Internat. J. Math. & Math. Sci.
VOL. 12 NO. 4 (1989) 685-692
685
PERIODIC BOEHMIANS
DENNIS NEMZER
Department of Mathematics
California State University, Stanlslaus
801 W. Monte Vista Avenue
Turlock, CA
95380
(Received November 14, 1988 and in revised form January 23, 1988)
A class of generalized functions, called periodic Boehmians, on the unit
ABSTRACT.
circle,
It is shown that the class of Boehmians contain all Beurling
An example of a hyperfunctlon that Is not a Boehmian is given. Some
is studied.
distributions.
growth conditions on the Fourier coefficients of a Boehmlan are given.
It is shown
hat the Boehmians, with a given complete metric topological vector space topology, is
not locally bounded.
KEY WORDS AND PHRASES.
Boehmians, delta sequence, convolution, Fourier coefficients.
1980 AMS SUBJECT CLASSIFICATION CODE. Primary 44A40, 46F15, 46F99, Secondary 42A16,
42A24.
I.
INTRODUCTION.
The construction of a class B of generalized functions,
first introduced by J. Mikusinski and P. Mikuslnskl [I].
called Boehmians, was
This construction, which is
purely algebraic, is similar to that of Mikuslnski operators [2].
will be concerned with Boehmians on the unit circle, denoted by
in [3].
The space of Boehmians is quite general.
real line (unit circle)
circle).
In
Section
contain all
4
it
will
distributions on the unit circle.
be
For example, the Boehmlans on the
Schwartz distributions on the real llne (unit
shown
that B(F) also
6-convergence, on
[2].
Type
,
contains
all
Beurling
Growth conditions on the Fourier coefficients of a
Boehmian were given in Theorems 5.14 and 5.15 of [3].
in Theorems 4.2 and 4.4.
In this paper, we
B(F), first introduced
These results will be extended
P. Mikusinski [4] also introduced a convergence, called
which is similar to Type
convergence on Mikuslnski operators
.
convergence is not topological (-see Section 3), but 6-convergence can be
It is known that the dual
of B, on the real line, contains only the trivial continuous linear functional. We
will show, in Theorem 3.5, that the dual of B(F)contains many continuous linear
used to induce a complete invarlant metric topology on
functlonals, but that B(F) is not locally bounded (Theorem 5.2).
In Section 2, the space B(F) is constructed.
can be found in [3] and [4], but is presented
In this section tw types of convergence
reader.
coefficients and Fourier series of a Boehmian are
Most of the material in Section 3
here
for the convenience of
the
on B(F)are discussed, the Fourier
defined, and it is shown (Theorems
3.7 and 3.8) that every Boehmlan is the sum of its Fourier series.
In Section 4,
D. NEMZER
686
sufficient conditions on a sequence of complex numbers to be the Fourier coefficients
of a Boelnlan are given (Theorem 4.2). It is also shown that the Fourier coefficients
satisfy a growth condition (Theorem 4.4).
An example .(Example 3) of a hyperfunctlon that is not a Boehmlan is
(n)
this example gives rise to a sequence
Inl ,
Also,
iven.
n" e(Inl)
as
Fourier coefficients of a Boehmian, answering a
In Section 5, by using the results in the previous section, we
(n)
such that
of complex numbers,
is
question posed in [3].
not the
The final section is
show that 8(r), endowed with a topology, is not a Banach space.
concerned with some remarks and open problems.
2.
NOTATION AND CONSTRUCTION OF 8.
The unit circle will
continuous
be denoted by
r.
C(F)
(Ll(r))
is
the collection of
(c(r))’"
cm(r)
(Integrable) complex-valued functions on r.
wiI1
all
the
be
collection of sequences of contlnuous (m times continuously dlfferentlable) complexvalued functions on
r.
Thus,
The convolution of f and g in c(r) is denoted by Juxtaposltlon.
(fg)(x)
f(x-t)g(t)dt.
I/2
-w
If f ,f
n
on
1,2,..., llm f n
c(r) for n
f will
mean
(in)
converges
uniformly
to
f
r.
A
sequence
of
continuous
valued
real
functlons,
will
(j),
be
called
an
approxlmate identity or a delta sequence if the following conditions are satisfied:
(i)
for each J, 1/2
6j(t)dt
I.
) 0.
(ii) for each J and all t,
(ill) Given a neighborhood, V, of l, there exists a positive integer N such that
is contained in V.
for all j ) N, the support of
6j(t)
6j
The collection of delta sequences will be denoted by A.
