I find that answers by all Researchers on Professor Ugur Kadak's question: "What do you think about the future of summability and its applications?" are correct and useful (especially for young researchers).

With a cordial greetings,

Jasmina Fatkić

What "alternative methods" are you referring to?  Likewise the terms "statistical convergence" and "statistical summability" are too vague to be able to draw any conclusions about how they relate to the list of "useful developments". It is true that there are several different definitions of convergence for  a sequence of functions, e.g. convergence pointwise, convergence uniformly (on a specified region such as an interval), convergence in measure, convergence Lp. In Fourier analysis convergence mean square or L1 are particularly important. In the case of a sequence of operators (depending on the space they are defined on) there are several different modes of convergence. For orthogonal series you have to have a space with an inner product. It is not possible to answer your inquiry unless you are more precise about the context.

Dear Professor Myers,

Thank you for your comments. As you know that ordinary convergence implies statistical convergence but not conversely. Again, there is a same relation between ordinary summability and statistical summability. Furthermore, under some conditions, (weighted) statistical convergence  implies (weighted) statistical summability. In recent years this topic has been used  in several branches of functional analysis, we refer to

(1) M. Mursaleen, V. Karakaya, M. Ertürk, F. Gürsoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput. 218 (2012) 9132-9137.

(2) U. Kadak, N. L. Braha, H. M. Srivastava, Statistical weighted $\mathcal B$-summability and its applications to approximation theorems, Appl. Math. Comput. (2017) doi: 10.1016/j.amc.2017.01.011.

(3) H. Aktuglu, Korovkin type approximation theorems proved via $\alpha\beta$-statistical convergence,  J. Comput. Appl. Math. 259 (2014) 174-181.

(4) K. Demirci, S. Orhan, Statistical relative approximation on modular spaces, Results. Math. DOI 10.1007/s00025-016-0548-5.

(5)  U. Kadak, Generalized weighted invariant mean based on fractional difference operator with applications to approximation theorems for functions of two variables, Results Math. (2016) DOI 10.1007/s00025-016-0634-8.

(6) U. Kadak, On weighted statistical convergence based on $(p,q)$-integers and related approximation theorems for functions of two variables, J. Math. Anal. Appl. 443 (2016) 752--764.

(7) U. Kadak, Weighted statistical convergence based on generalized difference operator involving $(p,q)$-gamma function and its applications to approximation theorems, J. Math. Anal. Appl. 448 (2017) 1633--1650

(8) N. L. Braha, H. M. Srivastava and S. A.  Mohiuddine, A Korovkin's type approximation theorem for periodic functions via the statistical summability of the generalized de la Vall\'{e}e Poussin mean, Appl. Math. Comput. 228 (2014) 162-169.

(9) N. L. Braha, V. Loku and H. M. Srivastava, $\Lambda^2$-Weighted statistical convergence and Korovkin and Voronovskaya type theorems, Appl. Math. Comput. 266 (2015) 675--686.

(10) F. Dirik, K. Demirci, Four-dimensional matrix transformation and the rate of $A$-statistical convergence of continuous functions, Comput. Math. Appl. 59 (2010) 2976-2981

(11) Orhan, K. Demirci, Statistical $A$-summation process and Korovkin type approximation theorem on modular spaces, Positivity 18 (2014) 669-686.

Apart from these papers, a lot of works have been done.  In particular, many authors point out different application areas in this topic. Baliarsing gives a new approach based on difference sequence spaces defining a new fractional order difference operatör (P. Baliarsingh, On a fractional difference operator, Alexandria Eng. J. 55 (2) (2016) 1811-1816.). This operatör has been extended by Kadak using (p-q)-calculus. In the year 2016, Prof. Mohiuddine introduced the notion of statistical weighted A-summability  [Statistical weighted A-summability with application to Korovkin's type approximation theorem, J. Inequal. Appl. 2016 (2016) Article ID 101]. Again, Kadak et al. have extended it to weighted B-statistical summability and present its applications (with figures) to Korovkin and Voronovskaja type approximation theorems [U. Kadak, N. L. Braha, H. M. Srivastava, Statistical weighted $\mathcal B$-summability and its applications to approximation theorems, Appl. Math. Comput. (2017) doi: 10.1016/j.amc.2017.01.011.] Finally, Demirci et al. have focused on statistical convergence and its applications to approximation theorems in modular function spaces [Statistical $A$-summation process and Korovkin type approximation theorem on modular spaces, Positivity 18 (2014) 669-686.]

In my opinion, this topic continues to be developed in various context as mentioned above. For these reasons, I wonder the future of summability and its applications. Thank you again

Sincerely yours

Before I criticize series expansions I first have to acknowledge that they have been useful in many cases. In some cases, the first few terms gives an adequate approximation to some quantity of interest. In some other cases, the coefficients in the series have an important significance. One example is in electrodynamics in which the coefficients describe the frequency dependence of some system. Another example is in quantum mechanics that gives a probability interpretation to such coefficients. Also, a series expansion can occasionally prove some theorems.

