Space Of Smooth Functions Research Articles

The Lavrentiev Phenomenon

The basic problem of the calculus of variations consists of finding a function that minimizes an energy, like finding the fastest trajectory between two points for a point mass in a gravity field or finding the best shape of a wing. The existence of a solution may be established in quite abstract spaces, much larger than the space of smooth functions. An important practical problem is that of being able to approach the value of the infimum of the energy. However, numerical methods work with very “concrete” functions and sometimes they are unable to approximate the infimum: this is the surprising Lavrentiev phenomenon. The papers that ensure the nonoccurrence of the phenomenon form a recent saga, and the most general result formulated in the early ’90s was actually fully proved just recently, more than 30 years later. Our aim here is to introduce the reader to the calculus of variations, to illustrate the Lavrentiev phenomenon with the simplest known example, and to give an elementary proof of the nonoccurrence of the phenomenon.

On the regularity of solutions to a class of nonlinear Volterra integral equations with singularities

We study the smoothness properties of solutions to nonlinear Volterra integral equations of the second kind on a bounded interval [0,b]. The kernel of the integral operator of the underlying equation may have a diagonal singularity and a boundary singularity. Information about them is given through certain estimates. To characterize the regularity of solutions of such equations we show that the solution belongs to an appropriately weighted space of smooth functions on (0,b], with possible singularities of the derivatives of the solution at the left endpoint of the interval [0,b].

Reeb spaces of smooth functions on manifolds II

Reeb spaces of smooth functions on manifolds II

Analysis of the critical CR GJMS operator

Abstract: The critical CR GJMS operator on a strictly pseudoconvex CR manifold is a non-hypoelliptic CR invariant differential operator. We prove that, under the embeddability assumption, it is essentially self-adjoint and has closed range. Moreover, its spectrum is discrete, and the eigenspace corresponding to each non-zero eigenvalue is a finite-dimensional subspace of the space of smooth functions. As an application, we obtain a necessary and sufficient condition for the existence of a contact form with zero CR $Q$-curvature.

On the Structure of the Spectrum and the Resolvent Set of a Toeplitz Operator in a Countably Normed Space of Smooth Functions

On the Structure of the Spectrum and the Resolvent Set of a Toeplitz Operator in a Countably Normed Space of Smooth Functions

Modified Green-Hyperbolic Operators

Green-hyperbolic operators - partial differential operators on globally hyperbolic spacetimes that (together with their formal duals) possess advanced and retarded Green operators - play an important role in many areas of mathematical physics. Here, we study modifications of Green-hyperbolic operators by the addition of a possibly nonlocal operator acting within a compact subset $K$ of spacetime, and seek corresponding '$K$-nonlocal' generalised Green operators. Assuming the modification depends holomorphically on a parameter, conditions are given under which $K$-nonlocal Green operators exist for all parameter values, with the possible exception of a discrete set. The exceptional points occur precisely where the modified operator admits nontrivial smooth homogeneous solutions that have past- or future-compact support. Fredholm theory is used to relate the dimensions of these spaces to those corresponding to the formal dual operator, switching the roles of future and past. The $K$-nonlocal Green operators are shown to depend holomorphically on the parameter in the topology of bounded convergence on maps between suitable Sobolev spaces, or between suitable spaces of smooth functions. An application to the LU factorisation of systems of equations is described.

Invertibility and Spectrum of the Riemann Boundary Value Problem Operator in a Countably Normed Space of Smooth Functions on a Circle

Invertibility and Spectrum of the Riemann Boundary Value Problem Operator in a Countably Normed Space of Smooth Functions on a Circle

Bounded composition operators on functional quasi-Banach spaces and stability of dynamical systems

In this paper, we investigate the boundedness of composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We prove that the boundedness of a composition operator strongly limits the behavior of the original map, and it provides an effective method to investigate the properties of composition operators using the theory of dynamical system. Consequently, we prove that only affine maps can induce bounded composition operators on any quasi-Banach space continuously included in the space of entire functions of one variable if the function space contains a nonconstant function. We also prove that any polynomial automorphisms except affine transforms cannot induce bounded composition operators on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.

