In this article, we shall discuss some recent developments and applications of the local spectral theory for linear operators on Banach spaces. Special emphasis will be given to those parts of operator theory, where spectral theory, harmonic analysis, and the theory of Banach algebras overlap and interact. Along this line, we shall present the recent progress of the theory of quotients and rest...

r∈Z f̂(r)e In the last lecture, we proved Fejér’s theorem f ∗ kn → f where the ∗ denotes convolution and kn (Fejér kernels) are trignometric polynomials that satisfy 1. kn ≥ 0 2. ∫ T kn = 1 3. kn(s) → 0 uniformly as n→∞ outside [−δ, δ] for any δ > 0. If X is a finite abelian group, then the space of all functions f : X → C forms an algebra with the operations (+, ∗) where + is the usual pointwis...

We introduce the notion of the Fourier and Fouier-Stieltjes algebra of a topological ∗-semigroup and show that these are commutative Banach algebras. For a class of foundation semigroups, we show that these are preduals of von Neumann algebras. 1. Definitions and Notations Let S be a locally compact topological semigroup and M(S) be the Banach algebra of all bounded regular Borel measures μ on ...

the product σ-algebra. Let A : E × E → E be A(x1, x2) = x1 + x2. For ν1, ν2 ∈ P(E), the convolution of ν1 and ν2 is the pushforward of the product measure ν1 × ν2 by A: ν1 ∗ ν2 = A∗(ν1 × ν2). The convolution ν1 ∗ ν2 is an element of P(E). Let I = R≥0. A convolution semigroup is a family (νt)t∈I of elements of P(E) such that for s, t ∈ I, νs+t = νs ∗ νt. From this, it turns out that μ0 = δ0. 2 A...

The success of deep learning often derives from well-chosen operational building blocks. In this work, we revise the temporal convolution operation in CNNs to better adapt it to text processing. Instead of concatenating word representations, we appeal to tensor algebra and use low-rank n-gram tensors to directly exploit interactions between words already at the convolution stage. Moreover, we e...

For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C0(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak∗-continuous. Given a discrete semigroup S, the convolution algebra l(S) also carries a coproduct. In this paper we examine preduals for l(S) making b...

Given a (reduced) locally compact quantum group A, we can consider the convolution algebra L(A) (which can be identified as the predual of the von Neumann algebra form of A). It is conjectured that L(A) is operator biprojective if and only if A is compact. The “only if” part always holds, and the “if” part holds for Kac algebras. We show that if the splitting morphism associated with L(A) being...

Morphological semigroups for functions on length spaces have recently studied [2], whose basic ingredients are the convolution in the (max,+)-algebra (or supremal convolution), the metric distance and a convex shape function. Links with Hamilton–Jacobi PDE on length spaces appears also naturally. These semigroups lead to powerful scale-space properties for multiscale filtering, regularization a...

We present extensions to the computer algebra and multimedia environmentMathematica to enable students to investigate signal processing theory. Students interact with the extensions at two levels. The lower level is a set of Mathematica programs called the Signal Processing Packages (SPP) that implement signals and systems not found in the core of Mathematica. The higher level is a set of inter...

Using the algebra of dioids, we further examine the analogy between timed event graphs and conventional linear systems by showing that some periodic inputs of the former behave M cosine inputs for the latter. In particular, we give a meaning to such notions M “phsse shift” and “amplification gain”, which allow ua to talk about the Black and Bode plots for d i m e t e event systems. In this theo...