

i l , . . . , i m + l = l

(i.e. as a multilinear function of 8 ' , . . . 8 ''~', with coefficients in _r where

gil . . . . . i,n§ depend only on ~, then f is said to be H-differentiable of order

m a t e .

/-/-differentiability of order m places both an analytic and an algebraic

condition upon f(~). Following Portmann's concept of /-/(ausdorff)-derivative

[2] we can define

Definition 1.2 If f(~) is H-differentiable of order m in an open set ~ of

d~ ' f~) .~, then the m-th order Hausdorff derivative of ~ - ~ of f(~) at ~ 6c)~ is defined

to be the ,, detached coefficient,, of the 8 ' , . . . , 8 '" in the m-th differential,

(1.2) d'~f(~) ~ gil...im+1 Ei," " " ~i,n+l. d ~'~ - - i ...... i,~+1:1

Another type of higher derivative is the iterated first order Hausdorff

df(~) ~ ( ~ , derivative, f"~'(~) defined inductively by f ' " (~ ) ------[f'~-~']'(~), f ' ( ~ ) - - d~ '

with an associated concept of iterative H-differentiability of order m.

These two concepts of higher differentiability are not equivalent. It will be

shown that if the component functions of f(~) have continuous partial derivatives

of order m in c)~ and if f(~) is H-differentiable (first order) in c)~, then F ' (~) ,

i-----1 . . . . . m - 1, is /-/-differentiable in ~Y{. On the other hand Volovel 'skaya

[5] has shown that if ~=~ is not semisimple, then even if f(~) is /-/-differentiable

in '~ and the component functions of f(~) are analytic in c~, then for some

m, f(~) may fail to be H-differentiable of m-th order (by failing the algebraic

requirement of the expressibility of the m-th differential as a homogeneous

multilinear function of 8' . . . . . 8'~').

However, if f(~) is H-differentiable of orders 1 and m in c){, then it will

be shown that f,m,(~) exists in c~ and that f ' ~ ' ( ~ ) : d'~'f(~)- d~, m,

w 2. The iterated /,/-derivative.

Theorem 2.1 Let f(~) be a function with domain and range in ~ , whose

component functions possess continuous partial derivatives of order m ~ 1 in

260 , , . F . R I N E H A R T and J A C K C. W I L S O N

an open set Qf of .~. If f (~) is H-differentiable (1) in ~ then f,r_l, (~.), 0 < r-~< rn

is H-differentiable in ~ , and f< '~(~)= oX7

Proof. Employing mathematical induction, we note that for r - 1, 2, the

assertion of the theorem is tautologically valid, since from 1.2 f ' ( ~ ) c a n be

computed by setting ~' - - ~ l , the identity of .<.:( in 1.1 with rn = 1 Assume the

theorem valid for r = i . ~ < r n - 1. Then with ~ = d~ = Zdxj*s , the differential J

Of<i-i, d(f'~-~'(~); d~) = E ------dx,JJ cj can be expressed in the form

j.k=l OX k

(2.1) s Of) ' - l ' (x , . . . . x.) ,. dx,~j = E g,,+a,,d~,~ = 2~ g,,qC,,,tCt~s~sdxk

j , k : l (~ X k p . q : l p ,q, j , L k

where the %, are the multiplication constants of .c'7 relative to the basis,

~ , . . . , ~,. Equations 2.1 are fulfillable in ~ if and only if the system of equations

Of,i-i, (2.2) JJ - - X g . . c.k,c,~j ( j , k = 1, n)

OX~ p.,7. t . . . .

are solvable for the g , , for { E ~ .

We wish to show that d(f<~'({); d{), which exists since i < m, is l ikewise

expressible in the form 2.1, i.e. that equations 2.2, with index i in place of

i - - 1, are solvable for the analogous g ; , . By 1.2, [ ' ( { ) can be computed by

setting d{ = ~ , the identity of ~r in (1.1) with m = 1, and f'~-~' in place of f .

Hence f'"(~.) = ~' Of;'-~' r=~ 0 x ~ --~" Thus we wish to show that the equations

( O f ) ' - " t = ~, g'..c.k,c,, s, ( j , k = 1 . . . . n) 0 (2.3) Ox , \ - J 7 , - i ,,. <,., '

are solvable for the g ~ . Now the solutions g . . of 2.2 are linear functions of o f r

.t y the - - - - , hence, since i < m, are cont inuously differentiable functions with

0 xk respect to x~, . . . x , . Hence, from 2.2,

Oxk = ~ ,,.~,, O x , . . . .

Og,,q Thus a solution g',,o = c)X~ to 2.3 is exhibited and f<"({) exists in ~ .

(~) The phrase .H-d i f f eren t iab l e . means H-differentiable of order l in the sense of

Definition 1.1.

H I G H E R A N D I T E R A T E D H A I 3 $ D O R F F D E R I V A T I V E S 261

Further

0 (f , i-l , 0 { O " - " f ( ~ ) ] _ O'f(~) f '"(~)-----d(f"-"({); ~ , ) = Ox-----[ ( ~ ) ) - Ox, ~ 0~5-~- ] Oxl "

d'f(~.) f,,,, w 3. The equali ty of d~m and (~).

Theo rem 3.1. Let f (~) be H-dif ferent iable in an open set ~7~ of ~ . If f (~)

is cont inuously (~) H-different iable of order m > /1 in c~, then the m-th i terated

d'nf(~) H-der ivat ive f '~ '(~) exis ts in '~ , and f " ' ( ~ ) - - d~" ' the m-th order derivat ive

of f(~) , in c~.

From the continuity of the m-th partial der ivat ives of the component

functions of f (~) , it fol lows from Theorem 2.1 that the m-th iterate derivat ive n n-t E O"~ f" ' (~) exists in c~. Further f " ' ( ~ ) : ~=~ 0x~" ~" But from definition 1.2, d" ( , ) d

is obtained from 1.1 by sett ing ~ ' : ~ " = . . . . ~ " ' : ~ . Hence

d'~f(~) ~ O"f~ f,m, - - -gx

Durham - Pella (U. S. A.), June 1962.

REFERENCES

[I] Hausdorff, F., Zur Theorie der Systeme complexer Zahlen, Berichte tiber die Verhandlungen der S~ichsischen Akademie der Wissenschaften zu Leipzig. Mathematisch-Naturwissen- schaftliche Klasse, vol. 52 (1900), pp. 43-61.

[2] Portmann, W. 0., A derivative for Hausdorff-analytic functior, s, Proceedings of the American Mathematical Society, vol. 10 {1959), pp. 101-105.

[3] Ringleb, F., Beitrdge zur Funktionentheorie in hyperkomplexen Systemen, Rendiconti del Circolo Matematico di Palermo, vol. 57 (1933), pp. 311-340.

[4] Trampus, A., Differentiability and analyticity of functions in linear algebras, Duke Math. Journ., vol. 27 (1960) pp. 431-442.

[5] Volovel'skaya, S. N., Analytic functions in non-semisimple associative linear algebras, Zapiski

Naucno-Issledovatel'skogo Instituta Matematiki i Mehaniki i Har'kovskogo Matema-

ticeskogo Obscestv, vol. 19 (1948), pp. 153-159. [6] Wagner, R. W., Differentials and analytic continuation in non-commutative algebras, Duke

Math. Jour., vol. 9 (1942), pp. 677-691.

(i) I.e., the partial derivatives of order m of the component functions are continuous.

17 - R e n d . C i r c . M a i e m . P a l e r m o - S e r i e n - T o m o X ! - A n n o 1962