Fourier Hyperfunctions Research Articles

Asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions

We study Fourier and Laplace transforms for Fourier hyperfunctions with values in a complex locally convex Hausdorff space. Since any hyperfunction with values in a wide class of locally convex Hausdorff spaces can be extended to a Fourier hyperfunction, this gives simple notions of asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions, which improves the existing models of Komatsu, B\"aumer, Lumer and Neubrander and Langenbruch.

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.

Analyticity of solutions to a quadratic system of Schrödinger equations without mass resonance condition

We consider the Cauchy problem for a quadratic system of Schrödinger equations in space dimensions n≥4 without mass resonance condition. Analytic smoothing effect for the quadratic system of Schrödinger equations under the mass resonance condition for data which satisfy exponentially decaying condition has been studied in [14]. In this study, we construct global solutions in the analytic Hardy space via the phase modulation operator, based on the scale critical Sobolev space H˙n/2−2 with data like the test functions of Fourier hyperfunctions, under the condition m1≤m2≤(1+4/n)m1 between two masses m1 and m2.

On quasianalytic classes of Gelfand–Shilov type. Parametrix and convolution

We develop a convolution theory for quasianalytic ultradistributions of Gelfand–Shilov type. We also construct a special class of ultrapolynomials, and use it as a base for the parametrix method in the study of new topological and structural properties of several quasianalytic spaces of functions and ultradistributions. In particular, our results apply to Fourier hyperfunctions and Fourier ultra-hyperfunctions.

On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth

We study function spaces consisting of analytic functions with fast decay on horizontal strips of the complex plane with respect to a given weight function. Their duals, so called spaces of (ultra)hyperfunctions of fast growth, generalize the spaces of Fourier hyperfunctions and Fourier ultrahyperfunctions. An analytic representation theory for their duals is developed and applied to characterize the non-triviality of these function spaces in terms of the growth order of the weight function. In particular, we show that the Gelfand-Shilov spaces of Beurling type $\mathcal{S}^{(p!)}_{(M_p)}$ and Roumieu type $\mathcal{S}^{\{p!\}}_{\{M_p\}}$ are non-trivial if and only if $$ \sup_{p \geq 2}\frac{(\log p)^p}{h^pM_p} < \infty, $$ for all $h > 0$ and some $h > 0$, respectively. We also study boundary values of holomorphic functions in spaces of ultradistributions of exponential type, which may be of quasianalytic type.

Hyperfunction semigroups

We analyze Fourier hyperfunction and hyperfunction semigroups with non-densely defined generators and their connections with local convoluted $C$-semigroups. Structural theorems and spectral characterizations give necessary and sufficient conditions for the existence of such semigroups generated by a closed not necessarily densely defined operator $A$.

Stationary hyperfunctional random process

The theory of a generalized random process, i.e. random process in distributions was introduced by Gel’fand and Vilenkin. They made use of the Bochner–Schwartz type theorems for (conditionally) positive distributions, Schwartz kernel theorem and theorems on translation-invariant positive-definite bilinear functionals. We introduced a generalized random process in (Fourier) hyperfunctions by making use of our parallel results of the Bochner–Schwartz type theorems for (conditionally) positive Fourier hyperfunctions, Schwartz kernel theorem, and theorems on translation-invariant positive-definite bilinear functionals in Fourier hyperfunctions. We have an important guiding theme to compare the generalized random process in distributions in and our results in the generalized random process in (Fourier) hyperfunctions as follows; 1. the measures appearing in the generalized random process in distributions are tempered or polynomially increasing. 2. the measures appearing in the generalized random process in (Fourier) hyperfunctions are of infraexponential growth.

Stability of quadratic functional equations in tempered distributions

We reformulate the following quadratic functional equation: as the equation for generalized functions. Using the fundamental solution of the heat equation, we solve the general solution of this equation and prove the Hyers-Ulam stability in the spaces of tempered distributions and Fourier hyperfunctions.

