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Abstract: A ladder structure of operators is presented for the associated Legendre polynomials and the sphericas harmonics. In both cases these operators belong to the irreducible representation of the Lie algebra with quadratic Casimir equals to . As both are also bases of square-integrable functions, the universal enveloping algebra of is thus shown to be homomorphic to the space of linear operators acting on the functions defined on and on the sphere , respectively. The presence of a ladder structure is suggested to be the general condition to obtain a Lie algebra representation defining in this way the “algebraic special functions” that are proposed to be the connection between Lie algebras and square-integrable functions so that the space of linear operators on the functions is homomorphic to the universal enveloping algebra. The passage to the group, by means of the exponential map, shows that the associated Legendre polynomials and the spherical harmonics support the corresponding unitary irreducible representation of the group . [Copyright &y& Elsevier]

We consider the boundary-value problem on a finite interval for a class of second-order operator-differential equations with a linear operator in one of its boundary conditions. We obtain sufficient conditions for the regular solvability of the boundary-value problem under consideration; these conditions are expressed only in terms of its operator coefficients. [ABSTRACT FROM AUTHOR]

To a function $f\backslash in\; L\_2[-\backslash pi,\backslash pi]$ and a compact set $Q\backslash subset[-\backslash pi,\backslash pi]$ we assign the supremum $\backslash omega(f,Q)\; =\backslash sup\_\{t\backslash in\; Q\}\backslash |f(\backslash ,\backslash cdot\backslash ,+\; t)-f(\backslash ,\backslash cdot\backslash ,)\backslash |\_\{L\_2[-\backslash pi,\backslash pi]\}$, which is an analog of the modulus of continuity. We denote by $K(n,Q)$ the least constant in Jackson's inequality between the best approximation of the function $f$ by trigonometric polynomials of degree $n-\; \backslash nomathbreak\; 1$ in the space $L\_2[-\backslash pi,\backslash pi]$ and the modulus of continuity $\backslash omega(f,Q)$. It follows from results due to Chernykh that $K(n,Q)\backslash ge1/\backslash sqrt2$ and $K(n,[0,\backslash pi/n])=1/\backslash sqrt2$. On the strength of a result of Yudin, we show that if the measure of the set $Q$ is less than $\backslash pi/n$, then $K(n,Q)>1/\backslash sqrt2$. [ABSTRACT FROM AUTHOR]