A HILBERT SPACE ON LEFT-DEFINITE
STURM-LIOUVILLE DIFFERENCE EQUATIONS
STURM-LIOUVILLE DIFFERENCE EQUATIONS
Abstract. We investigate the discrete Sturm-Liouville problems
where
is strictly positive, 
is nonnegative and 
may change sign.
If is positive, the 
-space weighted by is a Hilbert space and it is customary
to use that space for setting the problem. In the present situation the right-hand-side of the equation
does not give rise to a positive-definite quadratic
form and we use instead the left-hand-side to definite such a form. We prove in this paper that this form determines a
Hilbert space (such problems are called left-definite).
where
is strictly positive, is nonnegative and may change sign.
If is positive, the -space weighted by is a Hilbert space and it is customary
to use that space for setting the problem. In the present situation the right-hand-side of the equation
does not give rise to a positive-definite quadratic
form and we use instead the left-hand-side to definite such a form. We prove in this paper that this form determines a
Hilbert space (such problems are called left-definite).
AMS Subject classification: 39A70
Keywords and phrases: left-definite problems, difference equations, Hilbert spaces
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DOI: 10.12732/ijam.v27i2.6
Volume: 27
Issue: 2
Year: 2014
