Sufficient Conditions for Absolute Cesàro Summable Factor

Quasi-f-power increasing sequence has been used for infinite series to establish a theorem on a minimal set of sufficient conditions for absolute Cesàro φ − |C, α; δ; l|k summable factor. Further, a set of new and well-known arbitrary results have been obtained by using the main theorem. The presented main result has been validated by the previous result under suitable conditions. In this way, the Bounded Input Bounded Output (BIBO) stability of impulse response has been improved by finding a minimal set of sufficient conditions for absolute summability because absolute summable is the necessary and sufficient condition for BIBO stability. KeywordsAbsolute summability, Cesàro summability, Infinite series, Quasi-f-power increasing sequence.


Introduction
Summability is a field in which we study the non-converging series/integrals and assign a value (number) to it. In mathematics analysis, summability method is an alternative formulation of convergence of a series which is divergent in conventional sense. Thoughout the 19 th century, many mathematicians studied various divergent series and defined numerous summability methods namely Abel summability, Cesàro summability, Euler summability, Hausdroff summability, Nörlund summability, Riesz summability etc. The present paper is devoted to the study of Cesàro summability methods and a theorem has been established for the absolutely Cesàro summable factor of an infinite series. With the help of a minimal set of sufficient conditions for a system, the error can be minimized by using summability methods so that the output data is filtered and according to the user's interest. Let ∑ ∞ =0 be an infinite series with sequence of partial sums, = ∑ =0 and n th mean of the sequence { } is given by , s.t., (1) and Definition 1 (Flett, 1957). Let is n th ( , ) mean of order (0 < ≤ 1) of the sequence ( ), then the infinite series is | , ; | summable for ≥ 1 and ≥ 0, if where is given by Definition 2. If { } is a sequence of positive real number and the sequence of the mean { } satisfies the following condition: then the infinite series is − | , ; | summable for ≥ 1 and ≥ 0.
Bor has derived several theorems (1993, 2011a, 2011b, 2014, 2015, 2016) on applications of the various wider class sequences by applying the absolute summability techniques. In the same direction, Özarslan and Ari (2011), Özarslan and Yuvuz (2013) used the absolute matrix summability and Parashar (1981) applied (K, 1, α) summability for infinite series. Chandra and Jain (1988) used absolute product summability for Fourier series. Munjal (2016a, 2017) determined the theorems on generalized absolute Cesàro summability and in (2016b); they used triangle matrices for the minimal set of sufficient conditions of an infinite series to be bounded. Summability is very useful technique for solving the problem of discrete time signal (Richa and Kumar, 2019), Heat transfer problem (Singh et al., 2018) formulation with the help of Fourier series and for formation of iteration methods for differential equations (Chauhana and Srivastava, 2019) problems.

Known Result
A quasi-f-power increasing sequence (Sulaiman, 2006) is a positive sequence = { } with a constant = ( , ) ≥ 1 for all 1 ≤ ≤ such that A quasi-f-power increasing sequence converted to quasi--power increasing sequence (Leindler, 2001), if has a certain value as =0 in the condition (9). With the help of Cesàro summability of order , Bor (2015) has proved the following theorem for an infinite series.
Theorem 2.1 Let (Bn) be a quasi-f-power increasing sequence for some ς (0 <ς <1). Also suppose that there exists a sequence of numbers (Dn) such that it is ξ-quasi-monotone satisfying the following: ∑ .
If the conditions are satisfied, then the series ∑ is | , | summable for 0 < ≤ 1 and ≥ 1.

Main Result
Considering absolute Cesàro summability − | , ; , | factors and a quasi-f-power increasing sequence, we established the following theorem.
It is sufficient to show that By using Holder's inequality and Abel's transformation, we have Collecting (24) -(29), we have Hence, the proof of the theorem is complete.

Corollaries
then the series ∑ is − | , | summable for 0 < ≤ 1, ≥ 1 and is given by Proof: By using = 1 and = 0 in main theorem, we will get (31) and (32). We omit the details of the proof as it is similar to that of the main theorem 3.1.
Corollary 5.2 Let (Bn) be a quasi-f-power increasing sequence for some ς (0 <ς <1) and (Dn) be a ξ-quasi-monotone sequence of numbers satisfying (16-20) and the following condition: then the series ∑ is | , | summable for 0 < ≤ 1, ≥ 1 and is given by Proof: By using = , = 1 and = 0 in main theorem, we will get (34). We omit the details of the proof as it is similar to that of the main theorem 3.1.

Conclusion
The main result of this research article is a problem on generalized absolute summability factor of infinite series which make the system stable. A necessary and sufficient condition for a system to be BIBO (Bounded Input Bounded Output) stable is that the impulse response be absolutely summable, i.e., Summability techniques are trained to minimize the error. With the use of summability Technique, the output of the signals can be made stable, bounded and used to predict the behavior of the input data, the initial situation and the changes in the complete process.
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 have vast potential in dealing with the problems based on infinite series. Based on the investigation, it can be concluded that our result is a generalized form which can be reduced for several well-known summabilities. Our theorem is validated by Corollary 5.1, which is a mean result of Bor (2015) on the minimal set of sufficient condition for infinite series to be absolute summable.