Binary Operations in Metric Spaces Satisfying Side Inequalities

The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was done in the framework of iterated function systems and fractal theory. In this article we extract the main features of this association, and consider binary operations in metric spaces satisfying properties as idempotency and inequalities related to the distance between operated elements with the same right or left factor (side inequalities). Important examples are the logical disjunction and conjunction in the set of integers modulo 2 and the union of compact sets, besides the aforementioned fractal convolution. The operations described are called in the present paper convolutions of two elements of a metric space E. We deduce several properties of these associations, coming from the considered initial conditions. Thereafter, we define self-operators (maps) on E by using the convolution with a fixed component. When E is a Banach or Hilbert space, we add some hypotheses inspired in the fractal convolution of maps, and construct in this way convolved Schauder and Riesz bases, Bessel sequences and frames for the space.

Nonuniform biorthogonal wavelets on positive half line via Walsh Fourier transform

In this article, we introduce the notion of nonuniform biorthogonal wavelets on positive half line. We first establish the characterizations for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families in $$L^2({\mathbb {R}}^+)$$ L 2 ( R + ) . Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases.

Wavelet Numerical Solutions for a Class of Elliptic Equations with Homogeneous Boundary Conditions

This paper focuses on a method to construct wavelet Riesz bases with homogeneous boundary condition and use them to a kind of second-order elliptic equation. First, we construct the splines on the interval [0,1] and consider their approximation properties. Then we define the wavelet bases and illustrate the condition numbers of stiffness matrices are small and bounded. Finally, several numerical examples show that our approach performs efficiently.

Signal Representations Via SIP p-frames and SIP Bessel Multipliers in Separable Banach Spaces

Digital signals are often modeled as functions in Banach spaces, such as the ubiquitous [Formula: see text] spaces. The frame theory in Banach spaces induces flexible representations of signals due to the robustness and redundancy of frames. Nevertheless, the lack of inner product in general Banach spaces limits the direct representations of signals in Banach spaces under a given basis or frame. In this paper, we introduce the concept of semi-inner product (SIP) [Formula: see text]-Bessel multipliers to extend the flexibility of signal representations in separable Banach spaces, where [Formula: see text]. These multipliers are defined as composition of analysis operator of an SIP-I Bessel sequence, a multiplication with a fixed sequence and synthesis operator of an SIP-II Bessel sequence. The basic properties of the SIP [Formula: see text]-Bessel multipliers are investigated. Moreover, as special cases, characterizations of [Formula: see text]-Riesz bases related to signal representations are given, and the multipliers for [Formula: see text]-Riesz bases are discussed. We show that SIP [Formula: see text]-Bessel multipliers for [Formula: see text]-Riesz bases are invertible. Finally, the continuity of SIP [Formula: see text]-Bessel multipliers with respect to their parameters is investigated. The results theoretically show that the SIP [Formula: see text]-Bessel multipliers offer a larger range of freedom than frames on signal representations in Banach spaces.