On uniform Lebesgue constants of local exponential splines with equidistant knots

For a linear differential operator L _r of arbitrary order r with constant coefficients and real pairwise different roots of the characteristic polynomial, we study Lebesgue constants (the norms of linear operators from C to C) of local exponential splines corresponding to this operator with a uniform arrangement of knots; such splines were constructed by the authors in earlier papers. In particular, for the third-order operator L ₃ = D(D ² ? β ²) (β > 0), we find the exact values of Lebesgue constants for two types of local splines and compare these values with Lebesgue constants of exponential interpolation splines.     

Finite semigroups with slowly growing pn-sequences

The p _n -sequence of a semigroup S is said to be polynomially bounded, if there exist a positive constant c and a positive integer r such that the inequality p _n (S) ≤cn ^r holds for all n≥ 1. In this paper, we fully describe all finite semigroups having polynomially bounded p _n -sequences. First we give a characterization in terms of identities satisfied by these semigroups. In the sequel, this result will allow an insight into the structure of such semigroups. We are going to deal with certain ideals and the construction of ideal extension of semigroups. In addition, we supply an effective procedure for deciding whether a finite semigroup has polynomially bounded p _n -sequence and give some examples. Received March 5, 1999; accepted in final form November 1, 1999.     

Integral representation of local functionals

We study conditions under which a functional F(u, B) admits an integral representation of the form $$F(u, B) = \int\limits_B {f(x, D^k u(x)) dx.} $$      

Integral representation of linear operators

In this paper we obtain necessary and sufficient conditions in order that a linear operator, acting in spaces of measurable functions, should admit an integral representation. We give here the fundamental results. Let (T_i, _i) (i=1,2) be spaces of finite measure, and let (T,) be the product of these spaces. Let E be an ideal in the space S(T₁, ₁) of measurable functions (i.e., from |e₁||e₂|, e₁ S (T₁, ₁), e₂E it follows that e₁E). THEOREM 2. Let U be a linear operator from E into S(T₂, ₂). The following statements are equivalent: 1) there exists a-measurable kernel K(t,S) such that (Ue)(S)=K(t,S) e(t)d(t) (eE); 2) if 0e_nE (n=1,2,...) and e_n0 in measure, then (Ue_n)(S) 0 ₂ a.e. THEOREM 3. Assume that the function (t,S) is such that for any eE and for s a.e., the ₂-measurable function Y(S)=(t,S)e(t)d ₁(t) is defined. Then there exists a-measurable function K(t,S) such that for any eE we have (t,S)e(t)d ₁(t)=K(t,S)e(t)d ₁(t) ₁a.e.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 5–14, 1974.     

The growth irregularity of slowly growing entire functions

We show that entire transcendental functions f satisfying $\log M(r,f) = o(\log ^2 r),r \to \infty (M(r,f): = \mathop {\max }\limits_{|z| = r} |f(z)|)$ necessarily have growth irregularity, which increases as the growth diminishes. In particular, if 1 < p < 2, then the asymptotics $\log M(r,f) = \log ^p r + o(\log ^{2 - p} r),r \to \infty ,$ , is impossible. It becomes possible if “o” is replaced by “O.”     

Integral representation with weights I

 We describe a new approach to representation formulas for holomorphic functions, including the Cauchy-Fantappie-Leray formula, and provide a general method to generate weighted formulas. Received: 31 October 2001 / Published online: 28 March 2003 Mathematics Subject Classification (1991): 32 A 25. The author was partially supported by the Swedish Research Council.     

Integral representation of random set-valued measures

Abstract

We consider random set-valued measures with values in a separable Banach space. We prove two integral representation theorems using measurable multifunctions and set-valued integrals. The first theorem is valid for all separable Banach spaces, while the second holds for reflexive separable Banach spaces.