DEFINITION 2.1.
A--
AcCN(y)
{((fj), (j)):
Let
Two elements
by
x A be defined by
for each I and each
’~’
fiJ fJi }"
A are said to be equivalent, denoted
and J
gj 6i. A straightforward
is an equivalence relation on A.
The equivalence classes
((fj), (6j)) and ((gj), (oI)) of
((fj), (j)) ((gj), (aj)), if for all i-
calculation shows that
J
fiJ
will be called periodic Boehmians.
DEFINITION 2.2.
The space of periodic Boehmlans, denoted by
{[(fj)/(j)] ((fj), (Sj))
8
,
is defined as:
A}.
For convenience a typical element of 8 will be written as
[fj/j ].
By defining a natural addition, multiplication and scalar multiplication on 8,
i.e.
[fj/6j]
+
[g/aj]
[(fj j + gj.)/j ’]’i [fj/j ][gj/
[fj gj /
,] and
PERIODIC BOEHMIANS
a[fj/j]
Let
[afj/j],
(6j)
be viewed
e
as
C(F)
8becomes aa algebra.
Then D’(F) (Schwartz distributions on the unit circle) can
0 h.
B by identifying
subset of
a
[u*j/6j],
u with
where
u*6j
denotes
the
Let U be a class of sequences on a space X (with or without a topology).
We
6.
convolution of u and
3.
where a is a complex number,
687
[5]).
as d[stributions (see
CONVERGENCE AND FOURIER SERIES.
say
Xn
U
If the sequence
exists a topology
(X,Xl,X2,...)is
for X such that x
T
U
a
more
general
coustructlon
U is called topological if there
+z x.
A conve rgence used
convergence, is not topological [6].
x if and only if x
in Mikuslnskl operational calculus, called
In
in U.
of
Type
Boehmlans,
P.
Mikuslnskl
[4
defines
a
convergence, called A-convergence, and shows that A-convergence is topological.
In
fact, Mikusinski shows that B is an F-space (a complete topological vector space in
which the topology is induced by an invariant metric).
Before
define A-convergence, we
we
will
define
called 6-convergence.
Let a
n
exists
a delta sequence
each j
llmnanj
(j)such
The proof of the next lemma can be found in [4].
Let a
E c(r) for n=l,2,
c(r) and for each J,
fn
E
n
that for each n and J,
__anj
E
is 6-convergent to
n
that for each n and J,
This will be denoted by 6-1im
a6j.
LEdA 3.1.
_
1,2,..., we say that (a)
B for n
a e
6-11m a =if /6 ].
n
j j
DEFINITION 3.2.
convergence,
related
a
an
a6j,
an
j
if there
a
C(),
and
for
a.
If
limnan6
there exists ()
n
then
E
A such
fj,
A sequence (a
of Boehmlans is said to be A-convergent to a,
n
denoted by A-llm a =a, if there exists a delta sequence () such that for each n,
n
n
-a)
..iARK.
e c(r) and lira
(an-a) 6n"
Boehmians (a) is A-convergent to a if and only
n
A sequence of
subsequence of (a
n
0.
contains a subsequence 6-convergent to a,
The Fourier coefficients of an
LI(F)
if each
[4].
function are defined in the usual way.
That
if f e L (r) define
1/2
Ck(f
f
f(t)e -iktdt, for k
O, I,2,
A, then for
From the definition of a delta sequence it follows that if (6)
n
if for some
a
for
that
show
I.
to
easy
It
also
is
._C(6n
O. These observations
0
then C
and some k C
each k, lim
npositive integer
[fj/6j],
Jo
together with "the fact that
o k (j
o
o
if a
[fj/6j],
k (fj
o
o
then for each n and m,
fn 6m =fm6n’
make the
follot.ng definition possible.
DEFINITION 3.3.
Ck(a
that
Let a
Ck(fj)/Ck(j)
%(6j) O.
=[fj/6j]
where
for
e
For k
fixed
k,
0, +/-I, 2,..., define
is
the
smallest
index
such
688
D. NEMZER
The proofs of Theorems 3.4 and 3.7 can be found in [3].
Let a, an
THEOREM 3.4.
