A GENERALIZED DERIVATIVE
THOHAS :ULLIN1 CAIRNS
of Science
Oklahoma State tJniversit:y
Stillwater, Oklahoma
1953
Master of Science
Oklahoma State University
Stillwater, Oklahoma
1955
Submitted to the Faculty of the Graduate School of
The Oklahoma State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
May) 1960
1 1960
A GENERALIZED DERIVATIVE
Thesis Approved:
2
ii
PREFACE
My thanks are expressed to Professor R. B. Deal for his
and aas::lsta:nce in the vtr of this thesis, to Professor L. Johnson
for his counsel and encouragement throughout my
the of those vJ!:w constitute my
i'ii
program and to
committee.
TABLE OF CONTENTS
Section
I. INTRODUCTION •
II. A
nr.
ORDER GEi~ERli;.LlZED DERIV A'l'lVE •
Ll.!JFT GENERALIZED
iv
Page
1
~ 4
19
" 30
40
Table
I. ADDITIONAL PROPERTIES OF AND
. "
Page
17
32
I
This work on a generalized derivative ~s motivated by the absence
of strong techniques for numerical differentiation~ It is clear that the
technique for numerical differentiation of a function, f, is to fit f
fitting error may not at all re:f.lect a larger error in a derivative. Hence1
it seems ef:f'ective to define a derivative in terms of an integral, 'Which
approximation, give an approximate derivative of f which depends on the
nature of f on an entire interval instead of
an integral migbt well be put in a form to which one could apply a standard
numerical integration technique~ It also seems likely that in specific
applications one could establish criteria7 perhaps statistical, to de-termine
a 11best11 interval for the a:ppro:d.mation. A stud;y of such criteria
Some work has been done on generalized derivatives a.nd the te:rm has
been used to describe concepts in addition to the one described
here. For example, Kassimatis (5] defines the n~the generalized Riemrum
derivative of a measurable function, f, at x by
n
D:f(x) :::: ~~ (2h)""n I (-l )n-j(~)f(x+2jh. . nh) for h > o, n "" 1, 2, •••
,\::o
In the case of
1
2
D2:r(x) ""' lim (2hf2[f(x-2h) - 2:f(x) + f(x+2h)]
h0o
this is the same as Hobson 1 s [ 4] definition or the gene:t"B.lized second de-
:rivative. 'J:hese definitions provide generalized derivatives for some
functions which do not have derivatives in the ordinary sense.
The n'th Peano derivative (3] of a continuous function, :ts a n
+ E(h)hn were
n
€(h)h "" o.
n!
if
Laurent Schwartz has proposed the idea of a distribution as a gener-alization
on the idea of a :point :.t'unction. His distribution is a functional
defined on a set of testing f\mctions which have the properties that each
has derivat:lves of all orders which; as well as the testing ::function
vanishes at the ends of the interval [a, b). Then each fl.ll1ction, f 1 deter~
mines the 1:l.ll1ctional
b
f(x)~(x)dx
where ~ ranges over the set of testing fu11ctionsq Integration by parts
shows that
sense that it is the derivative of a distriootion ~rh1ch is a
tio:n on the idea of a point flmction. Ho-wever, the approach of Scll·Hartz
is not amenable to smoothing noisy data numerical approximation.
In this paper vie use the following procedure for defining a generw
alized deri\~tive. If a f\mction, f 1 is fit on some interval [x-h,x+h)
by a straight line in the sense that
rx+h 2
~-h [ f( § ) ~ a~ - b] d§
3
We define the generalized derivative
D:f(x) eo
H' the l:Lmit exists and if the mean;
aJ_so exists~ Cond.i t:Lons necessary :for the convergence of Df are much
\.Yeaker than for the existence of the dertvatiYe
{ if "Gb.e latter exists~
011e can st;;ate some theorems vJhieh have
of
nX
derivatives. :F'or 1r1e shO'<~ that if
in
in the
then D / g
~
D(f(x x)] ""
14e can d.etermine some conditions under which
x). L'l there are mean value
theorems similar to the familiar ones. lie can also discuss convergenee
of Df :in ter·ms o·.f derivab.ves an(l
means which are defined to as the :name the of' the
lines o:f best fit on the intervals to the El!ld lef't of any
'I'bese are a.nd i'le for
that under some conditions ) - ] 1<1here the
indicate means over a finite intert!al.
In a maJ:mer similar to that used to the
•de may define a nFth derivative. 'l'o facilitate
x+h] and examine the best :fit n 6th in the limit as
h zero. He are able to some
bet·ween this first deriva-tive
iterated n times.
