Uniform approximation by harmonic functions on compact subsets of ℝ3

We establish small energy H\"{o}lder bounds for minimizers $u_\varepsilon$ of \[E_\varepsilon (u):=\int_\Omega W(\nabla u)+ \frac{1}{\varepsilon^2} \int_\Omega f(u),\] where $W$ is a positive definite quadratic form and the potential $f$ constrains $u$ to be close to a given manifold $\mathcal N$. This implies that, up to subsequence, $u_\varepsilon$ converges locally uniformly to an $\mathcal N$-valued $W$-harmonic map, away from its singular set. We treat general energies, covering in particular the 3D Landau-de Gennes model for liquid crystals, with three distinct elastic constants. Similar results are known in the isotropic case $W(\nabla u)=\vert \nabla u\vert^2$ and rely on three ingredients: a monotonicity formula for the scale-invariant energy on small balls, a uniform pointwise bound, and a Bochner equation for the energy density. In the level of generality we consider, all of these ingredients are absent. In particular, the lack of monotonicity formula is an important reason why optimal estimates on the singular set of $W$-harmonic maps constitute an open problem. Our novel argument relies on showing appropriate decay for the energy on small balls, separately at scales smaller and larger than $\varepsilon$: the former is obtained from the regularity of solutions to elliptic systems while the latter is inherited from the regularity of $W$-harmonic maps. This also allows us to handle physically relevant boundary conditions for which, even in the isotropic case, uniform convergence up to the boundary was open.

We work in the Euclidean space R N , where, unless the contrary is stated, N≥2. The Euclidean norm of a point x∈R N is denoted by ∥x∥. If r>0, the, we denote the open ball of centre x and radius r by B(x, r) and we write B 0 (x, r)=B(x, r)\{x}

In this paper, we have construct a new version of the Bézier‐type Baskakov–Schurer–Szász–Stancu operators. For this new class of operators, uniform convergence is shown in any compact subset or positive real line. We prove Korovkin‐type theorem, Voronovskaya‐type theorem, and Grüss–Voronovskaya‐type theorem. Moreover, at the end, we express the behavior of the operators in the Lipschitz‐type space using the modulus of continuity and smoothness.

In Boos (1985) equicontinuity conditions are given which ensure the uniform convergence of densities in $\mathbb{R}^k$, given convergence in distribution. In the present note we show that such equicontinuity conditions in fact characterize uniform local convergence with no additional assumptions on the sequence of densities, or on the limit density. Versions of these results are also given when the distributions depend on an unknown parameter; these forms will be relevant for the uniform approximation of likelihood functions.

SummaryAssuming duality and certain analytic conditions on the potential density u(x,y), we show that any harmonic function is a constant and the invariant measure, if one exists, is unique.We will assume that Xt is a standard process in duality with another standard process X t ′ relative to some fixed reference measure m on the state space E. The reader is refered to [l, VI, Sec 1] for the precise definition of duality and to either [1] or [2] for other usual definitions. We will also assume that both X and X′ are transient; for any compact K, TKo θt → ∞ as t → ∞ a.s.. However, only the transience of X is used in Proposition 1 and only that of X′ is used in the rest of the discussion.For simplicity, any subset A of E and any function f defined on E is automatically assumed to be measurable with respect to the Borel field of E.There is a common potential density u(x,y) which is excessive in x and co-excessive in y and satisfies:Uf(x) = ∫u(x,y)f(y)m(dy) and U′f(y) = ∫m(dx)f(x)u(x,y) for any f ≥ 0, where U and U′ are, repectively, the potential operators of X and X′.A function h ≥ 0 on E is said to be harmonic if for any compact subset K of E, PKch = h.For Brownian motion in Rn with n ≥ 3, any harmonic function is a constant. Our first proposition generalizes this classical result under a general setting, which includes as special cases, not only Brownian motion, but also transient symmetric stable processes and one sided stable processes. Our proof which uses the general theory of Markov processes is quite short, compared with the complicated analytic proof for symmetric stable processes given in [3].

