On Properties of the Legendre Differential Expression

This paper surveys the known properties and presents many new results for the Legendre differential expression and the associated linear differential operators in its right-definite and left-definite Hilbert function spaces. The main tool for obtaining the best-possible form of these results is an integral operator inequality due to Chisholm, Everitt, and Littlejohn. Many of the results follow from an unpublished manuscript of Everitt and Marić and a forthcoming paper of Arvesú, Littlejohn, and Marcellán. A recent paper on left-definite differential operators by Vonhoff provides for an interesting comparison with the properties and results in this paper.

[1]  A. Krall,et al.  THE LEGENDRE POLYNOMIALS UNDER A LEFT DEFINITE ENERGY NORM , 1993 .

[2]  W. N. Everitt,et al.  An integral operator inequality with applications. , 1999 .

[3]  E. C. Titchmarsh,et al.  Reviews , 1947, The Mathematical Gazette.

[4]  L. Littlejohn,et al.  Properties of the operator domains of the fourth-order Legendre-type differential expressions , 1994 .

[5]  Å. Pleijel,et al.  On Legendre's Polynomials , 1976 .

[6]  Åke Pleijel.  On the boundary condition for the Legendre polynomials , 1976 .

[7]  Francisco Marcellán,et al.  On the Right-Definite and Left-Definite Spectral Theory of the Legendre Polynomials , 2002 .

[8]  C. Bennewitz.  A generalisation of Niessen's limit-circle criterion , 1977 .

[9]  Lance L. Littlejohn,et al.  A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations , 2002 .

[10]  I. M. Glazman,et al.  Theory of linear operators in Hilbert space , 1961 .

[11]  Hans G. Kaper,et al.  Characterizations of the Friedrichs extensions of singular Strum Liouville expressions , 1986 .

[12]  Rolf Vonhoff,et al.  A Left-Definite Study of Legendre’s Differential Equation and of the Fourth-Order Legendre Type Differential Equation , 2000 .

[13]  P. Goldbart,et al.  Linear differential operators , 1967 .

[14]  Edwin Hewitt,et al.  Eigenfunction expansions associated with second-order differential equations, Part II , 1959 .

[15]  Wn Everitt,et al.  Legendre polynomials and singular differential operators , 1980 .