In this sequel,
unless
Boehmlans
the
on
As
one.
trivial
In
Boehmians.
real
Burzyk and T.K.
llne have
the next
fact,
no
are
enough
separately, have shown that
linear
this
show
continuous
the one
be
for 8 will
topology
Boehme,
continuous
theorem will
there
the
stated,
otherwise
J.
by A-convergence.
induced
Suppose 6-11m a =a, then for each k,
n
8, for n=I,2,..’..
Ck(a).
limn k(a n
C
functlonals
is
the case
not
linear
other
than
the
for periodic
functlonals
to
separate
points.
THEOREM
Let a
3.5.
8, for
a
n
1,2,...
n
If A-lira a =a, then
n
for
each
Ck(a n_=Ck
)( a
k, lira
n
Follows from the remark following Definition 3.2 and Theorem 3.4.
PROOF.
n
THEOREM 3.7.
THEOREM 3.8.
[. Ck(a)e Ikt.
Let a e 8, then the Fourier series of a is
DEFINITION 3.6.
For each a
E
6-11m n
8, a
For each a e 8, a
k---n
n
A-llm n
Ck(a)e ikt.
Ck(a)eikt.
Follows from the remark following Definition 3.2 and Theorem 3.7.
PROOF.
GROWTH CONDITIONS ON THE FOURIER COEFFICIENTS.
4.
In the construction of Beurling distributions, see Example 2, a real-valued even
.
defined on the integers Z satisfying the following conditions is required.
function
(n) + re(m)
0 --m(0) < mCn+m)
2
CA)
(B)
<
(n)/n
.
for all n,m
E
Z.
n-Functions that satisfy conditions (A) and (B) will play an important role in this
section.
The proof of the next technical lemma is similar to the proof of Theorem 1.2.7 in
[7].
LEMMA 4.1. Suppose m is a real-valued even function satisfying conditions (A) and
(B). Let ml(n)
max
Then
ml also satisfies conditions (A) and (B).
PROOF. It is routine to show that
satisfies condition (A).
To show
{m(m):Iml
I
that
ml
sets
F,G, and H as follows: F
{n
G
satisfies condition
E
N:
0(n-l)=ml(n-l),
consider the set H.
i
m
Inl}.
< Jm <
i
m+l
H
which
im
--min {i
E’={neN:
)
2.
-I,
m
m
such that V
m
(If (n)
Let E
{n
re(n)
UV
(i-I)=1(i
m
m ),(Jm+1)
(B), partition the
E
N: (n)
set of nonnegative integers N into the
ml(n)}’
< ml(n), (n+l)=ml(n+l)}
N\(F U G). First,
and H
where for m=l,2,.., there exist i ,J
m
m
E
N with
{nN:
,j
(n)<(n) for n=im ,im+l
m
l(Jm+1)} 1(Jm+I)}. Consider a typlcal Vm for
the lemma is trivlal).
1(n) for all n or if Jm
{neN: i
Jm-im}.
m
and
+
im(n(im
m(n))m(im-l)/3},
n
i
+
m
and re(n)
and E ={neN:
<
Let
m(i -I)/3),
m
n=im-l+Xl-X2
)
’ Xl’X2eEl"
-m
PERIODIC BOEHMIANS
Let
I’I
cardinality of ". It then follows that E*c E’ and
IE*I
)IEI,
+
(=im-I
cases
689
or
IE’I
hence
/2.
)
=jm-lm),
we obtain
Jm
2
n=i
l(n)/n
.
n=i
108
4
nH
. .
. .
2
(n)/n
neG
(n)/n
2.
considering
and
estimates
m
(n)/n
2.
(4.1)
nell
Now, for each neG .(n)/n (n-l)/-(n-I)
Hence,
using
Jm
108
m
[ (n)/n 2
Thus,
By
2
2.
(n-l)/(n-l)
(
2.
(4.2)
neG
01(n)
Since for each neF (n)
(n)/n 2=
neF
(n)/n
2
(4.3)
neF
By combining (4.1), (4.2), and (4.3), we see that
I
satisfies condition (B).
Thus, the theorem is proved.
Before giving sufficient conditions for a sequence of complex numbers to be the
Fourier coefficients
Suppose
< (x) + (y) for all x,y
(t)/t2<
R,
,
0=(0)
function
--1 on
and
Mreover,
K.
I, the
that 0
such
(t)
O(e
-(t))
support
as
4
u(x+y)
n(l+It I)
and (t)
Then given any compact set K of R and any neighborhood V of
continuous
BJorck [7].
of a Boehmlan (Theorem 4.2), we need a result of
is a real-valued continuous function on R satisfying
Itl ,
of
is
where
K, there exists a
contained
in V,
denotes
the
Fourier
transform of
By using the above, we may construct an approximate identity
such that for
-(n)
each J C ()
0(e
THEOREM 4.2.
