Fourier Transform over the Spaces S′k

The Fourier Transform in Weighted Lorentz Spaces

Abstract Necessary conditions and sufficient conditions on weights u and w are given for the Fourier transform F to be bounded as a map between the Lorentz spaces Γq(w) and Λp(u). This may be viewed as a weighted extension of a result of Jodeit and Torchinsky on operators of type (1,∞) and (2, 2). In the case 0 < p ≤ 2 = q, the necessary and sufficient conditions are equivalent and give a simpl...

The Fourier Transform on Symmetric Spaces and Applications 1. the Fourier Transform

The symmetric spaces the title refers to are the spaces X = G=K where G is a connected semisimple Lie group with nite center and K is a maximal compact subgroup. The Fourier transform on X is deened by means of the Iwasawa decomposition G = NAK of G where N is nilpotent and A abelian. Let g; n; a; k denote the corresponding Lie algebras. We also need the group M = K A ; the centralizer of A in ...

Quantum Fourier Transform over Galois Rings 1

Galois rings are regarded as “building blocks” of a finite commutative ring with identity. There have been many papers on classical error correction codes over Galois rings published. As an important warm-up before exploring quantum algorithms and quantum error correction codes over Galois rings, we study the quantum Fourier transform (QFT) over Galois rings and prove it can be efficiently pref...

Regularity of the Fourier transform on spaces of homogeneous distributions

1. Introduction. In this paper we consider the action of the Fourier transform on spaces of regular homogeneous distributions on R n , by which we mean to include the nonisotropic (sometimes called quasihomogeneous [6]) case. Specifically , consider the group of dilations of R n with positive scaling exponents dn x n) for δ > 0 (we will write this map as δ • x for short). Dilation induces scali...

The Fourier Transform and Equations over Finite Abelian Groups