The notion of conjugate functions associated with ultraspherical expansions and their continuous analogues, the Hankel transforms, was introduced by Muckenhoupt and Stein [14], to which we refer the reader for general background and an excellent discussion of the motivation underlying these notions. The operation of passing from a given function to its conjugate is in many ways analogous to the passage from a function to its Hilbert transform, indeed, Muckenhoupt and Stein proved, among other things, that these operations acting on appropriate weighted Lebesgue spaces, Lp(𝝁), satisfy inequalities of M. Riesz type analogous to those satisfied by the Hilbert transform on the usual Lebesgue spaces, Lv( — ∞, ∞).

We shall give a simple characterization of a lattice homomorphism from a linear lattice E to a linear lattice F. This paper is motivated by the following two theorems in Kaplan [2] :

If ϕ is a lattice homomorphism, then ϕt(Fb) is an ideal in Eb.

(2) If ϕ is a lattice homomorphism, then ϕtt is a lattice homomorphism from ϕbb into ϕbb.

The main theorem is stated and proved in section 3. In section 1, we shall give notations and in section 2, we shall prove a main lemma. For details, we refer to Vulikh [3].

It is well known now that certain spherical methods in k (≧2) dimensions are rather poor for reconstructing a function from its Fourier transform. Consider a function f in L1(Rk), k ≧ 2,

and

where both integrals are integrals in Rk, the first over the whole space the second over the ball of radius R; x • y is the usual Euclidean inner product of x and y in Rk and \z\2 = z • z.

The classVk(p). We generalize the class Vk of analytic functions of bounded boundary rotation [8] by allowing critical points in the unit disc U.

Definition. Let f(z) = aqzq + . . . (q 1) be analytic in U. Then f(z) belongs to the class Vk(p) if for r sufficiently close to 1,

and

We note that (1.1) implies that / has precisely p — 1 critical points in U.

From the time that the basic existence and regularity problems for partial differential equations have been solved many interesting new variational and control problems could be studied. In general a differential equation or boundary value problem is used to define a class of admissible functions, and then the problem is that of finding the extrema of a given functional defined on that class of functions.

Following Herstein [2], we will call a ring R with identity von Neumann finite (vNf) provided that xy = 1 implies yx — WnR. Kaplansky [4] showed that group algebras over fields of characteristic zero are vNf rings, and further, that full matrix rings over such rings are also vNf. Herstein [2] has posed the problem for group algebras over fields of arbitrary characteristic. If group algebras over fields are always vNf, then it is easily seen that group algebras over commutative rings are always vNf. What conditions on the underlying ring of scalars would force the vNf property for all group rings over it?