In this study, we establish a Parseval equality and an expansion formula for a Sturm– Liouville operator on semi-unbounded time scales. AMS subject classification: 34L10.

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.