Some Inequalities Involving the Exponential Transform

Abstract

The exponential transform or modified Fourier transform is an integral transform where the Kernel function is K a ( ξ , t ) = a − i ξ t , and a ∈ ] 1 , + ∞ [ . This is a general case of the Fourier transform where the Kernel function is of the form K e ( ξ , t ) = exp ( − i ξ t ) . Joseph Fourier, the famous French Mathematician and Engineer, was the pioneer and he studied the properties in his seminal works. This important tool created an avenue of research later and it is very important in tackling problems in strong differential form and studying spectral properties of various ordinary and partial differential operators. In our article, we will obtain explicit bounds for the modified Fourier transform and derive some corollaries using in the Kernel function a multiple of the Euler-Mascheroni constant γ . The bounds obtained involve the Riemann zeta function for positive integers at the right-hand side.