Regularity, Local and Microlocal Analysis in Theories of Generalized Functions

We introduce a general context involving a presheaf and a subpresheaf ℬ of  . We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the ℬ-local analysis of sections of  . But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a “frequential microlocal analysis” and into a “microlocal asymptotic analysis”. The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. The microlocal asymptotic analysis is a new spectral study of singularities. It can inherit from the algebraic structure of ℬ some good properties with respect to nonlinear operations.      

Hölder Regularity of μ-Similar Functions

Given a positive measure μ, d contractions on [0,1] and a function g on ℝ, we are interested in function series F that we call “μ-similar functions” associated with μ, g and the contractions. These series F are defined as infinite sums of rescaled and translated copies of the function g, the dilation and translations depending on the choice of the contractions. The class of μ-similar functions F intersects the classes of self-similar and quasi-self-similar functions, but the heterogeneity we introduce in the location of the copies of g make the class much larger. We study the convergence and the global and local regularity properties of the μ-similar functions. We also try to relate the multifractal properties of μ to those of F and to develop a multifractal formalism (based on oscillation methods) associated with F.     

Uniform Approximation of Functions Continuous on a Compact Subset of ℂ and Analytic in Its Interior by Functions Bianalytic in Its Neighborhoods

We prove that an arbitrary function continuous on a compact set X and holomorphic in the interior of X can be approximated by functions bianalytic in neighborhoods of X with arbitrary accuracy.     

Statistical Convergence in Probability for a Sequence of Random Functions

In this paper, we introduce a new type of convergence for a sequence of random functions, namely, statistical convergence in probability, which is a natural generalization of convergence in probability. In this approach, we allow such a sequence to go far away from the limit point infinitely many times by presenting random deviations, provided that these deviations are negligible in some sense of measure. In this context, the set of values of a random function is considered as a probabilistic metric (PM) space of random variables, and some basic results are obtained using the tools of PM spaces.     

Regularity of the Laws of Shot Noise Series and of Related Processes

We investigate the regularity of shot noise series and of Poisson integrals. We give conditions for the absolute continuity of their law with respect to the Lebesgue measure and for their continuity in total-variation norm. In particular, the case of truncated series in addressed. Our method relies on a disintegration of the probability space based on a mere conditioning by the first jumps of the underlying Poisson process.     

The Strong Approximation of Functions by Fourier-Vilenkin Series in Uniform and H?lder Metrics

We will study the strong approximation by Fourier-Vilenkin series using matrices with some general monotone condition. The strong Vallee-Poussin, which means of Fourier-Vilenkin series are also investigated.     

Dimension Functions of Rationally Dilated GMRAs and Wavelets

In this paper we study properties of generalized multiresolution analyses (GMRAs) and wavelets associated with rational dilations. We characterize the class of GMRAs associated with rationally dilated wavelets extending the result of Baggett, Medina, and Merrill. As a consequence, we introduce and derive the properties of the dimension function of rationally dilated wavelets. In particular, we show that any mildly regular wavelet must necessarily come from an MRA (possibly of higher multiplicity) extending Auscher’s result from the setting of integer dilations to that of rational dilations. We also characterize all 3 interval wavelet sets for all positive dilation factors. Finally, we give an example of a rationally dilated wavelet dimension function for which the conventional algorithm for constructing integer dilated wavelet sets fails.     

Polynomial Approximation of Functions on a Quasi-Smooth Arc with Hermitian Interpolation

We construct polynomial approximations for continuous functions f defined on a quasi-smooth (in the sense of Lavrentiev) arc L in the complex plane which simultaneously interpolate f and its derivatives at given points of L.     

Uniform Convergence of Discrete Curvatures from Nets of Curvature Lines

We study discrete curvatures computed from nets of curvature lines on a given smooth surface and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.     

Uniform Spacing of Zeros of Orthogonal Polynomials

It is shown that for doubling weights, the zeros of the associated orthogonal polynomials are uniformly spaced in the sense that if cos θ _m,k , θ _m,k ∈[0,π] are the zeros of the m-th orthogonal polynomial associated with w, then θ _m,k −θ _m,k+1∼1/m. It is also shown that for doubling weights, neighboring Cotes numbers are of the same order. Finally, it is proved that these two properties are actually equivalent to the doubling property of the weight function.     

