The classical theorem of Müntz and Szász says that the span of

is dense in C[0,1] in the uniform norm if and only if . We prove that, if {λi} is lacunary, we can replace the underlying interval [0,1] by any set of positive measure. The key to the proof is the establishment of a bounded Remez-type inequality for lacunary Müntz systems. Namely if A ⊆ [0,1] and its Lebesgue measure µ(A) is at least ε > 0 then

where c depends only on ε and Λ (not on n and A) and where Λ:=infiλi+1/λi>1.

The inverse scattering problem we consider is to determine the surface impedance of a three-dimensional obstacle of known shape from a knowledge of the far-field patterns of the scattered fields corresponding to many incident time-harmonic plane acoustic waves. We solve this problem by using both the methods of Kirsch-Kress and Colton-Monk.

Let R be a left Noetherian ring with the ascending chain condition on right annihilators, let α be a ring monomorphism of R and δ an α-derivation of R. We prove that, if R is semiprime or α-prime, then R[X;α, δ] is semiprimitive (and left Goldie), and that J(R[X;α]) equals N(R)[X;α].

Let G be an affine Kac–Moody group over ℂ, and V∞ an integrable simple quotient of a Verma module for g. Let Gmin be the subgroup of G generated by the maximal algebraic torus T, and the real root subgroups.

It is shown that (the least positive imaginary root) gives a character δ∈Hom(G, ℂ*) such that the pointwise character χ∞ of V∞ may be defined on Gmin ∩ G>1.

For a measurable function f on the unit ball B in ℂn we define (M1f)(w), |w|<1, to be the mean modulus of f over a hyperbolic ball with center at w and of a fixed radius. The space , 0<p<∞, is defined by the requirement that M1f belongs to the Lebesgue space Lp. It is shown that the subspace of Lp spanned by holomorphic functions coincides with the corresponding subspace of . It is proved that if s>(n+1)(p−1−1), 0<p<1, then this subspace is complemented in by the projection whose reproducing kernel is . As corollaries we get an extension of the Forelli–Rudin projection theorem and we show that a holomorphic function f is Lp-integrable, 0<p<∞, over the unit ball B iff u = Ref is Lp-integrable over B. Finally, we sketch an alternative proof of the main result of this paper in the case 0<p<1.

In the present paper by means of the Schauder-Tychonoff principle sufficient conditions are obtained for Lp-equivalence of a linear and a nonlinear impulsive differential equations.

Precise asymptotic estimates for the eigenvalues of a uniformly right definite two parameter system of Sturm–Liouville problems are developed. The work extends recent results of B. P. Rynne.

This paper studies the asymptotic distribution of the multiparameter eigenvalues of a right definite multiparameter Sturm–Liouville eigenvalue problem. A uniform asymptotic analysis of the oscillation number of solutions of a single Sturm–Liouville type equation with potential depending on a general parameter is given; these results are then applied to the system of multiparameter Sturm–Liouville equations to give the asymptotic eigenvalue distribution for the system as a function of a “multi-index” oscillation number.

Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B(X) (K(X)). Let N(A) and R(A) denote, respectively, the null space and the range space of an element A of B(X). Set R(A∞)=∩nR(An) and k(A)=dim N(A)/(N(A)∩R(A∞)). Let σg(A) = ℂ\{λ∈ℂ:R(A−λ) is closed and k(A−λ)=0} denote the generalized (regular) spectrum of A. In this paper we study the subset σgb(A) of σg(A) defined by σgb(A) = ℂ\{λ∈ℂ:R(A−λ) is closed and k(A−λ)<∞}. Among other things, we prove that if f is a function analytic in a neighborhood of σ(A), then σgb(f(A)) = f(σgb(A)).

A correspondence is established between a class of coverings of an inverse semigroup S and a class of embeddings of S, generalising results of McAlister and Reilly on E-unitary covers of inverse semigroups.

Let S be a free semigroup (on any set of generators). When S is given the discrete topology, its Stone-Čech compactification has a natural semigroup structure. We give two results about elements p of finite order in βS. The first is that any continuous homomorphism of βS into any compact group must send p to the identity. The second shows that natural extensions, to elements of finite order, of relationships between idempotents and sequences with distinct finite sums, do not hold.

