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

Convolution theorem 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 [4], 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.