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This book gives an excellent and up-to-date overview on the convergence and joint progress in the fields of Generalized Functions and Fourier Analysis, notably in the core disciplines of pseudodifferential operators, microlocal analysis and time-frequency analysis. The volume is a collection of chapters addressing these fields, their interaction, their unifying concepts and their applications and is based on scientific activities related to the International Association for Generalized Functions (IAGF) and the ISAAC interest groups on Pseudo-Differential Operators (IGPDO) and on Generalized Functions (IGGF), notably on the longstanding collaboration of these groups within ISAAC.

Inhaltsverzeichnis

Frontmatter

On Temperate Distributions Decaying at Infinity

Abstract
We describe classes of temperate distributions with prescribed decay properties at infinity. The definition of the elements of such classes is given in terms of the Schwartz’ bounded distributions, and we discuss their characterization in terms of convolution and of decomposition as a finite sum of derivatives of suitable functions. We also prove mapping properties under the action of a class of Fourier integral operators, with inhomogeneous phase function and polynomially bounded symbol globally defined on \(\mathbb{R}^d\)
Alessia Ascanelli, Sandro Coriasco, André Süß

Transport in a Stochastic Goupillaud Medium

Abstract
This paper is part of a project that aims at modelling wave propagation in random media by means of Fourier integral operators. A partial aspect is addressed here, namely explicit models of stochastic, highly irregular transport speeds in one-dimensional transport, which will form the basis for more complex models. Starting from the concept of a Goupillaud medium (a layered medium in which the layer thickness is proportional to the propagation speed), a class of stochastic assumptions and limiting procedures leads to characteristic curves that are Lévy processes. Solutions corresponding to discretely layered media are shown to converge to limits as the time step goes to zero (almost surely pointwise almost everywhere). This translates into limits in the Fourier integral operator representations.
Florian Baumgartner, Michael Oberguggenberger, Martin Schwarz

Hilbert Space Embeddings for Gelfand–Shilov and Pilipović Spaces

Abstract
We consider quasi-Banach spaces that lie between a Gelfand–Shilov space, or more generally, Pilipovi´c space, \(\mathcal{H}\), and its dual, \(\mathcal{H}^\prime\) . We prove that for such quasi-Banach space \(\mathcal{B}\), there are convenient Hilbert spaces, \(\mathcal{H}_{k}, k=1,2\), with normalized Hermite functions as orthonormal bases and such that \(\mathcal{B}\) lies between \(\mathcal{H}_1\; \mathrm{and}\;\mathcal{H}_2\), and the latter spaces lie between \(\mathcal{H}\; \mathrm{and}\;\mathcal{H}^\prime\).
Yuanyuan Chen, Mikael Signahl, Joachim Toft

Blow-up Phenomena for Solutions of Discrete Nonlinear p-Laplacian Parabolic Equations on Networks

Abstract
This is an article to introduce discrete nonlinear p-Laplacian parabolic equations on networks and discuss the conditions under which blow-up occurs for the solutions. We first deal with the case p = 2, introducing a recent result about the blow-up phenomena for the solutions. Secondly, we deal with the general p-Laplacian case. In each case, we classify the parameters depending on the equations so that we can see when the solutions blow up or globally exist. Moreover, the blow-up time and blow-up rate are introduced for the blow-up solutions. The last part is devoted to the blow-up of Fujita type.
Soon-Yeong Chung

Generalized Function Algebras Containing Spaces of Periodic Ultradistributions

Abstract
We construct differential algebras in which spaces of (one-dimensional) periodic ultradistributions are embedded. By proving a Schwartz impossibility type result, we show that our embeddings are optimal in the sense of being consistent with the pointwise multiplication of ordinary functions. In particular, we embed the space of hyperfunctions on the unit circle into a differential algebra in such a way that the multiplication of real analytic functions on the unit circle coincides with their pointwise multiplication. Furthermore, we introduce a notion of regularity in our newly defined algebras and show that an embedded ultradistribution is regular if and only if it is an ultradifferentiable function.
Andreas Debrouwere

On General Prime Number Theorems with Remainder

Abstract
We show that for Beurling generalized numbers the prime number theorem in remainder form
$$ \pi \left( x \right) = Li\left( x \right) + O\left( {\frac{x} {{\log ^n x}}} \right)\,for\,all\,n\, \in \,{\Bbb N} $$
is equivalent to (for some a > 0)
$$ N\left( x \right) = ax + O\left( {\frac{x} {{\log ^n x}}} \right)\,for\,all\,n\, \in \,{\Bbb N} $$
where N and π are the counting functions of the generalized integers and primes, respectively. This was already considered by Nyman (Acta Math. 81 (1949), 299–307), but his article on the subject contains some mistakes. We also obtain an average version of this prime number theorem with remainders in the Cesàro sense.
Gregory Debruyne, Jasson Vindas

