Dohan Kim

Characterizations of the Gelfand-Shilov spaces via Fourier transforms

We give symmetric characterizations, with respect to the Fourier transformation, of the Gelfand-Shilov spaces of (generalized) type S and type W. These results explain more clearly the invariance of these spaces under the Fourier transformations.

Representation of quasianalytic ultradistributions

AbstractWe give the following representation theorem for a class containing quasianalytic ultradistributions and all the non-quasianalytic ultradistributions: Every ultradistribution in this class can be written as

An example of nonuniqueness of the Cauchy problem for the heat equation

u = P(\Delta )g(x) + h(x)

A quasianalytic singular spectrum with respect to the Denjoy-Carleman class

whereg(x) is a bounded continuous function,h(x) is a bounded real analytic function andP(d/dt) is an ultradifferential operator. Also, we show that the boundary value of every heat function with some exponential growth condition determines an ultradistribution in this class. These results generalize the theorem of Matsuzawa [M] for the above class of quasianalytic ultradistributions and partially solve a question of A. Kaneko [Ka]. Our interest lies in the quasianalytic case, although the theorems do not exclude non-quasianalytic classes.

Antosik-Mikusinski Matrix Convergence Theorem in Quantum Logics

Every periodic hyperfunction is a bounded hyperfunction and can be represented as an infinite sum of derivatives of bounded continuous periodic functions. Also, Fourier coefficients cα of periodic hyperfunctions are of infraexponential growth in Rn, i.e., cα 0 and every α ∈ Zn. This is a natural generalization of the polynomial growth of the Fourier coefficients of distributions. To show these we introduce the space BLp of hyperfunctions of Lp growth which generalizes the space D′ Lp of distributions of Lp growth and represent generalized functions as the initial values of smooth solutions of the heat equation.

Gevrey and Analytic Convergence of Picard'S Successive Approximations

We give an example of non trivial solution of the homogeneous Cauchy problem of the heat equation, which is, for each t, bounded in the space variables.

Stability of Jensen-Type Functional Equations on Restricted Domains in a Group and Their Asymptotic Behaviors

We give an elementary proof of the equivalence of the original definition of Schwartz and our characterization for the Schwartz space S. The new proof is based on the Landau inequality concerning the estimates of derivatives. Applying the same method, as an application, we give a better symmetric characterization of the Beurling–Björck space of test functions for tempered ultradistributions with respect to Fourier transform without conditions on derivatives.

Hyers-Ulam stability on a generalized quadratic functional equation in distributions and hyperfunctions

Making use of the FBI (Fourier-Bros-Iagolnitzer) transforms we simplify the quasianalytic singular spectrum for the Fourier hyperfunctions, which was defined for distributions by Hormander as follows; for any Fourier hyperfunction u , ( x 0 , ξ 0 ) does not belong to the quasianalytic singular spectrum W F M (u) if and only if there exist positive constants C, γ and N , and a neighborhood of x 0 and a conic neighborhood Г of ξ 0 such that for all x ∈ U , |ξ| ∈ Γ and |ξ| ≥ N , where M(t) is the associated function of the defining sequence M p . This result simplifies Hormander’s definition and unify the singular spectra for the C ∞ class, the analytic class and the Denjoy-Carleman class, both quasianalytic and nonquasianalytic.