Approximation of classes of smooth functions of several variables

The paper deals with imbedding theorems and harmonic approximation for classes of periodic functions of several variables. Necessary and sufficient conditions for imbedding are found for various classes of smooth functions. The paper contains asymptotic estimates for fractional order derivatives of multidimensional Dirichlet and Favard kernels, for Fourier best approximation and for the Kolmogorov diameter of smooth function classes.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 10, pp. 207–226, 1984.     

Uniqueness of Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ d

Summary. Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ ^d are studied. Introducing a linear operator L ₀ defined on a space of smooth local functions, we show the uniqueness of Dirichlet forms associated with self adjoint Markovian extensions of L ₀. We also discuss the ergodicity of the reversible process associated with the Dirichlet form. Received: 18 July 1996/In revised form: 13 February 1997     

Smooth numbers and the norms of arithmetic Dirichlet convolutions

In this paper we consider the problem of exactly evaluating the p-norm of a linear operator linked with arithmetic Dirichlet convolutions. We prove that a simply derived upper bound for this norm is actually attained for several different classes of arithmetic functions including completely multiplicative functions, but not for certain multiplicative functions. Our proof depends fundamentally on the asymptotic distribution properties of smooth numbers.     

Generalized Harmonic Functions and the Dewetting of Thin Films

This paper describes the solvability of Dirichlet problems for Laplace's equation when the boundary data is not smooth enough for the existence of a weak solution in H¹Ω. Scales of spaces of harmonic functions and of boundary traces are defined and the solutions are characterized as limits of classical harmonic functions in special norms. The generalized harmonic functions, and their norms, are defined using series expansions involving harmonic Steklov eigenfunctions on the domain. It is shown that the usual trace operator has a continuous extension to an isometric isomorphism of specific spaces. This provides a characterization of the generalized solutions of harmonic Dirichlet problems. Numerical simulations of a model problem are described. This problem is related to the dewetting of thin films and the associated phenomenology is described.     

Optimal estimates from below for Green functions of higher order elliptic operators with variable leading coefficients

Archiv der Mathematik - Estimates from above and below by the same positive prototype function for suitably modified Green functions in bounded smooth domains under Dirichlet boundary conditions...     

A spectral method for parabolic differential equations

We present a spectral method for parabolic partial differential equations with zero Dirichlet boundary conditions. The region Ω for the problem is assumed to be simply-connected and bounded, and its boundary is assumed to be a smooth surface. An error analysis is given, showing that spectral convergence is obtained for sufficiently smooth solution functions. Numerical examples are given in both ?² and ?³.     

Self-adjointness conditions for Dirichlet operators

We study operators associated with Dirichlet forms of quasi-invariant measures on a Hilbert space. We establish tests for the essential self-adjointness of such operators on smooth domains of definition in terms of the logarithmic derivative of the measure.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 284–289, February, 1990.     

Global existence for parabolic systems by Lyapunov functions

We present a result on the global existence of classical solutions for quasilinear parabolic systems with homogeneous Dirichlet boundary conditions in bounded domains with a smooth boundary. Our method relies on the use of Lyapunov functions.     

Dirichlet空间中的正交函数

主要考虑了Dirichlet空间中的正交函数.主要结论表明单位圆盘中的一个解析自映射是Dirichlet空间中的正交函数,当且仅当其所对应的计数函数是本质径向的;当且仅当其所对应的诱导测度是径向的.作为主要结论的一个应用,给出了单位圆盘上的单叶解析函数和闭单位圆盘上的解析函数是Dirichlet空间中正交函数的完全刻画.     

A weighted version of a rearrangement inequality

We show that if a positive absolutely continous measure causes a special relative isoperimetric inequality to hold, then Dirichlet-type integrals of sufficiently smooth real-valued functions decrease under an appropriate equimeasurable rearrangement.

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Sunto Si dimostra che se una misura, positiva e assolutamente continua, rende valida una certa disuguaglianza isoperimetrica relativa, allora integral del tipo di Dirichlet di funzioni sufficientemente regolari descrescono per effetto di un opportuno riordinamento equidistribuito.

     

A Monotonicity Property for Weighted Delaunay Triangulations

Given a triangulation of points in the plane and a function on the points, one may consider the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. In fact, the Dirichlet energy can be derived from a finite element approximation. S. Rippa showed that the Dirichlet energy (which he refers to as the “roughness”) is minimized by the Delaunay triangulation by showing that each edge flip which makes an edge Delaunay decreases the energy. In this paper, we introduce a Dirichlet energy on a weighted triangulation which is a generalization of the energy on unweighted triangulations and an analogue of the smooth Dirichlet energy on a domain. We show that this Dirichlet energy has the property that each edge flip which makes an edge weighted Delaunay decreases the energy. The proof is done by a direct calculation, and so gives an alternate proof of Rippa’s result.     

