Bilinear maps and convolutions
نویسنده
چکیده
Let X,Y, Z be Banach spaces and let u : X×Y → Z be a bounded bilinear map. Given a locally compact abelian group G , and two functions f ∈ L(G,X) and g ∈ L(G,Y ), we define the u -convolution of f and g as the Z -valued function f ∗u g(t) = ∫ G u(f(t− s), g(s))dμG(s) where dμG stands for the Haar measure on G . We define the concepts of vector-valued approximate identity and summability kernel associated to a bounded bilinear map, showing the corresponding approximation result in this setting. A Haussdorf-Young type result associated to a bounded bilinear map is also presented under certain assumptions on the Banach space X .
منابع مشابه
On continuous cohomology of locally compact Abelian groups and bilinear maps
Let $A$ be an abelian topological group and $B$ a trivial topological $A$-module. In this paper we define the second bilinear cohomology with a trivial coefficient. We show that every abelian group can be embedded in a central extension of abelian groups with bilinear cocycle. Also we show that in the category of locally compact abelian groups a central extension with a continuous section can b...
متن کاملBilinear algorithms for discrete cosine transforms of prime lengths
Abstract: This paper presents a strategy to design bilinear discrete cosine transform (DCT) algorithms of prime lengths. We show that by using multiplicative groups of integers, one can identify and arrange the computation as a pair of convolutions. When the DCT length p is such that (p−1)/2 is odd, the computation uses two (p−1)/2 point cyclic convolutions. When (p − 1)/2 = 2q with m > 0 and q...
متن کاملArens regularity of bilinear maps and Banach modules actions
Let $X$, $Y$ and $Z$ be Banach spaces and $f:Xtimes Y longrightarrow Z$ a bounded bilinear map. In this paper we study the relation between Arens regularity of $f$ and the reflexivity of $Y$. We also give some conditions under which the Arens regularity of a Banach algebra $A$ implies the Arens regularity of certain Banach right module action of $A$ .
متن کاملComparing 511 keV Attenuation Maps Obtained from Different Energy Mapping Methods for CT Based Attenuation Correction of PET Data
Introduction: The advent of dual-modality PET/CT scanners has revolutionized clinical oncology by improving lesion localization and facilitating treatment planning for radiotherapy. In addition, the use of CT images for CT-based attenuation correction (CTAC) decreases the overall scanning time and creates a noise-free attenuation map (6map). CTAC methods include scaling, s...
متن کاملA Novel Identity-based Group Signature Scheme from Bilinear Maps
We propose an identity-based group signature scheme from bilinear maps. The scheme has the security properties of group signatures. Our scheme is based on the CDHP assumption and bilinear maps between groups. The size of the group pubic key and the length of the signature are independent on the number of group members. Furthermore, a group member can sign many message using the same key pair.
متن کامل