II. A
Consider a fun.ction f( dei'ined
square on the in:terva.l [x -h1 x +h]. \ole may associate with
0 0
the function at the poin.t a straight line which, over the :interval
[x -h,
0
is the line of' fi 1:; by the least sq.ua:re criterion, 'lb.a.t
know that it is possible to find a unique pair aJ b suc.ll that
('X+h
= I [ r<
~~h
is a min:i..muJJ1 .. , If we make a change of' variables
the simpler expression
zero ..
h
[
'l'hese equations are 1Lne:ar in a and b
f h
a 5 ds
-h
for a and b,
) - - b
to and b and set them equal. to
} = a~ - b ]ds - 0
c:a:.n be vr:i.tten
obt.ain
or
or
2hb = ~~(x +£ )d~
-h 0
2h
5
We had a fe.mil:tar alternative to us for the
best
in the
line¢ hie can consider the functions 1 1 (
to be the
square
manner
on
elements of the space of all functions
X+£
0
the Hilbert axioms are satisfied. and the space is a space. Cons :i(ler
of the sp;:-:tce and t:he the
vecto:rs is -well knmm that the of f onto the
is that linear comlJi:natio:n of
+ b, 1,;rhiclt minim.:izes the distance to i'. The distance :Ln tenns of
[ ) -
the dis"l:;ance from f to the is the least square error ex~
The criterion i-Jhich lle determine a and b foll01vs as a result
of the fact that the difference be-r.ween f and its on the
is orthogonal to every element in the subspace, in particular the base
vectors. 'I.bxrt is,
rh
/ [ x +E) - ag - b]id£ = 0 . 0 '!..h
Thus we arrive at the same set of linear equations in a an.d b.
Several things are immediately clear. Our space is really L2 defined
x and h; and as a result, a dif't'erent best fit line. Our objective is
0
to treat
'fh 1::1 , 2h.L f'( x +£)as
. -h 0
as operators on a. space oi' f'unctions and so we Hill change our designation
of' the variables by dropping the subscript.. It is still necessary to
distinguish betveen the ind.ependent variable in the coefficients, a. and b,
and in the base el.eruents of the subspace. We ~>Jill prime the latter~
I:Ience1 for a gj.ven x and h the best fit line is
We will also use the :follmving notation:
x+h
(x-~ )f(S,}d~
and when we wish to emphasize t.he operator aspect of the latter> i..e~,
7
the maps :f: into
deri >.tErti ve of write or
mean of f on an
about x., A on the
notation or :point of view ..
:F.rom the second fom of the relation defining~ it is clear that fh
is absoJ.utely continuous~ If, for all p such that -h 4 p 'hi' f(x-rp) is
bounded1 that there exist .m a.OO M such t!w.t .m ~ f(x+p)' M3 then by the
I~ f is continuous on this interval, then there exists a p3 suCh that
... b, ( p < h and f'h ( x) ~ f( x+p).. From these properties we obtain the follow ..
ing ..
'lheorem I: If f(x+p) > 0 for .,h '- p' h a.OO f
intel"''!al, then fh ( x) > 0.
integrable on this
Corollaryt If f(x+p)) g(x+p) for ... h, ~ p~h and f and g are integrable
on this intel:"Val_, then fh (x) > g(x) ®
We may also see that Th :possesses properties of continuity and
associat;ed 1,r.lth :l:ts 1-ll:lenever we use as
an opemtor, we need require not o~ integrability, which will be assumed
hereafter, but if the domain of the operator is a set of 1\mct:lons defined,
taking care of what ba:p:pens at the end of the interval.. One possibility
is to ask that the 1\mctio:ns operated on by Th be defined and integrable
on the [~h+a,b+h] in order that the result of operation be
the functions are identica.1.1y
zero outside the interval. In practice it mig,.~t be more convenient to
handle this difficulty in some other way, but at least in the above two
the is linear and continuous in the sense
sup norm.
Dhf possesses similar elementary properties~ It is absolutely continuous
and Dh is also a continuous, linear operator if defined on an
set of i'unct:i.ons. If, as m ~ f( x+£) -' M in the interval
in question, we obtain by the mean value theorem
anct adding
The following theorems establish an important relationship between
the operator Dh and the ordi:na:ry derivative. lie know that if' f~ {x) > g 1 (x)
for all x in some then f.,. g is monotone increasing in interval.
Len1l'!lE!. 1: If f(x+~) is monotone increasing for @h ~; :'f:. h, then Dhf(x) > 0.
Proof! Suppose that f( x+g ) i:s monotone increasing for -h ~ ~ :f h. Then
by the second mean value theorem there exists a ; ~ such that -h ~; 0 ~ h and
~ ~ (n2[f(x+h) - f(x=h))- s2 [f(x+h} - f(x-h)]]
lr-h5
= _;, [f(x+h} - x@h) ][h2 -~ ; j 2 ].
4h5
It is clear that (1) f(x+h) ) f(x-h) because of the monotonicity of f, and
(2) h2 ) 5 ' 2 since s; t is interior to the :Luterval and h is the end point.
We have now established the following theorem~
9
Theorem II: Ii' f'{x+S)) (x+~) for all -h~ s ~h, then Dhf(x) ;>Dhg(x).
'l"heorem III: If f~ (x+;) llas a maxinrum and a minimum on -h ~ g ~ h7
Proof': Consider line, g(x) ""' (r.t1€kx fW (x+t; )]x + b. Obviously
g~ (x+;) > ft (x+g) in the interval. :then by the previous theorem
D11g(x)) i'(x). But g is a straight li11e and ope:cating on it with Dh
yield simply its slope, Dhg(x) ::::. max f~ (x+s). Hence, D11g(x) ~max f'' (x+~).