On the spectrum of isomorphisms defined on the space of smooth functions which are flat at 0

In this note we study the spectrum and the Waelbroeck spectrum of the derivative operator composed with isomorphic multiplication operators defined in the space of smooth functions in [0, 1] which are flat at 0.

Almost Sure Uniform Convergence of Stochastic Processes in the Dual of a Nuclear Space

Let \(\Phi \) be a nuclear space, and let \(\Phi '\) denote its strong dual. In this paper, we introduce sufficient conditions for the almost sure uniform convergence on bounded intervals of time for a sequence of \(\Phi '\)-valued processes having continuous (respectively, càdlàg) paths. The main result is formulated first in the general setting of cylindrical processes but later specialized to other situations of interest. In particular, we establish conditions for the convergence to occur in a Hilbert space continuously embedded in \(\Phi '\). Furthermore, in the context of the dual of an ultrabornological nuclear space (like spaces of smooth functions and distributions) we also include applications to convergence in \(L^{r}\) uniformly on a bounded interval of time, to the convergence of a series of independent càdlàg processes, and to the convergence of solutions to linear stochastic evolution equations driven by Lévy noise.

Characterizations of derivations on spaces of smooth functions

We provide a list of equivalent conditions under which an additive operator acting on a space of smooth functions on a compact real interval is a multiple of the derivation.

Pseudo-Fubini entire functions on the plane in the sense of Riemann

We prove the existence of a maximal dimensional vector space of real functions on the real plane all of whose nonzero members are bounded, entire, non-Lebesgue-integrable, and satisfy the equality of the two iterated integrals given in the conclusion of the Fubini theorem, with the additional property that these iterated integrals exist in the Riemann sense, but not in the Lebesgue one. Moreover, this vector space is dense in the space of smooth functions on the plane under its natural topology.

Time regularity of stochastic convolutions and stochastic evolution equations in duals of nuclear spaces

Let be a locally convex space and let be a quasi-complete bornological nuclear space (like spaces of smooth functions and distributions) with dual spaces and In this work we introduce sufficient conditions for time regularity properties of the -valued stochastic convolution where is the dual semigroup to a C 0-semigroup on is a suitable operator form into and M is a cylindrical-martingale valued measure on Our result is latter applied to study time regularity of solutions to -valued stochastic evolutions equations.

Extended Gevrey Regularity via Weight Matrices

The main aim of this paper is to compare two recent approaches for investigating the interspace between the union of Gevrey spaces Gt(U) and the space of smooth functions C∞(U). The first approach in the style of Komatsu is based on the properties of two parameter sequences Mp=pτpσ, τ>0, σ>1. The other one uses weight matrices defined by certain weight functions. We prove the equivalence of the corresponding spaces in the Beurling case by taking projective limits with respect to matrix parameters, while in the Roumieu case we need to consider a larger space than the one obtained as the inductive limit of extended Gevrey classes.

Density of smooth functions in Musielak–Orlicz spaces

We provide necessary and sufficient conditions for the space of smooth functions with compact supports $$C^\infty _c(\Omega )$$ to be dense in Musielak–Orlicz spaces $$L^\Phi (\Omega )$$ where $$\Omega $$ is an open subset of $${\mathbb {R}}^d$$ . In particular, we prove that if $$\Phi $$ satisfies condition $$\Delta _2$$ , the closure of $$C^\infty _c(\Omega )\cap L^\Phi (\Omega )$$ is equal to $$L^\Phi (\Omega )$$ if and only if the measure of singular points of $$\Phi $$ is equal to zero. This extends the earlier density theorems proved under the assumption of local integrability of $$\Phi $$ , which implies that the measure of the singular points of $$\Phi $$ is zero. As a corollary we obtain analogous results for Musielak–Orlicz spaces generated by double phase functional and we recover the well-known result for variable exponent Lebesgue spaces.

