Ultradifferentiable Functions Research Articles

Correction to: A simple proof of Kotake–Narasimhan theorem in some classes of ultradifferentiable functions

Correction to: A simple proof of Kotake–Narasimhan theorem in some classes of ultradifferentiable functions

On Orlicz classes defined in terms of associated weight functions

N-functions and their growth and regularity properties are crucial in order to introduce and study Orlicz classes and Orlicz spaces. We consider N-functions which are given in terms of so-called associated weight functions. These functions are frequently appearing in the theory of ultradifferentiable function classes and in this setting additional information is available since associated weight functions are defined in terms of a given weight sequence. We express and characterize several known properties for N-functions purely in terms of weight sequences which allows to construct (counter-) examples. Moreover, we study how for abstractly given N-functions this framework becomes meaningful and finally we establish a connection between the complementary N-function and the recently introduced notion of the so-called dual sequence.

Some results on ultradifferential convolution property of ultradistributions

We consider some sequential conditions for the convolvability of two ultradistributions, specifically the Roumieu ultradistributions defined on the space of all ultradifferentiable functions Dn{M_p} in the space Dn'{M_p} and show the similarities between these conditions in the sense of Mincheva-Kaminska S. The sequential conditions are based on the use of approximate identities which satisfy certain conditions to be a Banach algebra for the space Dn{M_p} . We present an equivalent result involving the convolution of the differentiability of two Roumieu ultradistributions with equivalent norm condition. 

Composition operators on Gelfand-Shilov classes

We study composition operators on global classes of ultradifferentiable functions of Beurling type invariant under Fourier transform. In particular, for the classical Gelfand-Shilov classes Σd,d>1, we prove that a necessary condition for the composition operator Cψ:f↦f∘ψ to be well defined is the boundedness of ψ′. We find the optimal index d′ for which Cψ(Σd(R))⊂Σd′(R) holds for any non-constant polynomial ψ.

Ultradifferentiable classes of entire functions

We study classes of ultradifferentiable functions defined in terms of small weight sequences violating standard growth and regularity requirements. First, we show that such classes can be viewed as weighted spaces of entire functions for which the crucial weight is given by the associated weight function of the so-called conjugate weight sequence. Moreover, we generalize results from M. Markin from the so-called small Gevrey setting to arbitrary convenient families of (small) sequences and show how the corresponding ultradifferentiable function classes can be used to detect boundedness of normal linear operators on Hilbert spaces (associated with an evolution equation problem). Finally, we study the connection between small sequences and the recent notion of dual sequences introduced in the Ph.D. thesis of J. Jiménez-Garrido.

On generalized definitions of ultradifferentiable classes

We show that the ultradifferentiable-like classes of smooth functions introduced and studied by S. Pilipović, N. Teofanov and F. Tomić are special cases of the general framework of spaces of ultradifferentiable functions defined in terms of weight matrices in the sense of A. Rainer and the third author. We study classes “beyond geometric growth factors” defined in terms of a weight sequence and an exponent sequence, prove that these new types admit a weight matrix representation and transfer known results from the matrix-type to such a non-standard ultradifferentiable setting.

A note on the barrelledness of weighted PLB-spaces of ultradifferentiable functions

In this note, we consider weighted PLB-spaces of ultradifferentiable functions defined via a weight function and a weight system, as introduced in our previous work (2022). We provide a complete characterization of when these spaces are ultrabornological and barrelled in terms of the defining weight system, thereby improving the main result of our above mentioned work. In particular, we obtain that the multiplier space of the Gelfand--Shilov space $\Sigma^r\_s(\mathbb{R}^d)$ of Beurling type is ultrabornological, whereas the one of the Gelfand--Shilov space $\mathcal{S}^r\_s(\mathbb{R}^d)$ of Roumieu type is not barrelled.

On invertibility of Duhamel operator in spaces of ultradifferentiable functions

On invertibility of Duhamel operator in spaces of ultradifferentiable functions

Differentiation Operator in the Beurling Space of Ultradifferentiable Functions of Normal Type on an Interval

Differentiation Operator in the Beurling Space of Ultradifferentiable Functions of Normal Type on an Interval

Quantizations and Global Hypoellipticity for Pseudodifferential Operators of Infinite Order in Classes of Ultradifferentiable Functions

We study the change of quantization for a class of global pseudodifferential operators of infinite order in the setting of ultradifferentiable functions of Beurling type. The composition of different quantizations as well as the transpose of a quantization are also analysed, with applications to the Weyl calculus. We also compare global \(\omega \)-hypoellipticity and global \(\omega \)-regularity of these classes of pseudodifferential operators.