The proof for the other inequality is similar.
) is conti:nuou.s for -h!:: [; ~h, then there exists
It is clear that there is an intim£J.. t e relationship between fi (x) and
derivative, from the mean value theorem discussed abmre, it is clear that
choosing a sufficiently
small h., Let us define the notation lim D"'f(x} ::;:; Df(x)1 and lim f" (x) "" f(x)~
>1-"> 0 u """" u
We see that under the conditions mentioned above Df(x) ~ fS(x). This is
an important result, but the same result can be obtained under weaker
conditions.
Theorem IV: Let f§ (.~<) exist and f(x+S,) be integrable on some interval
containing s ::= 0. 'l'hen Df(x) "" fa (x).
Proof: Differentiability implies that there exists a K such that for
This is also tru.e on some interval on which :f is integrable.. lfuen
By the mean :for h :in th.e
that of'
"" :f(x).,
3
4~ (K+ ~
j
would suf'f'ice to show
:f(x) then n.[x:f{~)d~ ~
definition
if'
a a>O such that O<~~h<a implies
and
The last two statem<"..nts by the first mean theorem.
Suppose f exists on some interval containing x and for all
1 rh 1 ro i.n the interval lim h f(xt +S)d~ = lim h.J (x+£ )ds = 1(
~-,_,., vo !,.~!> -h
Then f is continuous at x.
Proof: In the above intertal there is a o>O for every 610 such that
O<M.§; implies
and
]]; rhf(
hJ
0
- f(
where x+k+h is also in the interval~
Then the follm..ring are true for o<h, and k<.ot.
1
1 ('k
k~ f(X+!; )d!; "'
I k l-i. 0 f(x+~ )d~ - f( x+k) j< f,
and lf(x) - f(x+k) J<-~.
In particular., we :may show that
ancL
h
[ "" 0
f were non .. mcroos:Wg, then Dhf(x)( 0 on that :ix.rtel:"'lal end, hence, for some
x in the lin:d:t ..
~eorem VI: Iff end exist in an interval, [x ... h1 x+h]1 and
lim 1 .... J~f(x~+s) ... :r(x')]ds = o and Df(x')>o :tor all xi in the intel"''ral1
h""lC h.:: ... ]l
then Dhf(x)) o ..
Proof: Clearly Df(x) = Df(x). But by lemma 21 f(x) :is contin.ucrus&
'!hen the conclusion follows by lemma 3 and lemma L
'lheorem VII: If f' and Df exist in an :Wtenal [x ... b.1 ~b.] and if
1 fh .. 2
lim 2 [ f(x' +~) ... f(xi)] ds = 0 and m ~Df(x')' M for all x~ in the
h""l'C h ... h,
inte:L"VS.l1 then m!GDhf(x) ~M ..
Proof: '!his :follows immediately from theo~ VI ..
'lheorem VIII: If f and Df exist m an ::tnter.ral1 [
lim12 Jh[ f(x'+O- :f(x1 2 )] d~ = 0 and Df is cont.1n:uous
~~0 h -h
then Dh:f(x) = Df(x+k) for some k such that. ... h <k <.h ..
Proof: '!his follows immediately from theorem VIle
Theorem IX: Let. f(x), D:f(x), g(x) and Dg(x) exist. and let.
1 fh .. 2
lim-;:; [ :f(x+~) - f(x)] dE = 0
~ .... 0 h<:: -h
""' o ..
Proof:
D[f( ) - f(x)] ) "" x)
+ x) + g(x)Df(x)~
Now ) - H X+~} - .. ]dg I
]
) -
) - ) -
~ 2 ) - f(x)] ~ 0
H ) - "" 0
f(x)g(x)] ~ f(x)Dg(x) + g(x)Df(x)~
A final important property of these operators is th."tt if
f(x+p) = g(x+p) almost eve::cy-where for ~h ~ p~ h, then Thf(x) "" Th.g(x)
and Dhf(x) = Dhg(x)~
It is interesting to note that operating with 'l1h is a smoothing
14
continuous :functions into differen:tiable ones. Furthermore, Th :ll!S..:ps exponentials
into exponentials, sines into sines, and n 1th degree :polynomials
into n~tll :poly:nomals~ .For e::rem:ple,
h -h
(1) e = e
:::;
( 1
x~-· s:Ln h sin h
(3) "" if n is and
if n is odd;..e
Note that the first. term is Tl1e rest i.s a of n-2 in
+ h)
x,h) is a of n·~2 in x and n or n-1 in h.
with :is similar to a be
deduce~ from the into
sines into :into n-1 n st
For
( Sll'l X ;:;:; + Sll1 cos .,.
A"J
l.t::i.
f.
{3) \
"' l
.L.J
ii' n is and
J•O
~?,_ ,;;_,
~- )
t..__, if n is eveno
,J~o
:n-1
The first term is nx and for
that
where h) is a polynomial of ctegree n-3 in x and n or n-1 in h.
