Absolute Summability Factor |N, pn|k of Improper Integrals

Improper Integrals Smita Sonker , Alka Munjal #2 # Department of Mathematics, National Institute of Technology Kurukshetra, Haryana, India 1 smita.sonker@gmail.com 2 alkamunjal8@gmail.com Abstract— In this paper, we defined the summability for integrals and established a theorem on absolute Nörlund summability |N, pn|k factors of improper integral under sufficient conditions. Some auxiliary results (well-known) have also been deduced from the main results under suitable conditions. Keyword Absolute Summability, Nörlund summability, Improper Integrals, Inequalities for Integrals.

( dt t f is said to be summable |C, 1| k , k ≥1, if is Cesàro mean of s(x) and given by The Kronecker identity: The condition (5) can be written as Considering the (N, P n ) and (K, 1, α) summability, Parashar [9] obtained the minimum set of conditions for an infinite series to be (K, 1, α) summable. In 1986, Bor [1] found the relationship between two summability techniques |C, 1| k and |N̅ , p n | k and in [2], he used the |N̅ , p n | k for generalization of a theorem based on minimal set of sufficient conditions for infinite series. In 2016, Sonker and Munjal [10] determined a theorem on generalized absolute Cesàro summability with the sufficient conditions for infinite series and in [11], they used the concept of triangle matrices for obtaining the minimal set of sufficient conditions of infinite series to be bounded. In 2017, Sonker and Munjal [12] found the approximation of the function f ϵ Lip (α, p) using infinite matrices of Cesàro submethod and in [13], they obtained boundness conditions of absolute summability factors. In this way by using the advanced summability method, we can improve the quality of the filters.
Borwein [3] extended many results on ordinary and absolute summability methods of integral. C̨anak [4] and Totur [14] worked on the concept of Cesàro summability with a very interesting result for integrals. In the same direction, we extended the results of Mazhar [7] with the help of some new generalized conditions and absolute Nörlund summability |N, p n | k factor for integrals.

II. KNOWN RESULTS
In [6], Kishore has proved the following theorem concerning |C, 1| and |N, p n | summability methods. Özgen [8] obtained the following results for integrals.
be a positive monotonic non-decreasing function such that

III. MAIN RESULTS
In the present research article, we extended the result of Özgen [8] by using the |C, 1| k summability and some other concepts. With the help of functions and and Cesàro summability |C, 1| k , we established the following theorem. , is said to be summable |N, p n | k for k ≥1.

Note:
The above theorem can be proved by using the concept of example that and hence the introduction of the function Proof: It may be possible to choose the function may be chosen such that

IV. PROOF OF THE THEOREM
In order to prove the theorem, we need to consider only the special case in which is summable |C, 1| k . Our theorem will then follow by means of theorem 1. Let x T ) ( be the function of n th (C, 1) means of the integral is given by On differentiating both sides with respect to x, we get (21) ).
Hence proof of the theorem is complete.

V. COROLLARIES Corollary 1: Let p(0) > 0, p(x) ≥ 0 and p(x) be a non-increasing function. Let
be a positive nondecreasing function such that is said to be summable |N, p n | k for k ≥1.

Note:
The above corollaries can be derived by taking the following assumptions in the main result, (i) For corollary 1, we take ).
as a convex function.
(ii) For corollary 3, we take as a convex function.

VI. CONCLUSION.
The main result of this research article is an attempt to formulate the problem of absolute summability factor of integrals which make a more modified filter. Through the investigation, we concluded that the improper integral is absolute Nörlund summable under the minimal sufficient conditions. Further, this study has a number of direct applications in rectification of signals in FIR filter (Finite impulse response filter) and IIR filter (Infinite impulse response filter). In a nut shell, the absolute summability methods are a motivation for the researchers, interested in studies of improper integrals. [13] S. Sonker