Continuous Wavelet Transform of Schwartz Tempered Distributions in <i>S</i>′ (<inline-formula> <mml:math id="mm999" display="block"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:semantics> </mml:math> </inline-formula>)

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.

Keywords

• function spaces and their duals
• distributions
• tempered distributions
• Schwartz testing function space
• generalized functions
• distribution space
• wavelet transform of generalized functions
• Fourier transform