1: Let f(x) "" 1 for x ~ 0
is the
2:
'1\f(x)
ttrhf(x)
Let
x)
= -1 for x) and let h :::: 1 ..
= 0 :for x ;{ -1
3 2
=-= 2[x -1] for -1~ x ~1
- o for x
+ C. This the
:::: =l for x &; -1
"' -x for .. 1~ x.(l
- 1 for x > L
:::: 0 for x< -1
"" -1 for ·~1~ x ~1
"" 0 for x *-1.
:<:: X+ 1 for X ~0
"' ... x + J_ for x;>O, and let h""
3 x3
:::: -[-- for ~1~ x~l
2 3
""' 1 for x~ -1
:::: -1 for x~l.
2
1 X
:::: - 2 for -1 ~ x ~ 1
::::: X+ 1 for -1 ~x
= -x + 1 for x~L
d -T. f(x) -· 1 :f:'or x~ ~1 dx h
follows
"" x for x ~-1
Given in Table I are some additional properties o:f the operators Tn
and 1\ 'Which might be useful~ These fall into two categories, di::ff'eren=
tia:tion of Thf and Dhf and iterative operations with ~ and Dh.
11
TA:BI.E I
1. If :f(x+p) has a continuous deriva:tive for -h ~P ~h, then
d
T11:r• ( = a:xT11f(x), 'lbf~ (x) ::= f' 1 (x+!J.) for som.e 11 such that =h ~ !! $,.. and
d diThf(x) ::= f'(X+fJ.) for the !l above.
2. If x+p) has a derivative for all p such that -h f; p ~ h, and
d
:mff'(x+p)~M on this interval, then Thf' 1 (x) = dxThf(x), m~Thf'(x)~M,
d
and :m~ dxThf(x) ~~i.
3~ If f(x+p) is continuous for -h6p ~ then Thf(x) is differentiable
and,
d 1
in fact, dXThf(x) = 2ii[f{x+h) - f(x-h)].
4~ If f(x+p) a continuous second derivative for all p such tbat
-h£ p~h1 then Dhf'(x)""' fn(X+iJ.) for some !l such tl;lat -hf!l"-h, and
dd xDhf(x) "" f'n(X1!1) for the 1.1 abcve.
If f(x+p) has a second derivative for aJJ. p such that =h ~P~
m :S.f<~(x+!l) ~ M in this
d
interval, then D d 11f 1 (x) ""' a::xD11f(x), m ~ D11f 1 (x)~ M,
and m(:_ dxDhf(x) :6 I.L
If f( x+p} is continuous for all p such that - h ~ p ~
diff'erentiable.
1· Tj['lbf(x)) ::= [Tjf(x)] and Dj[Dhf(x)] = Dh(Djf( ].
then D, f(x) is
ll
8. If f(x+p) is continuous for all p such that .. 11-j~p!.h+j, then
Th(Tjf(x)] "" f(x+iJ.) for some !l such that -h-j :f !.1. f; h+j. In particular,
2Th :f( x) :::: Th [Thf( x)] has this property on the interval ( -2h, 2h }.
9., there exist :m and l~ such t.hat :m~ f(x+p)f H for all p such that
~h-j~p~h+j, then m~Th(Tjf(x)]~!.-1.
10~ If f(x) is such that :m ~ f"(x+p) ~M :for all p such that -h-j ~ p s;h+j,
18
J.L If f( x+p) has a continuous second for all p such that
.. h ... j ~ p ~ h+j.l' then DhiDj:f(x)] = :rn (x+J!) for some v. such that -h-j ~ p, ::;h+j~
12. If f(x+p) has a continuous n'th derivative for all p such that
~nh'-p~nh1 then D, f(x)::; f(n)(x~) for some p. such that -nh:!filf-nha
nu
13.. If f 1s such that m ~ f(n) ~ M for all p such that -nh ~ p ~ nh,
) has n cont:.inuous derivatives p such that
p~ (m+n)h1 then rPn[mThf(x)] = )
for some r;t such that
-(m+n)h' 1'.1 ~
If :f is such {., M for all p suc.l:l that
~· (k+n)h~ p ~ ( then m ~
the derivative
when it exists and which converges under some circUI'llStances ,_men the de-rilt'S.
tive does not e::xist. An interesting aspect is tat ve b.a:ve defined
the derivative in terms of an integraL We will call this generalized
derivative the ~~g;-<ieriva:tiven and f(x} the integral :mean of It is
evident that continuity of f(x) is not necessazy for the eY..istence of'
Hence, it is possible for lim Dhf(x) to exist without implying
1-.."1>0
the existence of a unique l:im.iting line" That is.1 even if the above expression
exists, f(x) might not exist and there need not be a line analo~
gous to the tangent line for the derivative. we will alter the
defi~ition of the g-derivative to include the requirement that f(x) also
exist to assure the existence of the limiting line y* ~ Df(x)(:x~=x) + f'(x)~
We can also define rigb;t and left hand g ... derivatives as well as right
developed the expression :for the best fit lir1e by the least squares crito
f( x+p) over the interval -h :;£ p ~h.. Let u.s now consider the best
fit line to f(:x{~) over the interval on p1 [01 h]. For a given value of
bm and x this line is, of' course,. in general not the same line as before.