Inverse Function Theorems for Generalized Smooth Functions

Abstract
Generalized smooth functions are a possible formalization of the original historical approach followed by Cauchy, Poisson, Kirchhoff, Helmholtz, Kelvin, Heaviside, and Dirac to deal with generalized functions. They are set-theoretical functions defined on a natural non-Archimedean ring, and include Colombeau generalized functions (and hence also Schwartz distributions) as a particular case. One of their key property is the closure with respect to composition. We review the theory of generalized smooth functions and prove both the local and some global inverse function theorems.
Paolo Giordano, Michael Kunzinger

The Stochastic LQR Optimal Control with Fractional Brownian Motion

Abstract
We consider the stochastic linear quadratic optimal control problem where the state equation is given by a stochastic differential equation of the Itô–Skorokhod type with respect to fractional Brownian motion. The dynamics are driven by strongly continuous semigroups and the cost functional is quadratic. We use the fractional isometry mapping defined between the space of square integrable stochastic processes with respect to fractional Gaussian white noise measure and the space of integrable stochastic processes with respect to the classical Gaussian white noise measure. By this mapping we transform the fractional state equation to a state equation with Brownian motion. Applying the chaos expansion approach, we can solve the optimal control problem with respect to a state equation with the standard Brownian motion. We recover the solution of the original problem by the inverse of the fractional isometry mapping. Finally, we consider a general form of the state equation related to the Gaussian colored noise, we study the control problem, a system with an algebraic constraint and a particular example involving generalized operators from the Malliavin calculus.
Tijana Levajković, Hermann Mena, Amjad Tuffaha

Multi-soliton Collision for Essentially Nonintegrable Equations

Abstract
We describe an approach to construct multi-soliton asymptotic solutions for essentially nonintegrable equations. As for equations, we assume the existence of an exact soliton type solution with an exponential decay rate, however we do not suppose the smallness of the interacting waves. The general idea is realized in the cases of two and three waves and for the gKdV-4 equation with small dispersion.
George Omel’yanov

Microlocal Solvability and Subellipticity of Several Classes of Pseudodifferential Operators with Involutive Characteristics

Abstract
In this paper pseudodifferential operators with involutive characteristics are considered in two different cases: elliptic subprincipal symbol and subprincipal symbol being a symbol of principal type near some characteristic point (i.e., vanishing at a part of the characteristic set). We prove (micro)local non-solvability results as well as subelliptic estimates in the second case when the loss of regularity is of the following type: \( \frac{{2k + 1}} {{k + 1}} = 1 + \frac{k} {{k + 1}},\,k \in {\Bbb N} \). For the operators of subprincipal type interesting results were proved recently by N. Dencker.
P. R. Popivanov

An Observation of the Subspaces of

Abstract
The spaces \( \mathcal{S}^{\prime}/\mathcal{P} \) equipped with the quotient topology and \( \mathcal{S}^{\prime}_\infty \) equipped with the weak-* topology are known to be homeomorphic, where \( \mathcal{P} \) denotes the set of all polynomials. The proof is a combination of the fact in the textbook by Treves and the well-known bipolar theorem. In this paper by extending slightly the idea employed in [5], we give an alternative proof of this fact and then we extend this proposition so that we can include some related function spaces.
Yoshihiro Sawano

Ultradifferentiable Functions of Class and Microlocal Regularity

Abstract
We study spaces of ultradifferentiable functions which contain Gevrey classes. Although the corresponding defining sequences do not satisfy Komatsu’s condition (M.2)’, we prove appropriate continuity properties under the action of (ultra)differentiable operators. Furthermore, we study convenient localization procedure which leads to the concept of wave-front set with respect to our regularity conditions. As an application, we identify singular supports of suitable spaces of ultradifferentiable functions as standard projections of intersections/unions of wave-front sets.
Nenad Teofanov, Filip Tomić

Matrix Parameterized Pseudo-differential Calculi on Modulation Spaces

Abstract
We consider a broad matrix parameterized family of pseudo-differential calculi, containing the usual Shubin’s family of pseudo-differential calculi, parameterized by real numbers. We show that continuity properties in the framework of modulation space theory, valid for the Shubin’s family extend to the broader matrix parameterized family of pseudo-differential calculi.
Joachim Toft

An Application of Internal Objects to Microlocal Analysis in Generalized Function Algebras

Abstract
We illustrate the use of internal objects in the nonlinear theory of generalized functions by means of an application to microlocal analysis in Colombeau algebras.
H. Vernaeve

Rotation Invariant Ultradistributions

Abstract
We prove that an ultradistribution is rotation invariant if and only if it coincides with its spherical mean. For it, we study the problem of spherical representations of ultradistributions on ℝn. Our results apply to both the quasianalytic and the non-quasianalytic case.
Đorđe Vučković, Jasson Vindas

Eigenvalue Problems of Toeplitz Operators in Bargmann–Fock Spaces

Abstract
In this paper we will derive a formula for the eigenvalues of Toeplitz operators with polyradial symbols in Bargmann–Fock spaces. Moreover we will clarify the relationship between Toeplitz operators in Bargmann–Fock spaces and Daubechies operators in L2(ℝn). As application of our results, we will give a new proof of the formula of the eigenvalues of Daubechies operators with polyradial symbols.
Kunio Yoshino
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