## Real Analysis Exchange

### Integration of Fourier-Legendre series

E. Russell Love

#### Abstract

It is well known than an ordinary (trigonometric) Fourier series “can” be integrated term by term; that is, whether the Fourier series of an integrable function $f$ is convergent or not, the series obtained by term by term integration of it is convergent to the corresponding integral of $f$. There is also a more general theorem in which $f$ and its Fourier series are first multiplied by a function of bounded variation before integrating. This paper aims to obtain similar theorems for the Fourier-Legendre series discussed by Love and Hunter in Proc. London Math. Soc (3) 64 (1992) 579-601 and by Love in that journal (3) 69 (1994) 629-672. Most of the paper is occupied by lemmas which lead to establishing the dominated convergence of the partial sums of the Fourier-Legendre series under certain conditions. For several reasons one of these conditions is that the interval of integration must be a closed subinterval of $(-1,1)$, the interval on which the Legendre functions $P_\nu^\mu (x)$ are defined; further, I have only had success with integration over subintervals of $[- \frac{1}{2} \sqrt{3}, \frac{1}{2}\sqrt{3}]$.

#### Article information

Source
Real Anal. Exchange, Volume 22, Number 1 (1996), 184-200.

Dates
First available in Project Euclid: 1 June 2012

https://projecteuclid.org/euclid.rae/1338515214

Mathematical Reviews number (MathSciNet)
MR1433607

Zentralblatt MATH identifier
0879.42017

Subjects
Primary: 26A33: Fractional derivatives and integrals

#### Citation

Love, E. Russell. Integration of Fourier-Legendre series. Real Anal. Exchange 22 (1996), no. 1, 184--200. https://projecteuclid.org/euclid.rae/1338515214

#### References

• A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, vol. 1, Bateman Manuscript Project, Mc Graw-Hill, 1953, ch. III.
• E. R. Love and M. N. Hunter, Expansions in series of Legendre functions, Proc. London Math. Soc. (3) 64 (1992), 579–601.
• E. C. Titchmarsh, Theory of Functions, 2nd ed., Oxford University Press, 1939, ch. XIII.