Academic Open Internet Journal |
Volume 16, 2005 |

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**Riesz basicity of the
system of exponents with degenerated coefficients.**

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**Sadiq G.Veliyev**

Nakhcivan Teachers Institute

AZ 7012,Nakhchivan,the Azerbaiyan Republic

E-mail;nmi@nakhcivan.az

System of exponents of the form

_{}

where _{}are complex-valued
functions, _{} is
degenerated in some points of the segment _{}, is considered in the paper. Necessary
and sufficient condition of Riesz basicity for this system in _{} is obtained for
definite conditions.

Bibl. 4 names.

Consider the following system of exponents with
degenerated coefficients _{}:

_{} (1)

where _{}-are complex-valued
functions on the segment _{}; _{} has the
presentation

_{} (2)

_{} are
the sets of real numbers. Earlier we studied the basicity in _{} of the system (1) for
definite conditions on the functions _{} and _{}. In offered paper we consider the problem
on Riesz basicity of the system (1) in _{}. Earlier the same problem relatively the
system of exponents _{} was
investigated in the work of V.F.Gaposhkin [1] and K.I.Babenko [2].

We do the following suppositions relatively the
functions _{} and
_{}.

1)
_{} are piecewise-Helder functions on the
segment _{}, _{} is the set of discontinuity
points of the function _{} on _{}, and moreover

_{};

2)
_{} are measurable functions on _{}, and satisfy the condition

_{}

Denote by _{} the
jumps of the function _{} at the points _{}, i.e. _{}.

The following theorem takes place.

Theorem. Let the conditions 1), 2) take place and the following inequalities

_{}

where _{} are
fulfiled. Then the system (1) forms Riesz basis in _{} if and only if _{}.

Proof. Consider the system

_{}

Then, as it follows from the results of the work [3],
system forms Riesz basis in _{} if the conditions of theorem are fulfiled.
Denote by _{} biorthogonal
system to system (3). It is quite obvious that biorthogonal to (1) system _{} has the form: _{}. Again, according to
the results of the work [3], it takes place: _{} in sufficiently small neighbourhoods of
the points _{}.
First of all we suppose that _{}. Then from the previous arguments and
from the presentation (2) for _{} it follows that _{}, such that _{}. From Riesz basicity of system
(3) in _{} it
follows the divergence of series _{}, where _{}are biorthogonal coefficients of the
function _{} on
system (3). Consequently, in this case the system (1) doesn’t form Bessel basis
and, also Riesz basis in _{}.

And now let the series _{} converges for some sequence of numbers _{}. As system (3) forms Riesz
basis in _{},
then _{} for
which _{} are biorthogonal
coefficients, i.e. _{} where

_{}.
If _{}, then it
is clear that _{} and
according to the conditions of theorem _{}. As a result we obtain, that for _{} the system (1) forms
Hilbert basis in _{}.

We consider the case, if _{} for some _{}. Then _{} such that _{}. Let _{} are biorthogonal coefficients of the
function _{} on
system (3). From Riesz basicity of (3) in _{} we have: _{}. It is quite obvious that this sequence
is biorthogonal coefficient of the functions _{} on system (1). From here it follows that
the system (1) is not Hilbert basis, as _{}. Really, we can take _{} such, that _{} for some _{}, where _{}.

According to the results of the paper [4] the conjugate
system for Hilbert basis is Bessel basis, consequently, _{} is complete in _{} and then it is
complete in _{}.
As a result for such sequence _{} from _{} there is no the function _{}, such that _{} are biorthogonal
coefficients of this function on system (1).

Let _{}. We take _{}. It is clear that _{}. From Riesz basicity of system
(3) it follows that _{} where
_{} are biorthogonal
coefficients of the function _{} on system (3), or function _{} on system (1).
Consequently, in this case the system (1) forms Bessel basis in _{}.

Theorem is proved.

The following corollary directly follows from the proof of this theorem.

Corollary. Let all conditions of theorem are fulfiled.
Then the system (1) forms Hilbert (Bessel) basis in _{} if and only if _{}.

Author is very greateful to prof. S.S.Mirzoyev for the attention to the work.

References:

1. Gaposhkin V.F. // Mat. sbornik, 1958, v.46 (88), ¹3.

2. Babenko K.I.// DAN SSSR, 1948, v.62, ¹2, p.157-160.

3. Bilalov B.T.// Different. Equations, 1990, v.26, ¹1, p.10-16.

4. Bari N.K.// Scientific notices of MSU, 1951, v.4, issue 148.

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