Vector valued Fourier hyperfunctions

Fourier transformation of Sato’s hyperfunctions

A new generalized function space in which all Gelfand-Shilov classes S ′0 α (α > 1) of analytic functionals are embedded is introduced. This space of ultrafunctionals does not possess a natural nontrivial topology and cannot be obtained via duality from any test function space. A canonical isomorphism between the spaces of hyperfunctions and ultrafunctionals on R is constructed that extends the...

Structural Theorems for Families of Fourier Hyperfunctions

Operator-Valued Bochner Theorem, Fourier Feature Maps for Operator-Valued Kernels, and Vector-Valued Learning

This paper presents a framework for computing random operator-valued feature maps for operator-valued positive definite kernels. This is a generalization of the random Fourier features for scalar-valued kernels to the operator-valued case. Our general setting is that of operator-valued kernels corresponding to RKHS of functions with values in a Hilbert space. We show that in general, for a give...

On the Vector Valued Fourier Transform and Compatibility of Operators*

is the dual group of G, and p ′ the conjugate exponent of p. An operator T between Banach spaces X and Y is said to be compatible with the Fourier transform F G if F G ⊗ T : Lp(G)⊗X → Lp′ (G ′ )⊗ Y admits a continuous extension [F , T ] : [Lp(G), X ] → [Lp′ (G ′ ), Y ]. We show that if G is topologically isomorphic with R×Z×F, where l and m are nonnegative integers and F is a compact group with...

On localization properties of Fourier transforms of hyperfunctions

In [Adv. Math. 196 (2005) 310–345] the author introduced a new generalized function space U(Rk) which can be naturally interpreted as the Fourier transform of the space of Sato’s hyperfunctions on Rk. It was shown that all Gelfand–Shilov spaces S′0 α (R k) (α > 1) of analytic functionals are canonically embedded in U(Rk). While the usual definition of support of a generalized function is inappl...