We ,,rill call the slope o:f this line D~f(x) and the ordinate of the point
of intersection of this line and the line x~ = x + h2 ! ~:fh( x ) ~ T+h f(x).
We may show that
.19
20
and
Now x is at the extreme lef't end of the interval and if we let h
zero; the inte:rval -will r;hri:r.!k to the x. It is also useful
to define another integral :mean which is the ordinate of the :point of
it is defined
2 1{ h < 2h- 3s )f'( X+~ )a.~.
~o
'l'he "best :fit line is the line
We now define the limits -which give us the
= lim D~:f(x)
h-'>.. u
and the means
and
In the same -way in which 1-1e defined the right hand g-derivative and
integra.]. :means, vre may obtain the left hand counterparts. He have
2l
D- f(x) = t~ Dhf'(x).,
(x) :::: lim i;( ~-'I>CI 1
(x) "' lim ~( :1' ._..,()
and the lim.Hing line is
ti.ves and the
1Uso t~he ri.ght and left-hru.1d. integ:r<:l..l means have analogies i.u the right
For a
expres::dons identifying different intervals
on p: [•h,} I':t €>...Ud (e 0) th respect (a) we re-vie1-
red the fact for a fixed h ·\~:e 't-.iere simp1y
a ccnst.rmt; and a straight is true of the i:o:ter-vals
(b) and (c), The exists between
the three: Let g(x+p) be a:ny function which :ts a straigh.t line on [ ]
and also on [ .. h., 0 j (possibly disc:o:.utinuous at 0). It is clear that g(x+p)
is e:Le:ment of space (a and in fact, all sible such func:tionr::
( strt'tight lines on each of [ and [ .. h,O ) a subspace of' thif:
space (a) ~ Suppose we call the L 2 space on [-h 1 h 1., R, a.nd the
of all possible func·t~ions of the nature g(x+p} described
22
Th.en3 as we stated, s2 H,. s1 H, and, furthermore, s1 s2 . T1:.let is, e:n:y
straight line on [-h1h] is also a straight line on both [ ... h,o] and h].
We can shov in such a case that the projection of an arbitrary vector,
A; on s1 is the same as the projection on s1 of the projection of l on s2•
In this case the implication is that if f(x+p) is fit for p on the interval
r
' h] by the fun.ctio.n which is a l:ine on [ o] and a
l:Ine on [O,h] and which fits best the least squares and lf
this best fit function is then f'it f'or p en h] a
then the same line is obtained as 1,;~ould be if :f(:x+p) were fit
a line f'or p on [-h,h]. This t .. b.a;t we can 11ri te and
in ·terms of their and left-hand.
the above
JJ.?t. us consider a function such as
x') for x 1 ~
x') -- x' + for X 1 ( x.
Then lve bave
! ) [ {' 0 rh
= / ~f.a.l(xl+!; + + i S[ yl ) + b} d~J
'-'=h Jo r
[a13
.,
J
'
= ·~ [ x 1-x) + ] + h" + (a r·( x 1 +
"" 2
+ (. br ) ~]~ (x' ) .
now that and le1"t-hand
parameters for ;;;;X for some f\:mction r, and some h. 'rhen
But this must also be Dh f('
x) a:nd ).
D f\f X ) =.-J:[- h e
x)] + x) =
We may l·ir:i.te a similar
+ Dh= . f(x)] + ~3 [-f'+ ,~ X ) = '+h h
contain~
x)).
In the first expression Dhf(x) is expressed in terms of
hand para;meters D~f(x), n;:r(x)sr ~(x), and. ~(x) and in the
of D~f(x) 1 Dh_f(x), ~(x) and
of and
23
imlnediately. We mll consider the means first.. Recall ·that
if f(x) is see
~f· ( ""2l :r.~, +c X>+
f'(x) =
(x)] if these if two of the
means are all are equal.,
It is useful to consider the
and D+ : f{x) ~ ~'le are
go us is
the ordinate of the best fit line at the center of the
that
slope of the line$ is true~
or
x)l.
are
h
+-
2
or
(x)] ~
If we keep the graphical representation in mind, some of the results are
easy to anticipate$
Theorem X: If :r;:(x) exists in the limit as h o, then :'e~(x) may not
be unbounded.
Proof: Sun~ose lim hl fhf (x+~ ~x ll,~o
0
exists. have
~... (x) 2fh == 2 (2h - 3~ )f(x+s )d~ 6
h 0
Now 2h"* is a monotone :f\mction. Hence by the second mean value theorem
there exists a iJ. such that 0 ' 1-.1.' h. and
1-Te have that there exis-t~s a K such that for everJ €.?0 there exists
a 5?0 such that
~~
h
f( = KfE·
Since 0 ~ 1J. ~
I~.[~( ~ K~~.
('!J.
~ Kl< < lOE.