On subadditivity-like conditions for associated weight functions

The aim of this article is to provide characterizations for subadditivity-like growth conditions for the so-called associated weight functions in terms of the defning weight sequence. Such growth requirements arise frequently in the literature and are standard when dealing with ultradifferentiable function classes defned by Braun-Meise-Taylor weight functions since they imply or even characterize important and desired consequences for the underlying function spaces, e.g. closedness under composition.

On the Moment Problem in the Spaces of Ultradifferentiable Functions of Mean Type

On the Moment Problem in the Spaces of Ultradifferentiable Functions of Mean Type

Weighted (PLB)-spaces of ultradifferentiable functions and multiplier spaces

We study weighted (PLB)-spaces of ultradifferentiable functions defined via a weight function (in the sense of Braun, Meise and Taylor) and a weight system. We characterize when such spaces are ultrabornological in terms of the defining weight system. This generalizes Grothendieck’s classical result that the space $${\mathcal {O}}_M$$ of slowly increasing smooth functions is ultrabornological to the context of ultradifferentiable functions. Furthermore, we determine the multiplier spaces of Gelfand-Shilov spaces and, by using the above result, characterize when such spaces are ultrabornological. In particular, we show that the multiplier space of the space of Fourier ultrahyperfunctions is ultrabornological, whereas the one of the space of Fourier hyperfunctions is not.

The global Kotake-Narasimhan theorem

In this paper we introduce the notion of global ultradifferentiable functions with respect to weight functions and include a discussion of its functional analytic theory and prove a characterization in terms of certain exponential decay of the Fourier-Broz-Iagolnitzer transform—a Paley-Wiener type theorem. As an application we investigate the regularity of global ultradifferentiable vectors proving a version of the Kotake-Narasimhan theorem in this setting.

Equality of Ultradifferentiable Classes by Means of Indices of Mixed O-regular Variation

We characterize the equality between ultradifferentiable function classes defined in terms of abstractly given weight matrices and in terms of the corresponding matrix of associated weight functions by using new growth indices. These indices, defined by means of weight sequences and (associated) weight functions, are extending the notion of O-regular variation to a mixed setting. Hence we are extending the known comparison results concerning classes defined in terms of a single weight sequence and of a single weight function and give also these statements an interpretation expressed in O-regular variation.

On the Fourier-Laplace transform of functionals on a space of infinitely differentiable functions on a convex compact

Classes of ultradifferentiable functions are classically defined by imposing growth conditions on the derivatives of the functions. Following this approach we consider a Fréchet-Schwartz space of infinitely differentiable functions on a closure of a bounded convex domain of multidimensional real space with uniform bounds on their partial derivatives. Our aim is to obtain Paley-Wiener-Schwartz type theorem connecting properties of linear continuous functionals on this space with the behaviour of their Fourier-Laplace transforms. Very similar problems were considered by M. Neymark, B.A. Taylor, M. Langenbruch, A.V. Abanin.

Correction to: Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality

Correction to: Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality

Reducibility of ultra-differentiable quasiperiodic cocycles under an adapted arithmetic condition

We prove a reducibility result for sl(2,R) quasi-periodic cocycles close to a constant elliptic matrix in ultra-differentiable classes, under an adapted arithmetic condition which extends the Brjuno-R{\"u}ssmann condition in the analytic case. The proof is based on an elementary property of the fibered rotation number and deals with ultra-differentiable functions with a weighted Fourier norm. We also show that a weaker arithmetic condition is necessary for reducibility, and that it can be compared to a sufficient arithmetic condition.

Regularity of global solutions of partial differential equations in non isotropic ultradifferentiable spaces via time-frequency methods

In this paper we study regularity of partial differential equations with polynomial coefficients in non isotropic Beurling spaces of ultradifferentiable functions of global type. We study the action of transformations of Gabor and Wigner type in such spaces and we prove that a suitable representation of Wigner type allows to prove regularity for classes of operators that do not have classical hypoellipticity properties.

Spectrum of the Cesàro Operator on the Ultradifferentiable Function Spaces $${\mathcal E}_\omega ({\mathbb {R}}_+)$$

The aim of this paper is to investigate the continuity, spectrum and point spectrum, and the dynamical behaviour of the Cesaro averaging operator $$\mathsf {C}$$ on the ultradifferentiable function spaces $${\mathcal E}_\omega ({\mathbb {R}}_+)$$ .