We extend to the degenerate case \(p\not = 2\), Simon’s approach to the classical regularity theory of harmonic maps of Schoen & Uhlenbeck, by proving a “p-Harmonic Approximation Lemma”. This allows to approximate functions with p-harmonic functions in the same way as the classical harmonic approximation lemma (going back to De Giorgi) does via harmonic functions. Finally, we show how to combine this tool with suitable regularity estimates for solutions to degenerate elliptic systems with a critical growth right hand side, in order to obtain partial \(C^{1,\alpha}\)-regularity of p-harmonic maps.

There exist functions, called U.L.S. (Universal Laurent Series), holomorphic on finitely connected domains Ω in C, whose Laurent-type partial sums approximate everything we can hope for, on compact subsets outside Ω ∪ {a 1,…,a}, for certain prescribed points a 1,…,a k. In this paper we prove that, under additional assumptions, for every U.L.S. there exists a subsequence of its Laurent-type partial sums, which converges to the function itself in the whole of Ω and which approximates everything we can hope for outside Ω ∪ {a 1,…,a, k}.

Harmonic approximation and Sarason's-type theorem

Let ƒ b e a mapping defined on a compact subset K of the finite complex plane C and taking its values on the extended plane C ⋃ ﹛ ∞﹜. For x a metric on the extended plane, we consider the possibility of approximating ƒ x-uniformly on K by rational functions. Since all metrics on C ⋃ ﹛oo ﹜ are equivalent, we shall consider that x is the chordal metric on the Riemann sphere of diameter one resting on a finite plane at the origin.

Uniformité en h dans la loi fonctionnelle limite uniforme les accroissements du processus empirique indéxé par des fonctions

Let {ϕ i } =∩ be continuous real functions on the compact set M⊂R. We consider the problem of best uniform approximation of the functionϕ by polynomials $$\sum\nolimits_{i = 1}^n {c_i \varphi _i }$$ on M. Let V(ϕ0, A) be a set of polynomials of best approximation on A ⫅ M. We show that $$V(\varphi _0 ,M) = \mathop \cap \limits_{A_{n + 1} } V(\varphi _0 ,A_{n + 1} )$$ , where An+1 represents all the possible sets of n+ 1 points {x1, ..., xn+1} in M, containing the characteristic set of the given problem of best approximation and for which the the rank of ∥ϕi ∥ (i=1, ...,n; j=1,..., n+1) is equal to n. This theorem is applied to a problem of uniform approximation where {ϕ i } =1 is a weakly Chebyshev system.

Let Ω ⊂ Rn be an open connected set such that if n = 2, it is assumed that Ω has a Green function. Let K be a compact subset of Ω. There are many equivalent ways to define the class H(K) of harmonic functions on K. In this note we present a new definition for the concept of harmonicity for a Borel measurable (not necessarily continuous) on K. We study this new class of functions and show that it coincides with the class of those quasi finely harmonic functions on the fine interior of K.

The main result can be stated roughly as follows: Let M M be an Alexandrov space, Ω ⊂ M \Omega \subset M an open domain and f : Ω → R f:\Omega \to \mathbb {R} a harmonic function. Then f f is Lipschitz on any compact subset of Ω \Omega . Using this result I extend proofs of some classical theorems in Riemannian geometry to Alexandrov spaces.

Let C[a,b] denote the set of all real-valued continuous function x defined on [a,b] such that x(a) = 0. In this thesis, we define the Wiener and Feynman integrals with complex parameters for functionals defined on C[a,b] , and we have a sufficient condition for the existence or the complex sequential Wiener integral for analytic and harmonic functionals. Moreover we have that this integral equals the ordinary Wiener integral, Finally, we define the limiting sequential Wiener and Feynman integrals, and we also establish an existence theorem for the limiting Feynman integral for analytic and harmonic functionals in such a way that it equals the ordinary Wiener integral.

We give a survey of Runge-type harmonic approximation theorems and the techniques used to prove them. The emphasis is on generalizations of Walsh’s classical theorem concerning uniform harmonic approximation on compact sets to the case where approximation takes place on certain non-compact sets: both uniform and tangential approximation are treated. We also give some applications of the theory to the construction of harmonic functions exhibiting various kinds of unexpected behaviour. The course is partly intended to provide preparatory material for S. J. Gardiner’ course “Harmonic approximation and applications”, published in this volume.