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\centerline{\bf ON EXPANSION OF LEBESGUE INTEGRABLE FUNCTIONS}
\centerline{\bf IN SERIES OF LEGENDRE FUNCTIONS}

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\centerline{\bf E.~R.~Love, and M.~N.~Hunter (Melbourne, Australia)}

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The Legendre functions considered are certain solutions $y =
P_{\nu}^{\mu}(x)$, on $- 1 < x < 1$ of the following equation
(see [1]):
\begin{equation}
\label{Legendre}
{d\over dx} \left((1 - x^2){dy\over dx}\right) + \left(\nu(\nu + 1)
- \frac{\mu^{2}}{1 - x^2}\right) y = 0.
\end{equation}

In our report we discuss the possibility for an integrable function
$f$ to be expanded in series of Legendre functions. The results
presented generalize those obtained in [2].

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{\bf Theorem 1.} If $(1 - t^2)^{-1\over 4} f(t)\in L(-1,1)$ and
$f$ satisfies the Dini condition at a certain $a\in (-1,1)$
(see e.g. [3]), $|{\rm Re}\, \mu| < {1\over 2}$,
$\nu$ is not a half of an odd integer, and
$$
a_n = (-1)^n \frac{\nu + n + {1\over 2}}{2\cos \nu\pi}
\int_{-1}^{1} f(t) P_{\nu + n}^{-\mu}(-t) dt,
$$
then
$$
f(x) = \sum_{-\infty}^{+\infty} a_n P_{\nu + n}^{\mu}(x),
$$
where $P_{k}^{\mu}$ are determined in (\ref{Legendre}).

{\bf Sketch of the proof.} ...

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{\bf Aknowledgement.} The work is partially supported by NSF.

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{\centerline {\bf Refrences}}

1. {\it Erdelyi A., Magnus W., Oderhettinger F., and Tricomi F.~G.}
{\it Higher Transcendential Functions,} vol. {\bf 1}. New York:
McGraw-Hill (1953).

2. {\it Love E.~R., Hunter M.~N.} Expansions in series of Legendre
functions. {\it Proc. London Math. Soc.} {\bf 64} (3) (1992),
579-601.

3. {\it Love E.~R., Hunter M.~N.} Expansions in series of Legendre
functions. In: {\it Boundary Value Problems, Special Functions
and Fractional Calculus} (Eds. I.~V.~Gai\-shun et al.) Minsk: BSU (1996),
204-214.

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