On the degree of approximation of conjugate functions in weighted Lipschitz class

The study of error estimates of periodic functions in https://www.w3.org/1998/Math/MathML"> L p ( p ≥ 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq1062.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> -spaces through Fourier series, although is an old problem and known as Fourier approximation in the existing literature, has been of a growing interests over the lastfour decades. The most common methods used for the determination of the degree of approximation of periodic functions are based on the minimization of the L^p -norm of https://www.w3.org/1998/Math/MathML"> f ( x ) − T n ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq1063.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where https://www.w3.org/1998/Math/MathML"> T n ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq1064.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is a trigonometric polynomial of degree n, and called the approximant of f. The degree of approximation of f, so obtained depends heavily on p. In this chapter, we look at a glance the previous work done by many investigators on the degree of approximation of https://www.w3.org/1998/Math/MathML"> f ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq1065.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , conjugate of a 2π-periodic function f belonging to the weighted https://www.w3.org/1998/Math/MathML"> W ( L p , ξ ( t ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq1066.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> -class and its subclasses such as https://www.w3.org/1998/Math/MathML"> L i p ( ξ ( t ) , p ) ,   L i p ( α , p )  and  L i p α , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq1067.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and obtain the degree of approximation of https://www.w3.org/1998/Math/MathML"> f ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq1068.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , belonging to Lipschitz class https://www.w3.org/1998/Math/MathML"> L i p   α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq1069.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> ( 0 < α ≤ 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq1070.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and weighted Lipschitz class https://www.w3.org/1998/Math/MathML"> W ( L p , ξ ( t ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq1071.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> by a trigonometric polynomial generated by the product matrix means of the conjugate Fourier series of f. The degree of approximation obtained in our theorems of this chapter is sharper than others and free from p. Some corollaries have also been deduced from our theorems. This chapter is an extended version of our paper (Singh & Srivastava 2013).