It is shown that an orthonormal wavelet basis for $L^2\left(ℝ\right)$ associated with a multiresolution is an unconditional basis for $L^p\left(ℝ\right)$, 1 < p < ∞, provided the father wavelet is bounded and decays sufficiently rapidly at infinity.

This paper deals with wavelet frames for a large class of distributions on euclidean n-space, including all compactly supported distributions. These representations characterize the global, local, and pointwise regularity of the distribution considered.

Let $E$ be a Banach (or quasi-Banach) space which is shift and scaling invariant (typically a homogeneous Besov or Sobolev space). We introduce a general definition of pointwise regularity associated with $E$, and denoted by $C_E^{\alpha }\left(x_0\right)$. We show how properties of $E$ are transferred into properties of $C_E^{\alpha }\left(x_0\right)$. Applications are given in multifractal analysis.

For a finite and positive measure space Ω,∑,μ characterizations of weak Cauchy sequences in $L_{\infty }(\mu ,X)$, the space of μ-essentially bounded vector-valued functions f:Ω → X, are presented. The fine distinction between Asplund and conditionally weakly compact subsets of $L_{\infty }(\mu ,X)$ is discussed.

We give new proofs that some Banach spaces have Pełczyński's property (V).

In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures $M_{ba}(\Sigma ,(X,Y))$. This generalizes a recent result of the author characterizing conditionally weakly compact subsets of the space of nuclear operator valued measures $M_{ba}(\Sigma ,₁(X,Y))$. This result has interesting applications in optimization and control theory as illustrated by several examples.

An analogue of the notion of uniformly separated sequences, expressed in terms of extremal functions, yields a necessary and sufficient condition for interpolation in Lp spaces of holomorphic functions of Paley-Wiener-type when 0 &lt; p ≤ 1, of Fock-type when 0 &lt; p ≤ 2, and of Bergman-type when 0 &lt; p &lt; ∞. Moreover, if a uniformly discrete sequence has a certain uniform non-uniqueness property with respect to any such Lp space (0 &lt; p &lt; ∞), then it is an interpolation...

In this article, we deal with weak convergence on sequences in real normed spaces, and weak* convergence on sequences in dual spaces of real normed spaces. In the first section, we proved some topological properties of dual spaces of real normed spaces. We used these theorems for proofs of Section 3. In Section 2, we defined weak convergence and weak* convergence, and proved some properties. By RNS_Real Mizar functor, real normed spaces as real number spaces already defined in the article [18],...

We prove invariance principles for partial sum processes in Besov spaces. This functional framework allows us to give a unified treatment of the step process and the smoothed process in the same parametric scale of function spaces. Our functional central limit theorems in Besov spaces hold for i.i.d. sequences and also for a large class of weakly dependent sequences.

It is proved that a Köthe sequence space is weakly orthogonal if and only if it is order continuous. Criteria for weak property ($\beta$) in Orlicz sequence spaces in the case of the Luxemburg norm as well as the Orlicz norm are given.

In this paper we consider the following Dirichlet problem for elliptic systems: $$\begin{array}{ccc}\hfill \overline{DA(x,u(x),Du(x\left)\right)}=& B(x,u(x),Du(x\left)\right),x\in \Omega ,u\left(x\right)=\hfill & \hfill 0,x\in \partial \Omega ,\end{array}$$ where $D$ is a Dirac operator in Euclidean space, $u\left(x\right)$ is defined in a bounded Lipschitz domain $\Omega$ in ${ℝ}^n$ and takes value in Clifford algebras. We first introduce variable exponent Sobolev spaces of Clifford-valued functions, then discuss the properties of these spaces and the related operator theory in these spaces. Using the Galerkin method, we obtain the existence of weak solutions to the scalar part of the above-mentioned...