ON DEFINING THE GENERALIZED FUNCTIONS f+-o - Downloads

Internat. J. Math. & Math. Sci. VOL. 16 NO. 4 (1993) 749-754

749

ON DEFINING THE GENERALIZED FUNCTIONS (z) AND "(x)

E.K. KOH and C.K. LI

Department of Mathematics amd Statistics University of Regina Regina, Canada $4S 0A2

(Received April 20, 1992 and in revised form August 10, 1992)

ABSTRACT.

In a previous paper (see [5]), we applied a fixed

-sequence and neutrix limit due to Van

5k and (’)k for k E (0,1) and k 2,3,.... In this paper, ve choose a fixed analytic branch such that z’(- < arg z _< ) is an analytic single-valued function amd define i(z) on a suitable function space In. We show that ’(z) E I’. Similar results on ((’)(z)) are obtained. Finally, we use the Hilbert integral o(z) f+-o dt where (t) D(R), to redefine (z) as a boundary value of ’(z i). The definition of f’(z) is independent of the choice of -sequence. der Corput to give meaning to distributions

-t

KEYWORDS AND PHRASES.

The -function, generalized functions, Hilbert integral.

1991

MATHEMATICS SUBJECT CLASSIFICATION: 46F10, 44A15, 30E20. 1. INTRODUCTION

The difficulties inherent in defining products, powers or nonlinear functions of generalized functions

(see e.g. [1], [2] and [3]). In [4], a definition for a is given using 5-sequences. However, they show that 2 does not exist. Similarly,

have not prevented their appearamce in the literature

product of distributions such objects as

v/ and

sin 5 appear to be meaningless.

In this paper,

we attempt to show a way out in

some sense.

In

a previous paper

[5],

we applied a fixed

give a meaning to distributions

5

and

(’)

-sequence

for k

and neutrix-limit due to Van der Corput to

(0,1) and k

2, 3,-... In fact, our definition is

a regularization of distributions which is similar to the method used to define some pseudofunctions

by

Hadamard’s finite part. A method is introduced here with the help of the Cauchy’s integral formula and the Hilbert integral transform. For M1 c,

(

5’(z) as the boundary vMue of n(z- i)

R +, we define a(z) on a suitable function space I=, and redefine as

0 +. The definition of ’(x) is independent of the choice

of the 5-sequence. No neutrix limit or Hadamard’s finite part are needed.

E.L. KOH AND C.K. LI

750

6(o E R +) number greater than 1. Let la

2. THE GENERALIZED FUNCTIONS

Let

a be a fixed positive

{0(z) 0(z) is analytic

on

Imzl < o},

We

supply la with the following convergence concept.

Let 0n(z) be in contained in

I

[l,.z[
= -’(0) In (1), we may think of

/

as the Cauchy representation of

z

6(z) (cf. [3]). By the Cauchy Theorem

on analytic functions, we know that this representation is unique up to an analytic function on the strip.

We shall take this as the essential Cauchy representation. This suggests the possibility of operating on the Cauchy representation to effect the same operation on 6(z). Thus we have

DEFINITION 1: The generalized function 6O(z) on la is given by

< 6,(z),,(z) >A= where C is defined

1 (Z)dz,

(2i) c

o

e R +, e I

(3)

above.

DEFINITION 2" If 6 and 6 e

= (2gi)+0 zO+0

(4)

It is ey to s that the index law and the power re of differentiation below e satisfi:

6(). 6.()

6+()

These properties suggest that the definitions above are reasonable.

(5)

751

GENERALIZED FUNCTIONS

Our first main result is THEOREM 1. ((z) belongs to the dual

l’

for a E R +, and

+oo

hO(z)=2isin

#"}(z)

.Co

nO

b"(z)=

for a

(n-a%l)n

2riC" b(,_,)(z

for n

(7)

# 1,2,3,.--,

(8)

1,2,3,....

(.- I) PROOF. Let (z) E Is and let ,=0 a,z" be the Taylor series which converges C. For a I, 2, 3,..-, equation (3) yields

(z)

to

9(z) uniformly

on

zn-adz n=0

c

c

(9)

C

z "-dz +

a

z "-dz

n=O

Since

is nMytic on

x-xis t

(- 1,0) nd (1,0).

{z "[Iz[

a-

{0)), by Cauchy’s Threm, we cn sume that C ps the

Integrating and combining the right hand side of (9), we obtain

an

< (), () > =0

2iCsin Since the Tylor series coecient

() This proves equation

except the

(n- 1)th

an

(0)

+ (_1)

n-+l"

()(0)

(_1) ((n)(z),(z) ), we infer that + {)(z) ,,3,.... ( + 1)’

ici

(7). To prove equation (8) we let

n, nd note that

terms in the series vnish

term. Thus,

()

2iCn (_ )

This concides with the result from the definition

-)()"

< n(z), (z) )= Cn c

Let m(z) 0 in I. There exists M ) 0, m m(z)] < M for MIm 1= It follows from the fct that (z) is nMytic on lz a that

(Zz

1,2,3,-.., where

I")(o)1 < M(a) 1

converges, foUows from (10) that (z) a From Cauchy’s integrM formula,

Since

it

(o)

g.

Clearly, "(z)

d. c

dr.

1.

(

a

(

a.

E.L. KOH AND C.K. LI

752

We define


1
0+lim Re < 6n(z- i),(z) > o+lim Re(2ri)n

J

e

0 +.

(z) d. (z i)

By Cauchy’s integral formula, tim Re < "(z) (z) >= ,-0+

("-)(i,)

n(z)--( _i)n_

(i).- (._ x) n=2k,

0 ()()

n

("-)(0) (.-

k= 1,2,3,.--

2 % 1,

0, 1,2,....

Thus, we obtn our main threm

THEOREM 3. for k

1,2,3,..-

and

$k+l(z) This theorem is quite similar

(-1)t8(t)(z)

(2x)2t(2k) to Theorem 1 of [5]. In

for k

0, 1,2,....

both cases, the even powers of (x) turn out to be

zero while the odd powers are expressible as a constant multiple of a derivative of

however, is obtMned without recourse to

a

(x). This threm,

delta sequence nor to Van der Corput’s neutrix Hmit, an idea

that requires a bit more machinery.

REFERENCES 1. S. Gasiorowicz, Elementary particle physics, J. Wiley and Sons

Inc., N.Y., 1966.

2. Chen Ching Yih, A survey of the budding nonlinear theory of distribution

1(1981), 3. J.H.

(in Chinese), J. of Math.

127-133.

Bremermann, Distributions, complex variables, and ing, Massachusetts, 1965.

Fourier transforms, Addison-Wesley, Read-

F54

E.L. KOH AND C.K. Ll

Roman Sikorski, Theory of distributions, the sequential approach, PWN-Polish Scientific Publishers, Warsawa, 1973.

4. Piotr Antosik, J. Mikusinski and

5. E.L. Koh and C.K. Li, On the distributions

k

and

(6’)k, to appear in

6. E.C. Titchmarsh, Introduction to the theory of Fourier

Math Nachr.

integrals, Clarendon Press, Oxford, 1948.

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