It details the elementary theory of infinite series; the basic properties of Taylor and Fourier series, series of functions, and the applications of uniform convergence; double series, changes in the order of summation, and summability; power series and real analytic functions; and additional topics in Fourier series, such as summability, Parseval's equality, and the convolution theorem
5) we used the Laplace transforms, and by applying the convolution theorem
It follows from the classical Titchmarsh convolution theorem
and uniqueness theorem for analytic functions that ker ([K.
By the inverse of a new integral transform and convolution theorem
(33) we find that:
According to the convolution theorem
, the convolution in one domain can be converted to the point-wise multiplication in another domain.
82)] based on the generalized convolution theorem
, together with the product formulas derived in [6, Theorem 6] and [7, Theorem 8], it follows that:
Employing the convolution theorem
of the discrete Fourier transform, the discrete convolution can by carried out as
for Fourier transform states that convolution in time domain equals point-wise multiplication in frequency domain:
Explanation to Step4: After finding the closeness index blending functions are calculated using convolution theorem
b) The convolution theorem
in the theory of two dimensional Laplace transform is given by (see, [2, p.
Alzer , who showed that the function g is strictly increasing on (0, [infinity]) by using the convolution theorem
for Laplace transformas.
The convolution theorem
is applied to get the response of the system on temporally Gaussian distributed laser radiation.