ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I I I (1969)
В. P. P arashar (Mandla)
On the field of operators of Ditkin and Prudnikov
1. Introduction. An operational calculus for the operator — 2—
has been developed by Ditkin and Prudnikov [1] on the lines as Miku- siński [2] has done for the operator d/dt. The procedure of Ditkin and Prudnikov is as follows:
Let L be the set of all locally integrable functions existing on the half-line 0 < t < oo and vanishing on the left-half of the real line and also satisfying the condition, J -— J \f(t)\dt< oo for any t0 > 0. M
о £ о
t I
denotes the set of all functions of the form J — § f{t)dt-\-C, where f(t)e L and C is a constant. M then forms a commutative algebra without zero divisors, with sums and scalar products defined in the usual way and product defined as,
t i
f ^ Y g l i ł — riW—ęyidri-,
о
0M is then extended to a quotient field.
In what follows, we shall define product in M as,
t i
Ц.1) f-g(t) = J d f J / ( 4 J)S[ ( l - 4) ( « - f ) ] * ,
0 0
and denote the quotient field thus obtained by m.
The object of this note is to study some analytic and algebraic prop
erties of m.
58 В. Р. P a r a s h a r
2. If h denotes the Heaviside’s unitary function, then,
( 2 . 1 )
Thus h acts as an integral operator in this calculus. The correspond
ing differentiation operator is defined as, В = l[h (here 1 is the multi
plicative identity of m). By induction we can prove that,
stant, f e L and g eM, we can imbed the set of numbers, L and M respectively in m.
d d The following theorem gives a relation between В and — t— :
dt dt
T h e o r e m 1. Let feM possess a locally integrable derivative of order 2k) furthermore let us suppose the limits
(
2
.2
)B n {(w—-l)!}2
By the mappings a о a — , f о f — and g о g — , where a is a con-
h h h
to exist. Then
P roo f. By direct computation we see that,
i.e.
h Hence
The theorem can now be proved by induction.
Relation (2.3) reduces to,
(2 .4 )
for functions which are continuous in all their derivatives up to к —1.
In (2.3) a single discontinuity at t = 0 has been assumed but the gen
eralization to the case of a finite number of discontinuities is not diffi
cult.
T ra n s la tio n o p erato r. Let
Then
S x{t) = 1 °, u ,
0 < ź < A, 0 < A< t.
If we denote the operator d
(*-A ) d
by B x, then d(t — X) d (t— A)
B x{Hx(t)f} = f ( t —X). Thus f = B xH^(t) acts as the translation operator in this calculus.
3. L e m m a 1. I f f(t)eM and
K T = max {\f(t)\}, 0
then
IA *)I<- z f f 1 {(те—l ) ! } 2 P ro o f. The result is true for те = 1.
If it is true for те, then
L f + 1 («)l = ! / ” (*)! • 1 / ( 0 1
K 'r'1 ("
{(» —!)!}■
Ę + 1 { ( » - ! ) !}2
t 1
J d ( J r T - ' f - ' d r , =
0 0
K n T+1tn {n\}2
(3.1) C orollary . F or feM , f n — > 0 as n -> oo uniformly for every interval 0 < t < T.
This is the analogue of a theorem of Ryll-Nardzewski [3] in the operational calculus • of Mikusiński.
From Lemma 1 and (3.1) we easily derive
O O
T h e o r e m 2. I f the radius o f convergence of the series an Xn, where
n= 0
« 0 , 0 ! , . . . are numerical coefficients and A a complex number, is positive.
60 В. Р. P a r a s h a r
then the series £ anXnf n, /elf, regarded as a series of two variables A and t
»=i
is uniformly convergent in every domain 0 < A < A0, 0 < £ < .
The theorem has an analogue in the operational calculus of Miku- sinski.
4. Linear mappings. Let Mv = { f : where M* = M — {0}. Clearly
m = { J {M p}.
P tM *
We now derive the following results:
(i) Mp is a module over I f .
For if geMp, then g = fjp for /elf.
Hence rg = rf[p eMp, where r elf.
(ii) Mp is not an algebra over M.
For if Tc, leM p, then Tc = fjp , l = glp. Hence k-l = (1 lp)(fglp)- Since in general fg jp iM , in general Tc • 1 4 MP.
Provide a relation < to M* defined by c < d if there exists b e lf such that be — d. The relation < directs the set M.
(iii) I f p < g, then Mp a Mq.
For if xeM p, then x = f/p for /elf.
Since q — Jcp for some 1c elf, x =//p = TcflqeMq. Hence xeM q.
The mapping p -> Mp thus defines a net in {Mv}. Let D be the set d d
of fe M such that D is closed with respect to the operator — t— . dt dt Let S be the subalgebra of m defined by S = {aem : aD c f>} and N = {p e lf*: — -D c f>).
Let MN = P {Ifp]-
P<lN
