In this paper, we will provide a simple method for starting with a given finite frame for an $n$-dimensional Hilbert space $mathcal{H}_n$ with nonzero elements and producing a frame which is $epsilon$-nearly Parseval and $epsilon$-nearly unit norm. Also, the concept of the $epsilon$-nearly equal frame operators for two given frames is presented. Moreover, we characterize all bounded invertible ...

The affine systems generated by Ψ ⊂ L(R) are the systems AA(Ψ) = {D A Tk Ψ : j ∈ Z, k ∈ Zn}, where Tk are the translations, and DA the dilations with respect to an invertible matrix A. As shown in [5], there is a simple characterization for those affine systems that are a Parseval frame for L(R). In this paper, we correct an error in the proof of the characterization result from [5], by redefin...

Let A ∈ Rd×d, d ≥ 1 be a dilation matrix with integer entries and | detA| = 2. We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction o...

this paper presents a proof of stirling's formula using haar wavelets and some properties of hilbert space, such as parseval's identity. the present paper shows a connection between haar wavelets and certain sequences.

We construct Parseval wavelet frames in $$L^2(M)$$ for a general Riemannian manifold M and we show the existence of unconditional $$L^p(M)$$ $$1< p <\infty $$ . This is made possible thanks to smooth orthogonal projection decomposition identity operator on , which was recently proven by Bownik et al. (Potential Anal 54:41–94, 2021). also characterization Triebel–Lizorkin $${\mathbf {F}}_{p,q}^s...

Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization...