Fourier Hyperfunctions Research Articles

Stability of Cubic Functional Equation in the Spaces of Generalized Functions

We reformulate the following additive functional equation with n-independent variables as the equation for the spaces of generalized functions. Making use of the fundamental solution of the heat equation we solve the general solutions and the stability problems of this equation in the spaces of tempered distributions and Fourier hyperfunctions. Moreover, using the regularizing functions, we extend these results to the space of distributions. 2000 MSC: 39B82; 46F05.

Stability of a quadratic Jensen type functional equation in the spaces of generalized functions

Making use of the fundamental solution of the heat equation we find the solution and prove the stability theorem of the quadratic Jensen type functional equation 9 f ( x + y + z 3 ) + f ( x ) + f ( y ) + f ( z ) = 4 [ f ( x + y 2 ) + f ( y + z 2 ) + f ( z + x 2 ) ] in the spaces of Schwartz tempered distributions and Fourier hyperfunctions.

Convolution and multiplication operators in Fourier hyperfunctions

We characterize the multiplication and convolution operators in the space ℱ′ of Fourier hyperfunctions. This generalizes the results of Schwartz on the multiplication and convolution operators on the space 𝒮′ of tempered distributions to the space ℱ′ of Fourier hyperfunctions.

Hermite functions and weighted spaces of generalized functions

We prove that the Hermite functions are an absolute Schauder basis for many weighted spaces of (ultra)differentiable functions and ultradistributions including the space of Fourier hyperfunctions. The coefficient spaces are also determined.

Stability of a Jensen type equation in the space of generalized functions

We reformulate and solve the stability problem of a Jensen type functional equation 3 f ( x + y + z 3 ) + f ( x ) + f ( y ) + f ( z ) − 2 f ( x + y 2 ) − 2 f ( y + z 2 ) − 2 f ( z + x 2 ) = 0 , in the spaces of some generalized functions such as tempered distributions and Fourier hyperfunctions.

A distributional version of functional equations and their stabilities

Generalizing the results in [J. Math. Anal. Appl. 286 (2003) 177–186; J. Math. Anal. Appl. 295 (2004) 107–114; Arch. Math., to appear; J. Math. Anal. Appl. 299 (2004) 578–586] that consider the Hyers–Ulam stability problems of several functional equations in the spaces of the Schwartz tempered distributions and the Fourier hyperfunctions we consider the stability problems of the functional equations in the space of distributions.

Hyers-Ulam stability theorems for Pexider equations in the space of Schwartz distributions

Making use of the fundamental solution of the heat equation we prove stabilities of the Pexider equation and the Pexider-exponential equation in spaces of generalized functions such as the Schwartz tempered distributions, Fourier hyperfunctions and Gelfand-Shilov generalized functions.

CONVOLUTORS FOR THE SPACE OF FOURIER HYPERFUNCTIONS

We define the convolutions of Fourier hyperfunctions and show that every strongly decreasing Fourier hyperfunction is a convolutor for the space of Fourier hyperfunctions and the converse is true. Also we show that there are no differential operator with constant coefficients which have a fundamental solution in the space of strongly decreasing Fourier hyperfunctions. Lastly we show that the space of multipliers for the space of Fourier hyperfunctions consists of analytic functions extended to any strip in which are estimated with a special exponential function exp.

Asymptotic hyperfunctions, tempered hyperfunctions, and asymptoticexpansions

We introduce new subclasses of Fourier hyperfunctions of mixed type, satisfying polynomial growth conditions at infinity, and develop their sheaf and duality theory. We use Fourier transformation and duality to examine relations of these asymptotic and tempered hyperfunctions to known classes of test functions and distributions, especially the Gel′fand‐Shilov spaces. Further it is shown that the asymptotic hyperfunctions, which decay faster than any negative power, are precisely the class that allows asymptotic expansions at infinity. These asymptotic expansions are carried over to the higher‐dimensional case by applying the Radon transformation for hyperfunctions.

Hyers–Ulam–Rassias stability of Cauchy equation in the space of Schwartz distributions

We reformulate and prove the Hyers–Ulam–Rassias stability of Cauchy equation in the space of Schwartz tempered distributions and Fourier hyperfunctions.

DENSENESS OF TEST FUNCTIONS IN THE SPACE OF EXTENDED FOURIER HYPERFUNCTIONS

We research properties of analytic functions which are exponentially decreasing or increasing. Also we show that the space of test functions is dense in the space of extended Fourier hyper-functions, and that the Fourier transform of the space of extended Fourier hyperfunctions into itself is an isomorphism and Parseval's inequality holds.

