Abstract
In this paper, we define a continuous wavelet transform of a Schwartz tempered distribution f ∈ S ′ ( R n ) with wavelet kernel ψ ∈ S ( R n ) and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of S ′ ( R n ) . It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.
Highlights
The n-dimensional wavelet inversion formula for tempered distributions will be proved very by using the structure Formula (9). This structure formula enables us to reduce the wavelet analysis problem relating to tempered distributions to the classical wavelet analysis problem of L2 (Rn )
We prove the following lemmas which will be used to prove the main inversion formula
Using the structure formula (9) for f, we find by distributional differentiation that
Summary
As studied in the earlier works (see, for example, [1,2,3,4,5,6,7,8,9,10,11,12], we define a Schwartz testing function space S(Rn ) to consist of C ∞ functions φ defined on Rn and satisfying the following conditions:. I, j, k, · · · take on all assumed values 1, 2, 3, · · · and all the lower suffixes in a term in Equation (3) are different It has been proved by Pandey et al [4,13] that a window function which is an element of L2 (Rn ) belongs to L1 (Rn ). Postnikov et al [21] studied computational implementation of the inverse continuous wavelet transform without a requirement of the admissibility condition. Our objective in this investigation is to extend the wavelet transform to Schwartz tempered distributions with real scale [ a 6= 0].
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