Let
be a real-valued even function satisfying conditions (A) and
/ -,
(B). Suppose (n) is a sequence of complex numbers such that
as
(j)
n-0(e(n))
then there exists a Boehmlan a e 8such that C (a)
n
Let 0*(n)
conditions (A) and (B).
PROOF.
l(n)
+
Inl,
n
is as in
Inl
for all n.
Lemma 4.1. Then
satisfies
Let *(t) be the piecewlse linear extension of
to R.
where
01
*
*
Since *(t) is increasing on
[0,=) and
7.
0(n)/n
2
<
=,
[ a(t)/t2dt <
=.
Also,
n=l
(t)
n(l +
C (6)
n j
0(e
Itl).
-*(n)
So there exists
(j)
.
e A such that for each
m
as
n
+-.
Let
t)
k= -m
ke
for u0,1,2
Then, for
690
D. NEMZER
each J, the sequence
Now, for
In
m)
C
(Pmj)converges
n (pm)=n.
uniformly.
Hence for some a e 8, 6-1ira
for each n C (a)
by Theorem 3.4
Therefore,
n
n
Pm=a.
This
establishes the theorem.
The following lemma is a special case of Theorem XV from [8].
LEMMA 4..
Let (t) be an increasing function for t)0, such that
.
(t)/t2dt
LI(F)
Suppose f e
such that C
-n
(f)
-(n)
O(e
)as
If f
n/.
vanishes almost everywhere on any interval, then f _= 0.
The next theorem shows that the Fourier coefficients of a Boehmlan cannot grow
too fast.
THEOREM 4.4.
[
(n)/n
Let : Z
.
2
n=l
Suppose
A, M, and
positive
ppose
Suppose (.) e A.
(n)
n
is a sequence of complex numbers such that there eist
lnl
oehmlan.
such
FQurler coefficients of a
PROOF.
R be an increasing function for n=0,1,2,.., and
)
that
Ae
)
e(n)
A e n) for all n
for all n
M.
)
Then
[
(n)/n
2
is
the
not
n
)
M.
t
Ikt
e
I
k
k=-n
Pn(t)
Now,
Extend (n) linearly to a funetlon (t) on [0,).
since (t) is increasing and
(n)
.
(t)/t2dt
=-,
en
by the previous
n=l
lamina, for each j there exists (n
k)
such that
Cnk(j) ,
-’nk
e
us,
for each
n
PnJ
3.7.
J’
[ k Ck(j )elkt
(t)
EXALES. I)
If (n)
[5]) is isomorphic to {a
converge and the result follows by
does not
n(l
+n),
then D’(r) (periodic
8: there exist
>
hwartz
0 such that C (a)
O(e
n
eorem
distributions
sn))
as
2) t m be an een real-alued unettn n the tnteger Z hich sattsftes
(n) + da) or ali n, e Z,
(n + )
() 0- (0)
[
(B)
(n)/n
2
< -.
n
(C) There
ali neZ.
(D)
exist
real
a
and
is concave down on
positive
b
such
that (n)
)a
+ b
n(l+In I)
for
n-0,1,2,....
The set of Beurllng distributions on the unit circle P’
function space with a locally convex topology).
is dual space of P
(a test
P’ is isomorphic to the collection
of all sequences of complex numbers of -slow growth [9]
(a sequence
(n)
of complex
numbers is said to be of u-slow growth if there exist constants M and a such that for
all neZ,
lnl
distributions.
Mea((n))).
So,
by
Theorem
4.2, 8 contains
all
periodic
Beurling
PERIODIC BOEHMIANS
691
>
0), where 0
a < I, in Theorem 4.2 (Theorem
The next example shows that
4.4), we obtain Theorem 5.14 (Theorem 5.15) in [3].
Theorem 4.2 (Theorem 4.4) is more general than Theorem 5.14 (Theorem 5.15).
a
cr+1
and for 0
is the
t/(nt) for t
t < e
3) For
(t)
let
e
By taking (n)
linear
function
Define
for t
n
whose
<
(re(n)
en, e
graph
passes
0, such that (0)
+I
a+l
Now,
satisfies condition (A).
(t), then
0 and (-t)
Thus, by Theorem 4.2,
let m (n) denote the restriction of m (t) to the integers.
