Smoothing of bump functions

Let X be a separable Banach space with a Schauder basis, admitting a continuous bump which depends locally on finitely many coordinates. Then X admits also a C^∞-smooth bump which depends locally on finitely many coordinates.     

On uniformly Gâteaux smooth C (n)-smooth norms on separable Banach spaces

Every separable Banach space with C ⁽ⁿ⁾-smooth norm (Lipschitz bump function) admits an equivalent norm (a Lipschitz bump function) which is both uniformly Gâteaux smooth and C ⁽ⁿ⁾-smooth. If a Banach space admits a uniformly Gâteaux smooth bump function, then it admits an equivalent uniformly Gâteaux smooth norm.     

James' Theorem Fails for Starlike Bodies

Starlike bodies are interesting in nonlinear functional analysis because they are strongly related to bump functions and to n-homogeneous polynomials on Banach spaces, and their geometrical properties are thus worth studying. In this paper we deal with the question whether James' theorem on the characterization of reflexivity holds for (smooth) starlike bodies, and we establish that a feeble form of this result is trivially true for starlike bodies in nonreflexive Banach spaces, but a reasonable strong version of James' theorem for starlike bodies is never true, even in the smooth case. We also study the related question as to how large the set of gradients of a bump function can be, and among other results we obtain the following new characterization of smoothness in Banach spaces: a Banach space X has a C¹ Lipschitz bump function if and only if there exists another C¹ smooth Lipschitz bump function whose set of gradients contains the unit ball of the dual space X*. This result might also be relevant to the problem of finding an Asplund space with no smooth bump functions.     

Smooth approximations without critical points

In any separable Banach space containing c ₀ which admits a C ^k-smooth bump, every continuous function can be approximated by a C ^k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set residual in some neighbourhood of zero.     

Smooth approximations

We prove, among other things, that a Lipschitz (or uniformly continuous) mapping f:X→Y can be approximated (even in a fine topology) by smooth Lipschitz (resp. uniformly continuous) mapping, if X is a separable Banach space admitting a smooth Lipschitz bump and either X or Y is a separable C(K) space (resp. super-reflexive space). Further, we show how smooth approximation of Lipschitz mappings is closely related to a smooth approximation of C¹-smooth mappings together with their first derivatives. As a corollary we obtain new results on smooth approximation of C¹-smooth mappings together with their first derivatives.     

The range of the derivative of a differentiable bumpL'image de la dérivée d'une fonction bosse différentiable

We study the range of the derivative of a Frechet differentiable bump. X is an infinite dimensional separable C^p-smooth Banach space. We first prove that any connected open subset of $\text{X}{}^1$ containing 0 is the range of the derivative of a C^p-bump. Next, analytic subsets of $\text{X}{}^1$ which satisfy a natural linkage condition are the range of the derivative of a C¹-bump. We find analogues of these results in finite dimensions. We finally show that $\text{f′(}\text{R}{}^2\text{)}$ is the closure of its interior, if f is a C²-bump on $\text{R}{}^2$. To cite this article: T. Gaspari, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 189–194.     

The range of the gradient of a LipschitzC 1-smooth bump in infinite dimensions

If a Banach space has a LipschitzC ¹-smooth bump function, then it admits other bumps of the same smoothness whose gradients exactly fill the dual unit ball and other reasonable figures. This strengthens a result of Azagra and Deville who were able to cover the dual unit ball. Supported in part by NSERC and the Canada Research Chair Programme. Supported in part by grants NATO CRG-973982, GAĈR 201-01-1198, and AV 1019003 (Czech Republic). Supported in part by NATO CRG-973982 and NSERC.     

A parametrix for the fundamental solution of the Klein-Gordon equation on asymptotically de Sitter spaces

In this paper we construct a parametrix for the forward fundamental solution of the wave and Klein-Gordon equations on asymptotically de Sitter spaces without caustics. We use this parametrix to obtain asymptotic expansions for solutions of (□−λ)u=f and to obtain a uniform Lp estimate for a family of bump functions traveling to infinity.     

Approximation with the output of linear control systems

In this paper we give a new proof that for controllable and observable linear systems every L²[0,T] function can be approximated in the L²[0,T] sense with an output function generated by an L²[0,T] input function. We also give a new characterization of how continuous functions on [0,T] are uniformly approximated by an output generated by a continuous input function. The relative degree of the transfer function of the system determines those functions that can be approximated. We further show that if the initial data is allowed to vary then every continuous function is uniformly approximated by outputs generated by continuous functions.     

Admissible majorants for model subspaces, and arguments of inner functions

Let Θ be an inner function in the upper half-plane ?⁺ and let K _Θ denote the model subspace H ² ? Θ H ² of the Hardy space H ² = H ²(?⁺). A nonnegative function w on the real line is said to be an admissible majorant for K _Θ if there exists a nonzero function f ∈ K _Θ such that {f} ? w a.e. on ?. We prove a refined version of the parametrization formula for K _Θ-admissible majorants and simplify the admissibility criterion (in terms of arg Θ) obtained in [8]. We show that, for every inner function Θ, there exist minimal K _Θ-admissible majorants. The relationship between admissibility and some weighted approximation problems is considered.     

The rate of approximation of Gaussian radial basis neural networks in continuous function space

There have been many studies on the dense theorem of approximation by radial basis feedforword neural networks, and some approximation problems by Gaussian radial basis feedforward neural networks(GRBFNs)in some special function space have also been investigated. This paper considers the approximation by the GRBFNs in continuous function space. It is proved that the rate of approximation by GRNFNs with n~d neurons to any continuous function f defined on a compact subset K(R~d)can be controlled by ω(f, n~(-1/2)), where ω(f, t)is the modulus of continuity of the function f .     

