Mathematics

In this paper we define a new class of continuous fractional wavelet transform (CFrWT) and study its properties in Hardy space and Morrey space. The theory developed generalize and complement some of already existing results.

We link conditional weak mixing and ergodicity of the tensor product in Riesz spaces. In particular, we characterise conditional weak mixing of a conditional expectation preserving system by the ergodicity of its tensor product with itself or other ergodic systems. In order to achieve this we characterise the components of the weak order units in the tensor product of two Dedekind complete Riesz spaces with weak order units.

In this paper, we study stability of M -compactness for l p sum of Banach spaces for 1≤p<∞ . We also obtain a characterization of M -compact sets in terms of statistically maximizing sequence, a notion which is weaker than a maximizing sequence. Moreover, we introduce the notion of I - M -compactness of a bounded subset M of a normed linear space X with respect to an ideal I and show that it is equivalent to M -compactness for non-trivial admissible ideals.

For m?�R we introduce the symbol classes S m , m?�R , consisting of smooth functions ? on R 2d such that | ??α ?(z)|??C α (1+|z | 2 ) m/2 , z??R 2d , and we show that can be characterized by an intersection of different types of modulation spaces. In the case m=0 we recapture the Hörmander class S 0 0,0 that can be obtained by intersection of suitable Besov spaces as well. Such spaces contain the Shubin classes ? m ? , 0<??? , and can be viewed as their limit case ?=0 . We exhibit almost diagonalization properties for the Gabor matrix of ? -pseudodifferential operators with symbols in such classes, extending the characterization proved by Gröchenig and Rzeszotnik. Finally, we compute the Gabor matrix of a Born-Jordan operator, which allows to prove new boundedness results for such operators.

Vilenkin groups, introduced by F. Ya Vilenkin, form a class of locally compact abelian groups. The present paper consists of the characterization of Parseval frame multiwavelets associated to multiresolution analysis (MRA) in the Vilenkin group. Further, we introduce the pseudo-scaling function along with a class of generalized low pass filters and study their properties in Vilenkin group.

The integral representation of Choquet operators defined on a space C(X) is established by using the Choquet-Bochner integral of a real-valued function with respect to a vector capacity.

We introduce a~\textit{Choquet-Sugeno-like operator} generalizing many operators for bounded functions and monotone measures from the literature, e.g., Sugeno-like operator, Lovász and Owen measure extensions, $\rF$-decomposition integral with respect to a~partition decomposition system, and others. The new operator is based on the concepts of dependence relation and conditional aggregation operators, but it does not depend on t -level sets. We also provide conditions for which the Choquet-Sugeno-like operator coincides with some Choquet-like integrals defined on finite spaces and appeared recently in the literature, e.g. reverse Choquet integral, d -Choquet integral, $\rF$-based discrete Choquet-like integral, some version of $C_{\rF_1\rF_2}$-integral, CC -integrals (or Choquet-like Copula-based integral) and discrete inclusion-exclusion integral. Some basic properties of the Choquet-Sugeno-like operator are studied.

We analyze the fine structure of Clark measures and Clark isometries associated with two-variable rational inner functions on the bidisk. In the degree (n,1) case, we give a complete description of supports and weights for both generic and exceptional Clark measures, characterize when the associated embedding operators are unitary, and give a formula for those embedding operators. We also highlight connections between our results and both the structure of Agler decompositions and study of extreme points for the set of positive pluriharmonic measures on 2-torus.

We classify the closed ideals of bounded operators acting on the Banach spaces ( ⨁ n∈N ℓ n 2 ) c 0 ⊕ c 0 (Γ) and ( ⨁ n∈N ℓ n 2 ) ℓ 1 ⊕ ℓ 1 (Γ) for every uncountable cardinal Γ .

We describe a closed operator functional calculus in Banach modules over the group algebra L 1 (R) and illustrate its usefulness with a few applications. In particular, we deduce a spectral mapping theorem for operators in the functional calculus, which generalizes some of the known results. We also obtain an estimate for the spectrum of a perturbed differential operator in a certain class.