Let be a 2π-periodic function, Lebesgue integrable on and belong to class. The Fourier series of is given by
with n th partial sum .
is called ‘conjugate series’ of the Fourier series (2.1) with n th partial sum .
A Housdorff matrix is a lower triangular matrix with entries
where Δ is the forward difference operator defined by and .
Let be an infinite series with n th partial sum .
If as , is said to be summable to the sum s by the Hausdorff matrix summability method ( means) (Boos and Cass ).
The Hausdorff matrix H is regular, i.e., H preserves the limit of each convergent sequence if and only if
where the mass function , , and . In this case, the have the representation
Let , .
If as , is said to be summable to s by the Euler method (Hardy , p.180).
The transform of the transform defines the transform of . It is denoted by . Thus,
If as , is said to be summable to s by the Euler-Hausdorff matrix summability means, i.e., means.
Thus, if the method of summability is superimposed on the method, another new method of summability is obtained.
The important particular cases of means are as follows:
if , ,
if and , ,
if and .
Let of a function be defined by
and the degree of approximation of a function by a trigonometric polynomial of order n, , under esssup norm be defined by (Zygmund , p.114)
We define the norm by , , and let the degree of trigonometric approximation be given by
Define the norm on class of functions by
The degree of approximation of a function f belonging to class under the norm is given by
A function if for (Titchmarsh , p.426).
if for , (def. 5.38 of McFadden ).
if (), where ξ is a modulus of continuity, that is, ξ is a non-negative, non-decreasing, continuous function with the properties , (Khan ).
If , class coincides with the class , and if , then reduces to Lipα class. Thus, it is obvious that .
Let u be a modulus of continuity such that is positive, non-decreasing.
Then . Thus, .
We write , ,