THE SPACE OF FOURIER HYPERFUNCTIONS AS AN INDUCTIVE LIMIT OF HILBERT SPACES

We research properties of the space of measurable functions square integrable with weight exp<TEX>$2\nu <TEX>$\mid$</TEX>x<TEX>$\mid$</TEX>$</TEX>, and those of the space of Fourier hyperfunctions. Also we show that the several embedding theorems hold true, and that the Fourier-Lapace operator is an isomorphism of the space of strongly decreasing Fourier hyperfunctions onto the space of analytic functions extended to any strip in <TEX>$C^n$</TEX> which are estimated with the aid of a special exponential function exp(<TEX>$\mu$</TEX>｜x｜).

Stability of Jensen equations in the space of generalized functions

Making use of heat kernel, we prove stabilities of the Jensen and Jensen–Pexider equations in a space of generalized functions like the spaces of tempered distributions and Fourier hyperfunctions.

Relativistic quantum field theory with a fundamental length

Since there are indications (from string theory and concrete models) that one must consider relativistic quantum field theories with a fundamental length the question of a suitable framework for such theories arises. It is immediately evident that quantum field theory in terms of tempered distributions and even in terms of Fourier hyperfunctions cannot meet the (physical) requirements. We argue that quantum field theory in terms of ultra-hyperfunctions is a suitable framework. For this we propose a set of axioms for the fields and for the sequence of vacuum expectation values of the fields, prove their equivalence, and we give a class of models (analytic, but not entire functions of free fields).

Bochner Schwartz Type Theorem for Conditionally Positive Definite Fourier Hyperfunctions

We prove the Bochner–Schwartz type theorem for conditionally positive definite Fourier hyperfunctions which generalizes the result of Gelfand-Vilenkin in their treatise Generalized functions, vol. IV for distributions.

Hyperfunction Quantum Field Theory: Localized Fields Without Localized Test Functions

This note addresses the problem of localization in quantum field theory; more specifically we contribute to the ongoing discussion about the most appropriate concept of localization which one should use in relativistic quantum field theory: through localized test functions or through the fields directly without localized test functions. In standard quantum field theory, i.e., in relativistic quantum field theory in terms of tempered distributions according to Garding and Wightman, this is done through localized test functions. In hyperfunction quantum field theory (HFQFT), i.e., relativistic quantum field theory in terms of Fourier hyperfunctions this is done through the fields themselves. In support of the second approach we show here that it has a much wider range of applicability. It can even be applied to relativistic quantum field theories which do not admit compactly supported test functions at all. In our construction of explicit models we rely on basic results from the theory of quasi-analytic functions.

Wiener-Type Tauberian Theorems for Fourier Hyperfunctions

Two Wiener-type Tauberian theorems concerning Fourier hyperfunctions are proved and commented. It is shownt that the shift asymptotics ( S -asymptotics) of a hyperfunction f is determined by the ordinary asymptotics of (f \star \mathcal K) (x) as x \rightarrow \infty , where \mathcal K is Hörmaner’s kernel. Moreover, Wiener-type theorems are used for the asympthotic analysis of solutions to some (pseudo-)differential equations.

Support and kernel theorem for Fourier hyperfunctions

Support and kernel theorem for Fourier hyperfunctions

Hyperfunction quantum field theory: Analytic structure, modular aspects, and local observable algebras

This paper addresses the following problem of relativistic quantum field theory: Given a relativistic quantum field, construct a net of local observable algebras over space–time with “natural” properties. A few years ago we started a project which suggests to look at this problem in the framework of relativistic quantum field theory in terms of Fourier hyperfunctions. Accordingly we present the relevant analyticity results, some modular aspects of our theory for the Bisognano–Wichman argument, and a concrete suggestion for the definition of the local observable algebras. Finally, we construct a class of models of hyperfunction quantum fields for which the algebras of bounded operators assigned to nonempty regions in space–time are not trivial. It is remarkable that our models are not tempered quantum fields and do not admit “test functions” of compact support.

Almost periodic hyperfunctions

We characterize the almost periodic hyperfunctions by showing that the following statements are equivalent for any bounded hyperfunction T . (i) T is almost periodic. (ii) T ∗ φ ∈ Cap for every φ ∈ F . (iii) There are two functions f, g ∈ Cap and an infinite order differential operator P such that T = P (D2)f + g. (iv) The Gauss transform u(x, t) = T ∗ E(x, t) of T is almost periodic for every t > 0. Here Cap is the space of almost periodic continuous functions, F is the Sato space of test functions for the Fourier hyperfunctions, and E(x, t) is the heat kernel. This generalizes the result of Schwartz on almost periodic distributions and that of Cioranescu on almost periodic (non-quasianalytic) ultradistributions to the case of hyperfunctions.