Stability of an additive 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 in Generalized Functions

We consider the following additive functional equation with n‐independent variables: in the spaces of generalized functions. Making use of the heat kernels, we solve the general solutions and the stability problems of the above equation in the spaces of tempered distributions and Fourier hyperfunctions. Moreover, using the mollifiers, we extend these results to the space of distributions.

Stability of Trigonometric Functional Equations in Generalized Functions

We consider the Hyers-Ulam stability of a class of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions, Fourier hyperfunctions, and Gelfand generalized functions.

THE STABILITY OF THE SINE AND COSINE FUNCTIONAL EQUATIONS IN SCHWARTZ DISTRIBUTIONS

We prove the Hyers-Ulam stability of the sine and cosine functional equations in the spaces of generalized functions such as Schwartz distributions, Fourier hyperfunctions, and Gelfand generalized functions.

Stability of Quartic Functional Equations in the Spaces of Generalized Functions

We consider the general solution of quartic functional equations and prove the Hyers-Ulam-Rassias stability. Moreover, using the pullbacks and the heat kernels we reformulate and prove the stability results of quartic functional equations in the spaces of tempered distributions and Fourier hyperfunctions.

A proof of Paley–Wiener theorem for Fourier hyperfunctions with support in a proper convex cone by the heat kernel method

In this article we shall give a new proof of Paley–Wiener theorem for Fourier hyperfunctions supported by a proper convex cone by the heat kernel method.

Stability of a Quadratic Functional Equation in the Spaces of Generalized Functions

Making use of the pullbacks, we reformulate the following quadratic functional equation: in the spaces of generalized functions. Also, using the fundamental solution of the heat equation, we obtain the general solution and prove the Hyers-Ulam stability of this equation in the spaces of generalized functions such as tempered distributions and Fourier hyperfunctions.

Stability of an Euler–Lagrange–Rassias equation in the spaces of generalized functions

Making use of the fundamental solution of the heat equation we reformulate and prove the stability theorem of a special case of the Euler–Lagrange–Rassias functional equation in the spaces of tempered distributions and Fourier hyperfunctions.

Characterization of Fourier hyperfunctions by solutions of the Hermite heat equation

Let E(x, ξ, t) be the Mehler kernel and let U(x, t) be the solution, with some appropriate growth estimate (*), of the Hermite heat equation in R×(0, T). Then there exists a unique Fourier hyperfunction u such that U(x, t)=⟨ u(ξ), E(x, ξ, t)⟩. Conversely for any Fourier hyperfunction u, U(x, t):=⟨ u(ξ), E(x, ξ, t)⟩ is the smooth solution of the Hermite heat equation in R×(0, T) satisfying the growth estimate (*).

Stability for quadratic functional equation in the spaces of generalized functions

In this paper, we consider the general solution of quadratic functional equation f ( a x + y ) + f ( a x − y ) = f ( x + y ) + f ( x − y ) + 2 ( a 2 − 1 ) f ( x ) for any integer a with a ≠ − 1 , 0 , 1 . Moreover we reformulate and prove the Hyers–Ulam–Rassias stability theorem of the above equation in the spaces of tempered distributions and Fourier hyperfunctions. The generalized Hyers–Ulam stability originated from the Th.M. Rassias's stability theorem that appeared in his paper [Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300].

Generalized Fourier expansion in kernels of convolution operators on Fourier hyperfunctions

We prove that the kernels of surjective convolution operators on Fourier hyperfunctions (and on Fourier ultra-hyperfunctions) admit a basis of exponential solutions. The corresponding coefficient spaces are explicitly determined.

Right inverses for partial differential operators\cr on Fourier hyperfunctions

We characterize the partial differential operators $P(D)$ admitting a continuous linear right inverse in the space of Fourier hyperfunctions by means of a dual $( \overline{\Omega})$-type estimate valid for the bounded holomorphic functions on the charac