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In this article, we express a numerical form of the convergence using the suitable modulus of smoothness for linear compositions of the Mellin convolution operators. Later, with the help of a different modulus of smoothness than before, called the logarithmic weighted modulus of smoothness in the weighted Mellin-Lebesgue spaces, a rate of convergence is obtained, and then the global smoothness preservation property is also expressed through the logarithmic weighted modulus of smoothness in the weighted Mellin-Lebesgue spaces. It is noteworthy that two different moduli of smoothness are used in this article. One is given on the weighted space of continuous functions, the other on the weighted Mellin-Lebesgue spaces comprising all Lebesgue measurable functions. Moreover, the striking results in this article are that the stated quantitative-type theorems are expressed in terms of the modulus of smoothness of themmth order Mellin derivative of the given function. In the last section, the results obtained are given for classical Mellin-Gauss-Weierstrass operators.

A note on linear compositions of the Mellin convolution operators in the weighted Mellin-Lebesgue spaces