It :foJJ.m.rs that
E -'E 1 and so even l:f ·~ does not convere;:e it is boundec1 0 and 1 and
< +E:.
If ''f-,+ ( converges as and x) does then
il.
(1) a dis and. (2) x) 1Jas an unbounded. dis~
Proof: Since -.:;~;-+h ( x ) converges as +
by hypothesis and the last theorem ~(x} has a bounded, oscillating dish
+ 'I'hen -2 Dh f(' is bounded and as h -~0 and must take
on nm:l=zero :for small and is1 therefore
unbounded. It not because if it 2h D+h f '( X) IJOUld
either properly or b~ve a limit and. likewise x)~ 'luis violates
the
Theorem XI: If but not to
+ x) becomes
Proof: ('. h ,;>:lnce 2
:follmn;: that
as h-4>0.
(x) "" x)J.
:must become w"lboumled.
-+.
and f' (x) exist and are not
becomes unbou:nded as h -+0.
'rheorem XII: If lim tr :.P~..+~,. ,f\ "" ')
¥>.40 .u
x) = o.
x)] = 0 and hence if
then
(
then
K
x)
Note that this does not ::ilrrply that D+f( exists. For example at
x = o, f(x) = (x is such that "' fA + (xJ' = 0 but D+h f(x) becomes un-bounded
i:n the limit.
and
Theorem lim. Db+f(x) exists.
k->ro _
1ben lim [fh+(x) -
1\-"10 .
x) converge or d1verge
Theorem XIV: If
(a)
'l.'heore:m XV:
x) =
h
2
f(
x) •
x)
x) ::c
rHverges1 then lim
V>..-'t <J
and' ~""{h 'x ) each.
x) = D- f(x) and -f+ (x)
converge
s.s h-~o.
Proof: The hypothesis states that the ri.ght and left hand best fit
lines merge into one in the limit. fue conclusion of the theorem follows
immed.iately from the facts tttt-tt
(l) all the parameters are continuous fUnctions of
(2) the function composed of the best fit lines on [-h,O] and [O,,ll]
is a better fit than any straight line on [ -h1 h] unless
26
li;ne:. Then in the li:mit the forrner is at as good a fit,.
ll.:;t us now
We are tempted to WB~t to the s ar.i·u~: about that
it is the average of the :righ.t and left ...
tland g-derivatives~ the f:Lrst above yields the infor~
mation that Dt(x} ~ ~fD+f(x) + D-:f(x)] the existence of all the
..::;
limits, if and only if lim ikrz;(.x.) - r; (x)J - o. Fu.:rthermore, if is
shows th.at
Df(x) ~ ~~ ~{r;:(x) - ~(x)].
It is interesting to note that even iff+( "" f+(x) and (x) ;::; :r (.x:), in
lim -1hIfh+(x) - f;:(x)] :::: lim ?~(:x} = f:(x}lk Recall that the
!'>/~;:) • M. ~~..,.. u ll
existence of Di:'(x) implies the existence of f{x) defLuition. !t is
:possible to write Dh f( x) in mar1y forms of
Another is
~:f(x} ""
we have shown two.
' . 1 ["'-( ) ~+
j + 4h fh x - 1h (x}] •
Theorem If f(:x+p.) is and left-hand d:tfferentiable at
Proof: There exists a K such that for every ~'>0 there exists a
o\ )0 such that 0 < ;,t < 5, implies I f'(x+~).:: :f(x) - KI.Z t.. Tnere exists an 1
j.\.
such that for every f:-)0 there exists a 5 . .)0 such that iJ. 'J01 ~( 6,_ implies
ll.:;t(x) ~~ f{~-J:t-t) _ Ll/"" _ ..,. -.., "' ~ For every e:c > 0 there exists a ~.> 0 such that ~~ (03
implies lf(x+~) - f(x)l< ~ • All t.rwse :follow from the hy:pothesis. From
these expressions we obtain by the mean value theorem for integrals
Subtrnctir!.g
or
x) +
1 eh
)< ~j f'(X+~ )d~ C::, 4f'(x) + 2h(K+t-},
0
< :f(x) +
/ 00
2h(Irt)- 3f'(x)<. 0:5/1 tf(x+§)dg (.,3f{x) + 2h(I.+t:)"
l:C" .. h
:f(x) -
g ... deriva:tives reta.in all the properties of' T11f &1d Dhf which are the
result-s of their be:Lng the parameters of' the best :fit line-. These are
primarily the continuity and. differe:rrl::.iabilit;y properties as 't<tell as the
f'rorn been
'!:Je eould for some
a:t; ~I'he same :f.toLd.s fo.r a.nd
~(x+p) and f{x+p).. It is also true tnat it f(x+p) is di:fferentiable
re::!.ative to such a set of' measure 2h at p:::01 then f(x+p) is also differentiable~
It is, of' course, not true that if on [-h,h] f(x+p) is contin·
uous at p:::O relative to a set of measure that f(x) .:: f(x), necessarily ..
Under these conditions it may be precisely at P=O at ~Ch f(x+p) is dis-continuous.
