Construction and Properties of Haar-Vilenkin Wavelets
The Haar wavelet based on Haar system, introduced by the Hungarian mathematician Alfred Haar [5 ], is the simplest example of wavelets. Recently, we have studied the concept of Haar-Vilenkin wavelet in [8 ] which is a generalization of Haar wavelet. We ha
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Construction and Properties of Haar-Vilenkin Wavelets Meenakshi and P. Manchanda
Abstract The Haar wavelet based on Haar system, introduced by the Hungarian mathematician Alfred Haar [5], is the simplest example of wavelets. Recently, we have studied the concept of Haar-Vilenkin wavelet in [8] which is a generalization of Haar wavelet. We have introduced a Special Type of Multiresolution Analysis [10] generated by Haar-Vilenkin wavelet which is a special case of matrix multiresolution analysis studied in [18]. In this paper we represent Haar-Vilenkin wavelets in discrete form by introducing Haar-Vilenkin matrices and expand a function in HaarVilenkin wavelet series. We have applied this method for solving ordinary differential equations in this paper. Keywords Wavelets · Haar-Vilenkin system multiresolution analysis · Matrices AMS classification 42A38 · 42A55 · 42C15 · 42C40 · 43A70
13.1 Introduction We have introduced the concept of Haar-Vilenkin wavelet and Haar-Vilenkin scaling function and have studied the basic properties of Haar-Vilenkin wavelet series and coefficients in [8]. Haar-Vilenkin wavelet is a generalization of Haar wavelet. Haar wavelet basis is the simplest and historically the first example of an orthonormal wavelet basis. Haar basis functions are step functions with jump discontinuities. Haar wavelet basis provides a very efficient representation of functions that consist of smooth, slowly varying segments punctuated by sharp peaks and discontinuities. Haar system is an orthonormal system such that each continuous function on [0, 1] has a uniformly convergent Fourier series with respect to this system. Meenakshi (B) Department of Mathematics, Lajpat Rai D.A.V. College, Jagraon, India e-mail: [email protected] P. Manchanda Department of Mathematics, Guru Nanak Dev University, Amritsar, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2017 P. Manchanda et al. (eds.), Industrial Mathematics and Complex Systems, Industrial and Applied Mathematics, DOI 10.1007/978-981-10-3758-0_13
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Meenakshi and P. Manchanda
The Haar wavelet is the function defined on the real line R as ⎧ ⎨ 1 x ∈ [0, 21 ) h(x) = −1 x ∈ [ 21 , 1) . ⎩ 0 otherwise It can be expressed in the form h(x) = χ[0, 21 ) (x) − χ[ 21 ,1) (x). By taking the translations and dilations of h(x), the system {h j,k (x)} j,k∈Z is referred as the Haar system on R where h j,k (x) = 2 j/2 h(2 j x − k), j, k ∈ Z. The Haar scaling function on the real line is p(x) = χ[0,1) (x). The collection { p j,k (x)} j,k∈Z is referred to as the system of Haar scaling functions: supp h j,k =
k k + 1 , , 2j 2j
for j, k ∈ Z form the family of dyadic intervals. The where the intervals 2kj , k+1 2j various properties of Haar system have been extensively studied. It has been shown that the system {h j,k (x)} j,k∈Z is an orthonormal system in L 2 (R) [2, 16, 18]. The family {h j,k } j,k∈Z is also associated with multiresolution analysis, for example: Let Sn = span {h j,k } j h j,k . The convergence is in the norm of the
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