(exp( (n)))
and (e
origin
I is not diicul o sho ha
Is any concave do increasing function for
satisfies condition (A), since, if
t
the
().
(-)
0 by
through
is the Fourier coefficients of a Boehmian when
>
I; but
(exp(l(n))),
by Theorem 4.4, is not the Fourier coefficients of a Boehmtan.
The above answers one (and partially another) question posed in [3].
n=e
Secondly, "How
(n)
sequence
Boehmian.
oo, is the sequence of
as [n
does 8 compare with the set of hyperfunctlons?"
I/n
llmsUPn lexp(l(n))l
(exp(l(n)))
I,
the
is
Fourier
Not every
coefficients
Fourier
for a
Since
of
coefficients
a
hyperfunction [I0], but is not the Fourier coefficients of a Boehmian.
5.
TOPOLOGY OF 8.
A topological vector space X is called a topological algebra if it Is an algebra
over the real or complex nabers and multiplication is continuous in each argument
It
separately.
is
difficult
not
to
show
that 8 endowed with A-convergence is
a
topological algebra.
Let X be an F-algebra (F-space which is an algebra over the real
DEFINITION 5.1.
Then X is called a p-normed algebra if there is a
or complex numbers) with metric d.
II’II:
function
R such that:
X
(ill) there exists 0
Zelazko
<
such that for all scalars a and each xX
p
[II] has proved that a complete metric topological algebra is locally
bounded if and only if it is a p-normed algebra.
THEOREM 5.2.
8 is not locally bounded.
Suppose 8 is locally bounded. Since 8 is a complete topological algebra,
on 8. Now for n=O,l,2
there exists an equivalent p-norm
PROOF.
II’II
n
let
an== aos
where s
0
(dj)
cN(F)qA’o
[j
()
a I/(i[)
and
positive integer M such that for n
llan-%II=II
for =0, 1,2
/j
n
n
z
Y.
=m+
.
=m+
,n,
for =0, I,2,...,n.
)
m
)
Given e
>
0 there
exists
M,
l / ,l II II <
That is, (a) is a Cauchy sequence and since 8 is complete, there exists an a
n
n
such that a
A-lira
n
[.
=0
as
So, by Theorem 3.5, for each k
E
8
a
.
692
Q_(a)
D. NEMZER
--(k)l!
Hence B must not be
which is impossible by Theorem 4.4.
e
4=0
locally bounded.
COROLLARY 5.3.
6.
B
is not normable.
SOME CONCLUDING REMARKS AND QUESTIONS.
squence (n)
(I) It can be shown that if the
is the Fourier coefficients of a
Boehmian a that satisfies the hypothesis of Theorem 4.2
limsuPnlnll/n
that
hyperfunctlon
hyperfu
and
is
a
hence
a
not
is
a
hyperfunction
Boehmian.
Is
(for some function ), then
[I0].
properly
3
Example
contained
in
presents
the
set
a
of
on
(2) Theorem 4.2 gives sufficient conditions for a sequence to be the Fourier
coefficients of a Boehmian. Are these conditions also necessary?
with A-convergence is an F-space [4].
(3)
Theorem 3.5 shows that
has enough
continuous linear functionals to separate points, but Corollary 5.3 shows that
not a Banach space.
Is
"B with
(4) What about boundary-value theory?
That is, what type of harmonic (analytic)
function in the unit disk has a Boehmian for its boundary value?
research
has
been
centered
8 is
A-convergence a Frechet space?
around
boundary-value
problems
of
Much mathematical
harmonic
(analytlc)
functions in the unit disk; see for example [9] and [I0].
REFERENCES
I.
J.
P., Quotients de suites et leurs applications
(1981), 463-464.
2. MIKUSINSKI, J., Operational Calculus, Pergamon Press, New York, 1959.
3. NEMZER, D., Periodic Generalized Functions, ock Mtn. J. Math., (to appear).
4. MIKUSlNSKI, P., Convergence of Boehmians, Japan J. Math. _9 (1983), 159-179.
5. SCHWARTZ, L., Theorie des distributions, Herman, Paris, 1966.
6. BOEHME, T.K., On Mikuslnski operators, Studia Math., 1969, 127-140.
7. BJORCK, G., Linear partial differential operators and generalized
distributions, Ark. Mat. 6 (1966), 351-407.
8. LEVlNSON, N., Gap and density theorems, American Math. Soc., New York, 1940.
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