Triharmonic CMC Hypersurfaces in Space Forms with at Most 3 Distinct Principal Curvatures

A k-harmonic map is a critical point of the k-energy defined on the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $$M^{n} (n\ge 3)$$ is a CMC proper triharmonic hypersurface with at most three distinct principal curvatures in a space form $$\mathbb {R}^{n+1}(c)$$ , then M has constant scalar curvature. This supports the generalized Chen’s conjecture when $$c\le 0$$ . When $$c=1$$ , we give an optimal upper bound of the mean curvature H for a non-totally umbilical proper CMC k-harmonic hypersurface with constant scalar curvature in a sphere. As an application, we give the complete classification of the 3-dimensional complete proper CMC triharmonic hypersurfaces in $$\mathbb {S}^{4}$$ .

Sequence space representations for spaces of smooth functions and distributions via Wilson bases

We provide explicit sequence space representations for the test function and distribution spaces occurring in the Valdivia-Vogt structure tables by making use of Wilson bases generated by compactly supported smooth windows. Furthermore, we show that these kind of bases are common unconditional Schauder bases for all separable spaces occurring in these tables. Our work implies that the Valdivia-Vogt structure tables for test functions and distributions may be interpreted as one large commutative diagram.

Sharp Hardy Inequalities via Riemannian Submanifolds

This paper is devoted to Hardy inequalities concerning distance functions from submanifolds of arbitrary codimensions in the Riemannian setting. On a Riemannian manifold with non-negative curvature, we establish several sharp weighted Hardy inequalities in the cases when the submanifold is compact as well as non-compact. In particular, these inequalities remain valid even if the ambient manifold is compact, in which case we find an optimal space of smooth functions to study Hardy inequalities. Further examples are also provided. Our results complement in several aspects those obtained recently in the Euclidean and Riemannian settings.

Supercuspidal representations of GLn(F) distinguished by an orthogonal involution

Let F be a non-archimedean locally compact field of residue characteristic p≠2, let G=GLn(F) and let H be an orthogonal subgroup of G. For π a complex smooth supercuspidal representation of G, we give a full characterization for the distinguished space HomH(π,1) being non-zero and we further study its dimension as a complex vector space, which generalizes a similar result of Hakim for tame supercuspidal representations. As a corollary, the embeddings of π in the space of smooth functions on the set of symmetric matrices in G, as a complex vector space, is non-zero and of dimension four, if and only if the central character of π evaluating at −1 is 1.

Gravitational-wave geodesy: Defining false alarm probabilities with respect to correlated noise

Future searches for a gravitational-wave background using Earth-based gravitational-wave detectors might be impacted by correlated noise sources. A well known example are the Schumann resonances, which are extensively studied in the context of searches for a gravitational-wave background. Earlier work has shown that a technique termed "gravitational-wave geodesy" can be used to generically differentiate observations of a gravitational-wave background from signals due to correlated terrestrial effects, requiring true observations to be consistent with the known geometry of our detector network. The key result of this test is a Bayes factor between the hypotheses that a candidate signal is astrophysical or terrestrial in origin. Here, we further formalize the geodesy test, mapping distributions of false-alarm and false-acceptance probabilities to quantify the degree with which a given Bayes factor will boost or diminish our confidence in an apparent detection of the gravitational-wave background. To define the false alarm probability of a given Bayes factor, we must have knowledge of our null hypothesis: the space of all possible correlated terrestrial signals. Since we do not have this knowledge we instead construct a generic space of smooth functions in the frequency domain using Gaussian processes, which we tailor to be conservative. This enables us to use draws from our Gaussian processes as a proxy for all possible non-astrophysical signals. As a demonstration, we apply the tool to the SNR = 1.25 excess observed for a 2/3-power law by the LIGO and Virgo collaborations during their second observing run.