Local Uniformization and Free Boundary Regularity of Minimal Singular Surfaces

We continue the study of minimal singular surfaces obtained by a minimization of a weighted energy functional in the spirit of J. Douglas’s approach to the Plateau problem. Modeling soap films spanning wire frames, a singular surface is the union of three disk-type surfaces meeting along a curve which we call the free boundary. We obtain an a priori regularity result concerning the real analyticity of the free boundary curve. Using the free boundary regularity of the harmonic map, we construct local harmonic isothermal coordinates for the minimal singular surface in a neighborhood of a point on the free boundary. Applications of the local uniformization are discussed in relation to H. Lewy’s real analytic extension of minimal surfaces.     

Extensions of Zeta Functions – Examples and a Study of the Double Sine Functions

We discuss zeta extensions in the sense of Kurokawa and Wakayama, Proc. Japan Acad. 2002, for constructing new zeta functions from a given zeta function. This notion appeared when we introduced higher zeta functions such as higher Riemann zeta functions in Kurokawa et al., Kyushu Univ. Preprint, 2003, and a higher Selberg zeta functions in Kurokawa and Wakayama, Comm. Math. Phys., 2004. In this article, we first recall some explicit examples of such zeta extensions and give a conjecture about functional equations satisfied by higher zeta functions. We devote the second part to making a detailed study of the double sine functions which are treated in a framework of the zeta extensions.Mathematics Subject Classification (2000) 11M36.Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340012, and by Grant-in-Aid for Exploratory Research No. 13874004. This is based on the talk at The 2002 Twente Conference on Lie Groups 16–18 Dec. University of Twente, Enschede, The Netherlands.     

Convexity and Smoothness of Scale Functions and de Finetti’s Control Problem

We continue the recent work of Avram et al. (Ann. Appl. Probab. 17:156–180, 2007) and Loeffen (Ann. Appl. Probab., 2007) by showing that whenever the Lévy measure of a spectrally negative Lévy process has a density which is log-convex then the solution of the associated actuarial control problem of de Finetti is solved by a barrier strategy. Moreover, the level of the barrier can be identified in terms of the scale function of the underlying Lévy process. Our method appeals directly to very recent developments in the theory of potential analysis of subordinators and their application to convexity and smoothness properties of the relevant scale functions.     

Regularity of Tensor Product Approximations to?Square Integrable Functions

We investigate first-order conditions for canonical and optimal subspace (Tucker format) tensor product approximations to square integrable functions. They reveal that the best approximation and all of its factors have the same smoothness as the approximated function itself. This is not obvious, since the approximation is performed in L ².     

Approximation of the Classes B p,θ Ω of Periodic Functions of Many Variables in Uniform Metric

We obtain order estimates for the best M-term trigonometric approximations and approximations by Fourier sums for the classes B _p, of periodic functions of many variables in the uniform metric.     

Minimization of Isotonic Functions Composed of Fractions

In this paper, we introduce a class of minimization problems whose objective function is the composite of an isotonic function and finitely many ratios. Examples of an isotonic function include the max-operator, summation, and many others, so it implies a much wider class than the classical fractional programming containing the minimax fractional program as well as the sum-of-ratios problem. Our intention is to develop a generic “Dinkelbach-like” algorithm suitable for all fractional programs of this type. Such an attempt has never been successful before, including an early effort for the sum-of-ratios problem. The difficulty is now overcome by extending the cutting plane method of Barros and Frenk (in J. Optim. Theory Appl. 87:103–120, 1995). Based on different isotonic operators, various cuts can be created respectively to either render a Dinkelbach-like approach for the sum-of-ratios problem or recover the classical Dinkelbach-type algorithm for the min-max fractional programming.     

A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations

In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L^\frac21-r( 0,T;[(X)\dot]_r( \mathbbR³) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.     

Local Geometry of Levi-Forms Associated with the Existence of Complex Submanifolds and the Minimality of Generic CR Manifolds

Let M be a generic CR manifold in \BbbC^m+d\Bbb{C}^{m+d} of codimension d, locally given as the common zero set of real-valued functions r ¹,…,r ^d . Given an integer δ=1,…,d, we find a necessary and sufficient condition for M to contain a real submanifold of codimension δ with the same CR structure. We also find a necessary and sufficient condition and several sufficient conditions for M to admit a complex submanifold of complex dimension n, for any n=1,…,m. We use the method of prolongation of an exterior differential system. The conditions are systems of partial differential equations on r ¹,…,r ^d of third order.     

Regularity properties of a class of hybrid methods

IntroductionIt is important that the discrete dynamical system given by a numerical method appliedto a continuous dynamical system can have the same dynamical properties as the underlyingcontinuous system. Recently, many authors[1--71 have investigated the conditions under whichspurious solutions are not introduced by time discretization, and many interesting results aboutRunge-Kutta methods, linear multistep methods and general linear methods applied to dynamical systems of ordinary different…