Excluding the above applications, a series expansion to me is a last resort. It is an act of desperation when I have to crank out some numbers and don't know any other way to do it. It is a number-crunching routine. In terms of insight, I don't get any by looking at a series. Is a series of functions increasing in the argument? Is it decreasing? Does it have a relative maximum or minimum? I can't tell from a casual inspection of a series. I will try anything (variational methods or any other approximations that I can think of) before resorting to a series. I resort to a series when I have given up trying to find a better way and have to crank out some numbers.

Dear Professor Edmonds.

Thank you for your reply.  I agree with you that summability includes different applications for each working area. One of my purposes is to investigate the contribution in such different areas.To help motivate the early ideas of summability, I offer some comments from Hardy.

"Newton and Leibniz, the first mathematicians to use infinite series systematically, had litlle temptation to use divergent series. The temptation became greater as analysis widened, and it was soon found that they were useful, and that operations performed on them critically often led to important results which could be verified independently."

Hardy states that "The summation of an infinite series, more widely applicable, than the classical definition of Cauchy".

In my opinion, these methods (especially matrix methods and almost convergence) can be extended to other areas. For example, the strong summability which was given in the case of Cesaro method based on Fourier series by Hardy and Littlewood. In the case of double sequences, these methods are very often matrix methods defined by four dimensional matrices. Also, we say that these methods originated in other fields like function theory and applied mathematics.

Thank you again

Sincerely

Dear Professor Alomari

Thank you for your original evaluations. In the near future, I guess, we will see the summability and its applications in most disciplines.

Sincerely

Your topic is already considered to be an important topic. But if you want to get more support from people like me, you need to explain how we can gain insight from a series expansion. "Insight" means, to me, that we can make some qualitative predictions (e.g., whether some function is increasing or decreasing, or has a relative minimum or maximum) without having to calculate any numbers. I can't see those properties when looking at an infinite series. If you can help me to see those properties, I will be very grateful.

Dear Professor Edmonds

I want to remind some important features based on the infinite series. In view of the following features of infinite series, we may gain insight from a series expansion.

As you know, a continuous function of n variables attains a maximum or a minimum in the interior of a region, only at those values of the variables for which the n partial derivatives either vanish simultaneously (stationary points) or at which one or more of these derivatives cease to exist. The proof involves examining the Taylor Series expansion at the points where the partial derivatives either vanish or cease to exist.

Here, we see that Taylor or Maclaurin series are the main motivation for the formal study of infinite series.

(1) More relevant to calculus is the fact that numerical integration methods use e.g. Simpson's method, the error analysis of which is related to Taylor series.

(2) One of the best applications of infinite series is in harmonic analysis. For instance,  a periodic function can be determined as an infinite series of well known trigonometric functions.

(3) By using just one function, we cannot solve some differential equations. But, by the help of an infinite series (of powers of x), we can approximate i.e. Taylor series expansion.

(4) Furthermore, we know that power series is the another best way to motivate infinite series.

We can give various examples of application in connection with the infinite series.

The main object of my question is related to the development of various summability concepts and how these concepts lead to a number of approximation by lnear operators. Also, I have focused on the development of matrix summability (statistically), in particular, four dimensional regular matrix methods by the weighted mean.  For more various approaches, we refer to

• U. Kadak, N. L. Braha, H. M. Srivastava, Statistical weighted B-summability and its applications to approximation theorems, Appl. Math. Comput. 320 (2017)
• U. Kadak, Generalized weighted invariant mean based on fractional difference operator with applications to approximation theorems for functions of two variables, Results Math. (2016) DOI 10.1007/s00025-016-0634-8.
• U. Kadak, On weighted statistical convergence based on (p,q)-integers and related approximation theorems for functions of two variables, J. Math. Anal. Appl. 443 (2016) 752--764.
• U. Kadak, Weighted statistical convergence based on generalized difference operator involving (p,q)-gamma function and its applications to approximation theorems, J. Math. Anal. Appl. 448 (2017) 1633--1650

Thanks...

Ugur 

Hello Researchers ! what is actual use of absolute summability in BIBO stability? I am attaching a useful image which increase the importance of summability. what is deep concept of the relation of BIBO stability and absolute summability? Want to understand yours valuable comments and knowledge about this image.

The theory of summability has many uses throughout analysis and applied mathematics. Engineers and physicists working with Fourier series or analytic continuation will also find the concepts of summability theory valuable to their research.

Kind regards

I find that answers by all Researchers on Professor Ugur Kadak's question: "What do you think about the future of summability and its applications?" are correct and useful (especially for young researchers).

With a cordial greetings,

Jasmina Fatkić