Minimal kernels of weakly complete spaces

Let X be a weakly complete space i.e. X a complex space endowed with a C^k-smooth, k?0, plurisubharmonic exhaustion function. We give the notion of minimal kernelΣ¹=Σ¹(X) of X by the following property: x∈Σ¹ if no continuous plurisubharmonic exhaustion function is strictly plurisubharmonic near x. The study of the geometric properties of the minimal kernels is the aim of present paper. After stating that the minimal kernel Σ¹ of a weakly complete space can be defined by a single plurisubharmonic exhaustion function ?, called minimal, using the characterization in terms of Bremermann envelopes, we prove the following, crucial, result: if X is a weakly complete manifold and ? a minimal function for X, the nonempty level sets Σ_c¹=Σ¹∩{?=c} have the local maximum property. In the last section we discuss the special case of weakly complete surfaces. We prove that if dim_cX=2 and c is a regular value of a minimal function ? then the nonempty level sets Σ_c¹=Σ¹∩{?=c} are compact spaces foliated by holomorphic curves.     

On the largest principal angle between random subspaces

Formulas are derived for the probability density function and the probability distribution function of the largest canonical angle between two p-dimensional subspaces of Rⁿ chosen from the uniform distribution on the Grassmann manifold (which is the unique distribution invariant by orthogonal transformations of Rⁿ). The formulas involve the gamma function and the hypergeometric function of a matrix argument.     

Upper bounds for the formula size of symmetric Boolean functions

We prove that the complexity of the implementation of the counting function of n Boolean variables by binary formulas is at most n ^3.03, and it is at most n ^4.47 for DeMorgan formulas. Hence, the same bounds are valid for the formula size of any threshold symmetric function of n variables, particularly, for the majority function. The following bounds are proved for the formula size of any symmetric Boolean function of n variables: n ^3.04 for binary formulas and n ^4.48 for DeMorgan ones. The proof is based on the modular arithmetic.     

Compactly supported positive definite radial functions

We provide criteria for positive definiteness of radial functions with compact support. Based on these criteria we will produce a series of positive definite and compactly supported radial functions, which will be very useful in applications. The simplest ones arecut-off polynomials, which consist of a single polynomial piece on [0, 1] and vanish on [1, ∞). More precisely, for any given dimensionn and prescribedC ^k smoothness, there is a function inC ^k (? ⁿ ), which is a positive definite radial function with compact support and is a cut-off polynomial as a function of Euclidean distance. Another example is derived from odd-degreeB-splines.     

Nodal solutions for singularly perturbed equations with critical exponential growth

The existence and concentration behavior of nodal solutions are established for the equation −?²Δu+V(z)u=f(u) in Ω, where Ω is a domain in R², not necessarily bounded, V is a positive Hölder continuous function and f∈C¹ is an odd function having critical exponential growth.     

Recognition of triangles by covariogram

The paper considers the problem of recognition of triangles by orientation-dependent chord length distribution. The following results are obtained: 1. The explicit form of the covariogram and orientation-dependent chord length distribution function for a triangle. 2. The explicit form for the chord length distribution function for a triangle. 3. The length of the maximal chord of a triangle is continuous function on direction u ∈ S ¹ (S ¹ is the space of all directions in the plane). 4. If we have orientation-dependent chord length distribution function for an everywhere dense set of S ¹, then we can uniquely recognize the triangle with respect to reflections and translations. 5. For any finite subset A ? S ¹, there are two non-congruent triangles with the same values of orientationdependent chord length distribution functions on A.     

C-GV triviality of function germs and Newton polyhedra

We provide estimates on the degree of C l GV determinacy ( G is one of Mather’s groups R or K ) of function germs which are defined on analytic variety V and satisfies a non-degeneracy condition with respect to some Newton polyhedron. The result gives an explicit order such that the C l geometrical structure of a function germ is preserved after higher order perturbations, which generalizes the result on C l G triviality of function germs given by M.A.S.Ruas.     

The class number of cyclotomic function fields

Let k be a rational function field over a finite field. Carlitz and Hayes have described a family of extensions of k which are analogous to the collection of cyclotomic extensions {Q(ζm)| m ≥ 2} of the rational field Q. We investigate arithmetic properties of these “cyclotomic function fields.” We introduce the notion of the maximal real subfield of the cyclotomic function field and develop class number formulas for both the cyclotomic function field and its maximal real subfield. Our principal result is the analogue of a classical theorem of Kummer which for a prime p and positive integer n relates the class number of Q(ζpn + ζpn?1), the maximal real subfield of Q(ζpn), to the index of the group of cyclotomic units in the full unit group of Z[ζpn].     

On the enumeration of certain weighted graphs

We enumerate weighted simple graphs with a natural upper bound condition on the sum of the weight of adjacent vertices. We also compute the generating function of the numbers of these graphs, and prove that it is a rational function. In particular, we show that the generating function for connected bipartite simple graphs is of the form p₁(x)/(1-x)^m+1. For nonbipartite simple graphs, we get a generating function of the form p₂(x)/(1-x)^m+1(1+x)^l. Here m is the number of vertices of the graph, p₁(x) is a symmetric polynomial of degree at most m, p₂(x) is a polynomial of degree at most m+l, and l is a nonnegative integer. In addition, we give computational results for various graphs.