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...

It is known that superpositions of ridge functions (single hidden-layer feedforward neural networks) may give good approximations to certain kinds of multivariate functions. It remains unclear, however, how to effectively obtain such approximations. In this paper, we use ideas from harmonic analysis to attack this question. We introduce a special admissibility condition for neural activation fu...

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...

We study the convergence properties of social Hegselmann–Krause (HK) dynamics , a variant HK model opinion dynamics where physical connectivity graph that accounts for extrinsic factors could prevent interaction between certain pairs agents is incorporated. As opposed to original (which termina...

In this paper we establish a surprising new identity for Parseval frames in a Hilbert space. Several variations of this result are given, including an extension to general frames. Finally, we discuss the derived results.