Degree of Approximation of Functions Through Summability Methods

The theory of approximation is a very extensive field and the study of the theory of trigonometric approximation is of great mathematical interest and of great practical importance. Positive approximation processes play an important role in approximation theory and appear in a very natural way dealing with approximation of continuous functions, especially one, which requires further qualitative properties, such as monotonicity, convexity, and shape preservation and so on. The theory of summability arises from the process of summation of series and the significance of the concept of summability has been strikingly demonstrated in various contexts, e.g., in Fourier analysis, fractional calculus, analytic continuation, quantum mechanics, probability theory, approximation theory, nonlinear analysis, and fixed point theory. The methods of almost summability and statistical summability have become an active area of research in recent years. This chapter contains two sections. In the first section, an attempt is made to obtain a theorem on the degree of approximation of functions belonging to the $$Lip\, (\alpha , r)$$ L i p ( α , r ) -class, using almost Riesz summability method of its infinite Fourier series, so that some theorems become particular case of our main theorem. In the second section, a theorem concerning the degree of approximation of the conjugate of a function f belonging to $$Lip\, (\xi (t), r)$$ L i p ( ξ ( t ) , r ) -class by Euler (E, q) summability of conjugate series of its Fourier series has been established which in turn generalizes the results of Shukla [Certain Investigations in the theory of Summability and that of Approximation, Ph.D. Thesis, 2010, V.B.S. Purvanchal University, Jaunpur (Uttar Pradesh)] and others.