Robin spectral rigidity of nearly circular domains with a reflectional symmetry

This is a note on a paper of De Simoi–Kaloshin–Wei. We show that by combining their techniques with the wave trace invariants of Guillemin–Melrose and the heat trace invariants of Zayed for the Laplacian with Robin boundary conditions, one can extend the Dirichlet/Neumann spectral rigidity results of De Simoi–Kaloshin–Wei to the case of Robin boundary conditions. We will consider the same generic subset as did by De Simoi–Kaloshin–Wei of smooth strictly convex ?₂-symmetric planar domains sufficiently close to a circle, however we pair them with arbitrary ?₂-symmetric smooth Robin functions on the boundary and of course allow deformations of Robin functions as well.     

Structural properties of Dirichlet series with harmonic coefficients

An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sums, and special values of L-functions subject to the parity obstruction. This in turn leads to new representations of Catalan’s constant, odd values of the Riemann zeta function, and polylogarithmic quantities. Consequently, a dichotomy result is deduced on the transcendentality of Catalan’s constant and a series with hyperharmonic terms. Moreover, making use of integrals of smooth functions, we establish Diophantine-type approximations of real numbers by values of an infinite family of Dirichlet series built from representations of harmonic numbers.     

Operators with Singular Trace Conditions on a Manifold with Edges

We establish a new calculus of pseudodifferential operators on a manifold with smooth edges and study ellipticity with extra trace and potential conditions (as well as Green operators) at the edge. In contrast to the known scenario with conditions of that kind in integral form we admit in this paper ‘singular’ trace and Green operators. In contrast to standard conditions in the theory of elliptic boundary value problems (like Dirichlet or Neumann conditions) our singular trace conditions, in general, do not act on functions that are smooth up to the boundary, but admit a more general asymptotic structure. Their action is now associated with the Laurent coefficients of the meromorphic Mellin transforms of functions with respect to the half-axis variable, the distance to the edge.     

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR

In this work, we investigate the solvability of the boundary value problem for the Poisson equation, involving a generalized Riemann-Liouville and the Caputo derivative of fractional order in the class of smooth functions. The considered problems are generalization of the known Dirichlet and Neumann problems with operators of a fractional order.     

锥中与稳态的薛定谔算子相关的广义Martin函数无穷远处的控制

本文研究了稳态的薛定谔算子的Dirichlet问题和Martin函数的边界行为.利用广义Martin表示和稳态的薛定谔算子对应的常微分方程基本解,在具有光滑边界的锥形区域中获得了与稳态的薛定谔算子相关的广义Martin函数无穷远处广义调和控制的一些刻画,推广了拉普拉斯算子情形的结果.     

The Nehari problem for infinite-dimensional linear systems of parabolic type

A complete solution is obtained to the suboptimal Nehari extension problem for transfer functions of parabolic systems with Dirichlet boundary control and smooth observations. The solutions are given in terms of the realization (–A, B, C), whereA is a uniformly strongly elliptic operator of order two with smooth coefficients defined on a bounded open domain ofR _d ,B=AB _D andB _D is the Dirichlet map associated with Dirichlet boundary conditions andC is a bounded observation map fromL ₂() to the output spaceY. The approach is to solve an equivalentJ-spectral factorization problem for this particular realization.     

半平面上随机Dirichlet级数和Dirichlet空间

在 [1 ]的基础上 ,我们讨论了复平面上的单位圆上的Dirichlet空间和半平面上Dirichlet空间的关系 ,并找出了Dirichlet级数或一般随机Dirichlet级数属于a .s .属于Dirichlet空间和Lipr(0 相似文献   

Dirichlet-like space and capacity in complex analysis in several variables

For a Kähler manifold X, we study a space of test functions W^∗ which is a complex version of W1,2. We prove for W^∗ the classical results of the theory of Dirichlet spaces: the functions in W^∗ are defined up to a pluripolar set and the functional capacity associated to W^∗ tests the pluripolar sets. This functional capacity is a Choquet capacity. The space W^∗ is not reflexive and the smooth functions are not dense in it for the strong topology. So the classical tools of potential theory do not apply here. We use instead pluripotential theory and Dirichlet spaces associated to a current.     

Pointwise fourier inversion and related eigenfunction expansions

Necessary and sufficient conditions are found for the convergence at a pre-assigned point of the spherical partial sums (resp. integrals) of the Fourier series (resp. integral) in the class of piecewise smooth functions on Euclidean space. These results carry over unchanged to spherical harmonic expansions, Fourier transforms on hyperbolic space, and Dirichlet eigenfunction expansions with respect to the Laplace operator on a class of Riemannian manifolds. © 1994 John Wiley & Sons, Inc.