ABSTRACTS
IVAN P. DIMITROV
1. ON THE EQUIVALENT WEAKLY MIDPOINT LOCALLY UNIFORMLY CONVEX
RENORMING OF THE SPACE l∞
Geoge A. Alexandrov, Ivan P. Dimitrov
Abstract
In this paper is proved that the space l∞ has no equivalent weakly midpoint locally unuformly
convex norm .
MATHEMATICS AND EDUCATION IN MATHEMATICS, 1985, Proceedins of the Fourteenth
Spring Conference of the Union of Bulgarian Mathematicians, Sunny Beach, April 6-9, 1985, Sofia,
BAC, 1985, p. 189-191 (in Russian).
2. SOME NON-METRIZIABLE SPACES OF HARMONIC FUNCTIONS
Balikov G.I. and I. P. Dimitrov
Abstract
Let D be the unit ball in R2 and F be an arbitrary subset of S2 .We denote by H(D ∪ F ) the
space of all harmonic functionson D ∪ F , equipped with inductive limit topology. Let H̃(D ∪ F )
be the space of restrictions of the functions from H(D ∪ F ) equipped with it’s natural topology.
In this paper we prove that H̃(D ∪ F ) is isomorphic to a sequence space.
Transform Methods & Special Functions, Varna’96, Proceedings of Second International Workshop, 23-30 August 1996, p. 24-31.
3. ABOUT ZAHARIUTA - VOGT’S CONDIIONS
Gurko. I. Balikov, Ivan. P. Dimitrov
Abstract
The subject of this paper is to show that for a class of Fréchet spaces there are simpler
equivalent expressions for Zahariuta - Vogt’s topological invariants, in our opinion, more natural
and suitable for applications.
25−th International Summer School ”‘Applications of mathematics in engineering and economics”’,
Heron press, Sofia 1999, p.75-77.
4. SOME NOTES ON ZAHARIUTA-VOGT’S CONDITIONS
Balikov G.I. and I. P. Dimitrov
Abstract
In this paper we present a non-trivial examples of functional spaces, satisfying conditions DN
and DN ( Zahariuta-Vogt’s conditions ). Also we set an open problem, concerning basis existence
in a functional space of special type.
Fractional Calculus & and Applied Analysis, Vol.2, 5, (1999), p. 583-589.
5. SOME NOTES ON THE SOLUTIONS OF NONLINEAR DIFFERENTIAL
EQUATIONS WITH COEFFICIENTS FROM A SPACE OF
ULTRADIFFERENTIABLE FINCTIONS
Balikov G.I. and I. P. Dimitrov
Abstract
As it is well-known the solution y(t) of the equation
0
y = f (t, y), y(0) = a, t ∈ [−1; 1], y ∈ [c; d]
is also analytic ( respectively infinitely differentiable ) function in the neighbourhood of zero if
the right side f is analytic (infinitely differentiable) function. We consider an ordinary differential
equation of type
s
X
0
y =
aj (t)y j , y(0) = a,
j=o
where aj (t), j = 1, 2, ..., s k belong to a space of ultradiffereniable functions, defined on [−1; 1].
We do not try to find conditions on functions aj (t) and constant a which guarantee an existing
of solution on the whole segment [−1; 1]. Suppose that y(t) is defined on [−1; 1], we prove that it
belongs to the space of coefficients.
Proceedings of 28−th International Summer School on Applications of Mathematics in Engineering
and Economics, p. 75-79, Sofia 2002.
6. ABOUT SOME PROPERTIES OF A CLASS OPERATORS IN SPACES OF
ULTRADIFFERENTIABLE FINCTIONS
Balikov G.I. and I. P. Dimitrov
Abstract
In this work we consider operators of type L(y) = dy/dt − f (t, y), t ∈ [−1; 1], where the
function f (t, y) is, generally speaking, non linear with respect to y. Suppose y(t) and f (t, y) are
ultradifferentiable functions, defined on [−1; 1] and respectively on the rectangle [−1; 1] × [a; b],
a, b ∈ R, we give a short description for some properties of L(y).
Proceedings of 29−th International Summer School on Applications of Mathematics in Engineering
and Economics, p. 63-66, Sofia 2003.
7. SOME SPECIAL SPACES OF ANALYTIC FUNCTIONS AND SEQUENCE SPACES
I.P. Dimitrov
Abstract
For each τ0 > 0 we denote by S(τ0 ) = {z ∈ C : | Im z| < τ0 } the strip parallel to the real
b 0 ) we denote the Frèshet spaces of all analytic functions, defined in
axes. By H(τ0 ) and H(τ
S(τ0 ) and having representation there in series of Hermite polynomials, respectively in series of
Hermite functions.
We introduce the sequence spaces:
b2 (τ0 ) := { b
E
a = {b
an }∞
an ∈ C, |b
a|τ < ∞, 0 < τ < τ0 }
n=0 : b
E2 (τ0 ) := { a = {an }∞
n=0 : an ∈ C, kakτ < ∞, 0 < τ < τ0 } ,
(1)
where
|b
a|τ :=
kakτ :=
P∞
n=0
1/2
√
|b
an |2 exp( 2τ 2n + 1 )
and
1/2
√
√
2
, In = n!2n π.
n=0 |an | In exp( 2τ 2n + 1 )
(2)
P∞
b2 (τ0 ) and E2 (τ0 ) are nuclear Frèshet spaces, which are
Here we prove that the spaces E
isomorphic each other. Moreover, we show that the sequence b
a = {b
an }∞
n=0 belongs to the space
P
∞
b
b
b2 (τ0 ) if and only if the function fb(z) =
H
(z)
∈
H(τ
)
,
respectively
b
a
E
0
n=0 n n
P∞ the sequence
a = {an }∞
n=0 belongs to the space E2 (τ0 ) if and only if the function f (z) =
n=0 an Hn (z) ∈
H(τ0 ) .
Applications of Mathematics in Engineering and Economics, AIP Conf. Proc.1410, 221-229 (2011).
Published by American Institute of Phisics.
8. AN AUTOMORPHISM OF THE SPACE OF ANALYTIC FUNCTIONS, HAVING
EXPANSIONS IN HERMITE POLYNOMIALS
I.P. Dimitrov
Abstract
For each fixed τ0 > 0 we denote by S(τ0 ) = {z ∈ C : | Im z| < τ0 } a strip parallel to the
real axes. By H(τ0 ) we denote the Frèshet space of all analytic functions, defined in S(τ0 ) and
having representation there in series of Hermite polynomials. Let α be a complex number such
that −1/2 < Re α < 1/2. We consider the mapping
P (α) : H(τ0 ) −→ H(τ0 ),
(3)
defined by the equality
f (z) = P (α) (F )(z) =
1
Γ(α + 1/2)
1
Z
(1 − t2 )α−1/2 F (zt) dt, F ∈ H(τ0 ),
(4)
0
where Γ(.) is the Gamma-function.
We prove that the operator P (α) is an automorphism of the space H(τ0 ).
Applications of Mathematics in Engineering and Economics, AIP Conf. Proc.1497, 304-3011 (2012).
Published by American Institute of Phisics.