If a differential equation with meromorphic coefficients has a certain form where the growth of one of the coefficients dominates the growth of the other coefficients in a finite union of angles, then we show that this puts restrictions on the deficiencies of any meromorphic solution of the equation. We use the spread relation in the proofs. Examples are given which show that our results are sharp in several ways. Most of these examples are constructed from the quotients of solutions of w″ + G(z)w = 0 for certain polynomials G(z) and from meromorphic functions which are extremal for the spread relation.

The relationship between the prime ideals and the primal ideals of a Banach algebra is investigated. It is shown that the closure of the prime radical of a Banach algebra may be properly contained in the intersection of the closed primal ideals of the algebra.

An algorithm is given for determining presence or absence of injectively (at the fundamental group level) immersed tori (and constructing them, if present) in a branched cover of S3, branched over the figure eight knot, with all branching indices greater than 2. Such tori are important for understanding the topology of 3-manifolds in light of (for example) the Jaco-Shalen–Johannson torus decomposition theorem and the fact that the figure eight knot is universal, i.e., that all 3-manifolds are representable as branched covers of S3, branched over the figure eight knot.

The algorithm is principally geometric in its derivation and graph-theoretic in its operation. It is applied to two examples, one of which has an incompressible torus and the other of which is atoroidal.

Semigroup presentations have been studied over a long period, usually as a means of providing examples of semigroups. In 1967 B. H. Neumann introduced an enumeration method for finitely presented semigroups analogous to the Todd–Coxeter coset enumeration process for groups. A proof of Neumann's enumeration method was given by Jura in 1978.

In Section 3 of this paper we describe a machine implementation of a semigroup enumeration algorithm based on that of Neumann. In Section 2 we examine certain semigroup presentations, motivated by the fact that the corresponding group presentation has yielded interesting groups. The theorems, although proved algebraically, were suggested by the semigroup enumeration program.

Lazarsfeld proved a bound for the excess dimension of an intersection of irreducible and reduced schemes. Flenner and Vogel gave another approach for reduced, non-degenerate schemes which are connected in codimension one, using the intersection algorithm of Stückrad and Vogel and defining a new multiplicity k. Renschuch and Vogel considered a condition to ensure that there is no degeneration for more than two schemes. We define an integer which enables us to unify these methods. This allows us to generalize the result of Flenner and Vogel to non-reduced schemes by comparing the multiplicities j and k. Using this point of view we give applications to converses of Bézout's theorem; in particular we investigate the Cohen-Macaulay case.

The paper studies dense Q-subalgebras of Banach and C*-algebras. It proves that the domain D(δ) of a closed unbounded derivation δ of a Banach unital algebra A automatically contains the identity and is a Q-subalgebra of A, so that SpA(x) = SpD(δ)(x) for all x∈D(δ). The paper shows that every finite-dimensional semisimple representation of a Q-subalgebra is continuous. It also shows that if π is an injective *-homomorphism of a dense locally normal Q*-subalgebra B of a C*-algebra, then ‖x‖≦‖π(x)‖ for all x∈B. The paper studies the link between closed ideals of a Banach algebra A and of its dense subalgebra B. In particular, if A is a C*-algebra and B is a locally normal *-subalgebra of A, then I→I∩B is a one-to-one mapping of the set of all closed two-sided ideals in A onto the set of all closed two-sided ideals in B and .

We investigate linear independence of integer translates of a finite number of compactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some lp space (1 ≦ p ≦ ∞) and we are interested in bounding their lp-norms in terms of the Lp-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

Let Ω be an open connected subset of the unit disc U, let E = U\Ω and let {Ωk} be a Whitney decomposition of U. If z(Q) is the centre of the “square” Q, if T is the unit circle and t = dist.(Q, T), we consider

where Ek = E ∩ Qk and c(Ek) is the capacity of Ek. We prove that the set E is minimally thin at τ ∈ T in U if and only if W(τ)< ∞. We study functions of type W and discuss the relation between certain results of Naim on minimal thinness [15], a minimum principle of Beurling [3], related results due to Dahlberg [7] and Sjögren [16] and recent work of Hayman-Lyons [15] (cf. also Bonsall [4]) and Volberg [19]. For simplicity, we discuss our problems in the unit disc U in the plane. However, the same techniques work for analogous problems in higher dimensions and in more complicated regions.

In this paper, we investigate the complex oscillation theory of

where A, F≢0 are entire functions, and obtain general estimates of the exponent of convergence of the zero-sequence and of the order of growth of solutions for the above equation.