However 1 we
p, s only from theset of measure ?.h.. The sa.:me is true of differentiation
ve :make it. re::!.ative to the set of measure 2h ..
f(x+p) - lim f(x~)
"" lim .M,-4€>
f-'1~
holds for right and left-hand differentiation of re::!.ative to
such a set.
Table II gives some additional properties of left and right-hand
g .. derivatives ..
TABLE II
:JIIUID G-DERIV ATT>!ES
1. n+r+(x) ~ D+f(x}.
2. T+[T+ f(x)] == T+[T-:f(x)l =: x).
3. If f is differentiable at x, then f'~{
= D+ f(x).
4. If f is right continuous at x, then is also.
5. r:r f is right differentiable at x, 'then is also.
If lim f(x+p) exists, then 1+(x) exists and they are equal.
r~o
If f is right continuous at x, then f(
8. If lin!.
f-'90
X+p) it is to
~ "" f (x).
x).
29
The met that a useful has been
slope of a line best fit over some interval by
least squa.res criterion suggests that a stmilar relationship might exist
betvreen the second derivative and the best fit quadratic., Furthermore,
since the criterion of best fit is least squares, one might antic:l.:pate
tha.t a simplification in the approach might be obtained if \ve 'lte:re to fit
with Legendre pol;ynomials~ Both of these are true.. Ho~:.,rever, in the same
'.;ray that we :fit the line over a general interval (x-h, x+h] we must adapt
of Legendre polynomials are well kr~orm~
(1) They are orthogonal on the bi·wUX:tit interval.,
( 2) lJ.'l:le expansion of an Lebesque square integrable
f'unction in L2 in tenns of Legendre polynomials :mini.m·i.zes
the integral of the square of error~
(3) The orthogonality conditions that such an e:r,:;pans ion
requires that increasing t:he degree of the polynom:ial fit
by one necessita·tes calculating only one new coefficient.
Furthermore, the Legendre polynomials satisfy the familiar Legendre
equation and can be obtained by orthogonaliz:i.ng the functions 11 x, x2, •
using the Schmidt orthogonalization process and the 12 inner product.
The func·tions which we will define as genera.lized Legeru:lre polynomials
30
1dl1 be obta:iltable
or, more
f'\h
[ g] = j t(x+~ )g(
'=h
the Schmidt process
They 1-Till be polynomials the primed variables (x~-x) like the
~;rt:rnigb.t line discussed previously, 001d v.l.ll also be normalized.
Let us denote by P the fa.miJ..ia:r nth degree Legendre polynomial, and
n
p the nth degree generalized Legendre n
Fixing x and h gives us an space on
p 's form a basis for the space. That is,
i
)? .(p
J
the
If' we a function f in these polynomials, the n~th coefficient
rh
is given by = [f, ] = J. f(x+~)p11 (!;
=0.
mentioned above which carry over from
the bi-unit interval to the
relations in revised form and also a nev form of Rodriguez' Formula.
Given ill Table III is a listing of the analogous relations, the form on
the bi-unit interval followed by the form on the general interval.
1.
3·
4.
6.
RECUllSION
:for:u:n:lia :
$) "" (n+l}P ( ¢)
n
III
-x)
33
Given below is a comparison of the first three Legendre polynomials
'
(1) p (x) "" 1, p (xf-x) =[~Jt; 0 0
I
(2) P1 (x) "" x, p (x' .. x) -[_lrx• ... x •
1 ~ 2h !I
~ [ 5 Ji "'
( 3) -- =.1}, :x' "".::!: -- 2 lj. 2 2h
as of ) is
""
±'(xi) =:: (X~
then
h
~- f(
1 [~] h
::::
h s
1 [ ~ J h
·-·- 2 ?~
He l:!c&:ve used lot.;re:r case a.ns here coe:i'ficie.nts.
ir.t thE: coeff~l.cien:t of each
vle :J.ntrod:uce
as be the coe1'fici.ent of
That is
p0 (x'-x) :;;:: b oo'
(xg "" bll(x~
( v~ \, ,,. "' b20 + b22(x~ '
¢
- b no
to the expansion :Ls
+
0 1 ' b (' t )n+l l(x'-x; + • ~ " + n+l n+l x .. x; ~
:n+ " J
T'nis coefficient :i.s a function of x and h. lt is clear that the
recursion relations listed a'bove serve simply to establish relations between
the various b .. ~ s and could be written in tern.s of these ..
