In this paper, Haar wavelet-based neural network is described for the identification and control of discrete-time nonlinear dynamical systems. Wavelets are suited to depict functions with local nonlinearities and fast variations because of their intrinsic properties of finite support and self-similarity. Due to the orthonormal properties of Haar wavelet functions, wavelet neural networks result...

The linear scale-space kernel is a Gaussian or Poisson function. These functions were chosen based on several axioms. This representation creates a good base for visualization when there is no information (in advanced) about which scales are more important. These kernels have some deficiencies, as an example, its support region goes from minus to plus infinite. In order to solve these issues se...

This paper describes and compares application of wavelet basis and Block-Pulse functions (BPFs) for solving fractional integro-differential equation (FIDE) with a weakly singular kernel‎. ‎First‎, ‎a collocation method based on Haar wavelets (HW)‎, ‎Legendre wavelet (LW)‎, ‎Chebyshev wavelets (CHW)‎, ‎second kind Chebyshev wavelets (SKCHW)‎, ‎Cos and Sin wavelets (CASW) and BPFs are presented f...

We present a technique to make a close enough recognition of a function, based on recognition coefficients. The recognition coefficients are determined using inner products 5 of the function to be recognized with a complete set of functions. This is how generalized Fourier coefficients are calculated. However they may not give a series representation of the function to be recognized as the comp...

Hankel operators lie at the junction of analytic and real-variables. We will explore this junction, from the point of view of Haar shifts and commutators. 1. Haar Functions We consider operators which satisfy invariance properties with respect to two well-known groups. The first group we take to the translation operators (1.1) Try f (x) := f (x − y) , y ∈ R . Note that formally, the adjoint ope...

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It is known that the space of real valued, continuous functions C(B) over a multidimensional compact domain B ⊂ R , k ≥ 2 does not admit Haar spaces, which means that interpolation problems in finite dimensional subspaces V of C(B) may not have a solutions in C(B). The corresponding standard short and elegant proof does not apply to complex valued functions over B ⊂ C. Nevertheless, in this sit...

Generalizing Christensen's notion of a Haar-null set and Darji's Haar-meager set, we introduce study the Haar-$\mathcal I$ in Polish group. Here $\mathcal is an ideal subsets some compact metrizable space $K$. A Borel subset $B\subset X$ group $X$ called if there exists continuous map $f:K\to such that $f^{-1}(B+x)\in\mathcal for all $x\in X$. Moreover, $B$ generically witness functions $\{f\in...