Pointwise convergence of bounded cascade sequences

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For an initial function , a cascade sequence is constructed by the iteration where is defined by In this paper, under a condition that the sequence is bounded in , we prove that the following three statements are equivalent: (i) converges . (ii) For , there exist a positive constant and a constant such that (iii) For some converges in . An example is presented to illustrate our result.

     

A multiplicity theorem for the Neumann problem

Here is a particular case of the main result of this paper: Let be a bounded domain, with a boundary of class , and let be two continuous functions, , with 0$">, , with n$">. If

and if the set of all global minima of the function has at least connected components, then, for each 0$"> small enough, the Neumann problem

admits at least strong solutions in .

     

Compact operators on the Bergman space of multiply-connected domains

If is a smoothly bounded multiply-connected domain in the complex plane and where we show that is compact if and only if its Berezin transform vanishes at the boundary.

     

Sobolev spaces and the Cayley transform

The generalised Cayley transform  from an Iwasawa -group into the corresponding real unit sphere  induces isomorphisms between suitable Sobolev spaces and . We study the differential of  , and we obtain a criterion for a function to be in  .

     

A note concerning the index of the shift

Let be a finite, positive Borel measure with support in such that - the closure of the polynomials in - is irreducible and each point in is a bounded point evaluation for . We show that if 0$">and there is a nontrivial subarc of such that

-\infty,\end{displaymath}">

then for each nontrivial closed invariant subspace for the shift on .

     

Sufficient conditions for one domain to contain another in a space of constant curvature

As an application of the analogue of C-S. Chen's kinematic formula in the 3-dimensional space of constant curvature , that is, Euclidean space , -sphere , hyperbolic space (, respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space , or a -sphere or a hyperbolic space .

     

Pseudo-differential operators and maximal regularity results for non-autonomous parabolic equations

In this paper, we show that a pseudo-differential operator associated to a symbol ( being a Hilbert space) which admits a holomorphic extension to a suitable sector of acts as a bounded operator on . By showing that maximal -regularity for the non-autonomous parabolic equation is independent of , we obtain as a consequence a maximal -regularity result for solutions of the above equation.

     

Existence of positive solutions for a semilinear elliptic problem with critical Sobolev and Hardy terms

Let , let and let be a bounded domain with a smooth boundary . Our purpose in this paper is to consider the existence of solutions of the problem:

where

     

A lower bound for sums of eigenvalues of the Laplacian

Let be the th Dirichlet eigenvalue of a bounded domain in . According to Weyl's asymptotic formula we have

The optimal in view of this asymptotic relation lower estimate for the sums has been proven by P.Li and S.T.Yau (Comm. Math. Phys. 88 (1983), 309-318). Here we will improve this estimate by adding to its right-hand side a term of the order of that depends on the ratio of the volume to the moment of inertia of .

     

On the minimum of several random variables

For a given sequence of real numbers , we denote the th smallest one by . Let be a class of random variables satisfying certain distribution conditions (the class contains Gaussian random variables). We show that there exist two absolute positive constants and such that for every sequence of real numbers and every , one has

-

where are independent random variables from the class . Moreover, if , then the left-hand side estimate does not require independence of the 's. We provide similar estimates for the moments of as well.

     

On the index and spectrum of differential operators on

If is an system of differential operators on having continuous coefficients with vanishing oscillation at infinity, the Cordes-Illner theory ensures that is Fredholm from to for all or no value We prove that both the index (when defined) and the spectrum of are independent of

     

The Wiener transform on the Besicovitch spaces

In his fundamental research on generalized harmonic analysis, Wiener proved that the integrated Fourier transform defined by is an isometry from a nonlinear space of functions of bounded average quadratic power into a nonlinear space of functions of bounded quadratic variation. We consider this Wiener transform on the larger, linear, Besicovitch spaces defined by the norm . We prove that maps continuously into the homogeneous Besov space for and , and is a topological isomorphism when .

     

The structure of quantum spheres

We show that the C*-algebra of a quantum sphere , 1$">, consists of continuous fields of operators in a C*-algebra , which contains the algebra of compact operators with , such that is a constant function of , where is the quotient map and is the unit circle.

     

On quasi-complete intersections of codimension

In this paper, we prove that if , , is a locally complete intersection of pure codimension and defined scheme-theoretically by three hypersurfaces of degrees , then for using liaison theory and the Arapura vanishing theorem for singular varieties. As a corollary, a smooth threefold is projectively normal if is defined by three quintic hypersurfaces.

     

An elementary proof of the law of quadratic reciprocity over function fields

Let and be relatively prime monic irreducible polynomials in (). In this paper, we give an elementary proof for the following law of quadratic reciprocity in :

where is the Legendre symbol.

     

``Lebesgue measure' on , II

Let be the set of real numbers, and define . We construct a complete measure space where the -algebra contains the Borel subsets of , and is a translation-invariant measure such that for any measurable rectangle , if , then , where is Lebesgue measure on . The measure is not -finite. We prove three Fubini theorems, namely, the Fubini theorem, the mean Fubini-Jensen theorem, and the pointwise Fubini-Jensen theorem. Finally, as an application of the measure , we construct, via selfadjoint operators on , a ``Schrödinger model' of the canonical commutation relations: , , .

     

A topological property of integrable systems

If we are given real-valued smooth functions on which are in involution, then, under some mild hypotheses, the subset of where these functions are linearly independent is not simply connected.

     

Weighted inequalities for iterated convolutions

Given a fixed exponent , , and suitable nonnegative weight functions , , an optimal associated weight function is constructed for which the iterated convolution product satisfies

for all complex valued measurable functions with . Here and for each , . Analogous results are given when is replaced by and also when the convolution on is taken instead to be . The extremal functions are also discussed.

     

Homogeneous polynomials on strictly convex domains

We consider a circular, bounded, strictly convex domain with boundary of class . For any compact subset of we construct a sequence of homogeneous polynomials on which are big at each point of . As an application for any circular subset of type we construct a holomorphic function which is square integrable on and such that where denotes unit disc in .

     

A characterization of Möbius transformations

Let be an integer and let be a domain of . Let be an injective mapping which takes hyperspheres whose interior is contained in to hyperspheres in . Then is the restriction of a Möbius transformation.