::l..J
We will now determine what happens to this gene1~ized ~~159AL~L
in the l.ir'11i t as h goes to zero~ T'ne term containing the th
d.eg:ree polynomial may be >-?ritten as
:if' n is even~ !<'or n odd
independent of' :x aud h. The entire first factor is a fl.mction of x and h
j
a np: n (x'-x) ""
since the first limit exists. We
(x~
if l:tm
h_~?"'
j_tl the 1.irni t ~~s h
zero since
to the
a fin:l:te i:f
to this coe:fficdent is f.rom
in (x'
35
= 0 for all 1ll < n. T:.11at
m
of ( in
terms in
in h.,
fact tbzft if f is
the non=zero contri~
any coefficient of
Theorem XVI:D:: If x+p) ~ 0 for a.ll p such ·that "'h ~P ~ then the
of in 1~he
Proof: ~· formula
'l'l1en
s)ds ""
I
)1:
"" h
(
h
h
f(
2
n
'" 1] •
1
as a result of' It i:s clear
by parts until we a:r:rive at
+ .n
... f" (
-where the sign of t.he last term is [:!] if n is (:;.~] ~
that
end
+ ~[(~)2 ~ 11 n-l
11
If we continue taking cleriva.tives, irl ecvery term. appears a :power of
~ ') ((h)"'· "" 1], and there one su.c::h term in each de:r:i"'rat;ive wh:Lch is one degree
less than the term of minimum degree in the :preceding derivative,. In fact:~
will be
'l'his tem
~ow est
n ,,
"
wery other te:r-1n J.n the expansion of
j l"\h
:f(
-h
ex{:ept the one of least degree is zero when evaluated at h or -h. Hence
12 · n12 n Now, on the interval [-h,h], [(h) - 1]~0. Hence (-1) [(h)· ... 1] ~o.
Since we know that Kn, h) 0 and, by hypothesis, jl ( x+s ) > o, it :follows
that
rh
/ f'(x+~ )p ( S)d~ > o.
v-n. n
Corollary: Given a function with the same properties as in the
previous theorem, t.he coefficient of (x•-x)n in the Legendre expansion
is greater than zero.
Proof: This follows immediately from the theorem and the fact that
b n,n >0 for all n.
Theorem XIX: Let f and g be fUnctions such. that ~(x+p) > gn(x+p)
for -h ~ p ~h. Then
and by the previous theorem
rn eh
I f(x+Op (~)dt ~J' g(x+~)p (t)d~.
·...1-h n -h n
Theorem XX: Suppose m.f:t:P(x+p)~M :t'or all -h~p~h .. 'I'hen
m ~n! a ll ~ M, where a b is the coefficient of (x'-x) ln :pn{x'-x) n n,n n n,n
:in the I.egend:re expansion of' f( x) •
38
g(x') "" (x~
h(x~) co: is clear
Now let
J
be respectively the best fit n 1 h. By
the theorem But
1: If' r(:x+p) is continuous for all p that -h~ p ~
'Where f3 is the
n
of the nt
f(x+p) is n
some neighborhood of zero. Then
best fit n~th
for all p in
'Where ~ is as in
n
establishes the same kind o:f
the the derivative of f and i;he best fit n 1th
first a..nd the best
straight line. In fact, it defines a generalized ntthe derivative.
Suppose the best fit th degree polynomial to x+p) for p such
that -h.f. p~ his
Recall that f3 is also the
n
coefficie:r:rt; of ( ""x) in
conclude and
iterated n
that
respectively1
(1) for every p such that -nh f. p!:: m ~ .:::Pnf(x) ~ M.
(2) for every p such that -h 6:: p ~h, :m ~D~f(x) !S: M:.
of f exists and is continuous on some inte!"V1'il
x) "" f(n) (x). 'lliis is typical of the
relationship between D~f(x) and nDhf(x) for functions of various differ~
entiability characteristics.
L Achieser,
Company:
N. L
New
2. Halperin, Israel.
University of Toronto
BIBI.IOG!lAPHY
Frederick Ungar Publishing
3. - Jones, U.S.: On a generalized derivative: Quarterly
Journal of Mathematics, Oxford Series, (2)4, pp. 190-197~ 1953.
4.
5. Kassimatis,
Canadian
6. 11cShane, E. J. Intem:.at.ion: Princeton Press, 1947.
4l
VITA
Thomas vHlliam Cairns
Candidate for the Degree of
Doctor of Philosophy
Thesis: A GENERALIZED DElUVA'l'IVE
Major: Mathematics
Biographical:
Born: The writer was born at Hutchinson, Kansas, Nov~nber 13~ 1931$
the son of Edmund A. and Gladys Cairns.
Undergraduate Study: He attended elementary and secondary school at
Hutchinson and graduated from Hutchinson High School in 1949.
From September,l949 to Maytl953 he attended Oklahoma State
University and was granted a Bachelor of Science degree with a
major in physics.
Graduate Study: He attended Oklahoma State University from
tember» 1953 to August) 1954 and was granted a Master of
Science degree in Mathematics in January of 1955. In
tember» 1956 he was discharged from the army and returned to
pursue studies leading to the Doctor of Philosophy degree. He
was in residence until August~ 1959,
Experiences: The writer entered the States Army in
September, 1954 and "-iSS discharged in September~ 1956. During
his service career he spent eighteen months in research with
the National Security Agency with the title of 11mathematician11 ,
In September, 1959 he accepted an appointment to the position
of Assistant professor with the mathematics department of the
University of Tulsa~
ions: The vrriter is a member of Phi Phi, Pi Mu Epsilon,
Omicron Delta Kappa and Sigma Pi societies and
is a member of the American Mathematical Society, Mathematical
Association of America and the Society of Industrial and Applied
Mathematics.