American Journal of Mathematical Analysis
Volume 8, 2020 - Issue 1
Website: http://www.sciepub.com/journal/ajma

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Research Article

Open Access Peer-reviewed

Suresh Kumar Sahani^{ }

Received May 06, 2020; Revised June 08, 2020; Accepted June 15, 2020

This paper briefly discusses the uniform (N, p, q) summability of Fourier series and its conjugate series. We prove that if and be positive (i.e. monotric function of t) and and is monotonic sequence of constant with their non-vanishing partial sums and tending to infinity as m, n if = 0** **as n 0 as n Where 0 and as t uniformly in a domain E in which f(*x*) is bounded then the **F**ourier series (1.4) is summable (N, p, q) uniformly in E to the sum f(*x*). Also, If (2.4) uniformly in E then (1.5) is summable (N, p,q) uniformly in the domain E to the sum (2.5) whenever the integral exist uniformly in E.

The old hazy notion of convergence of infinite series was placed on sound foundation with the appearance of Cauchy’s monumental work course “d Analysis Algebrique” in 1821 and Abel’s ^{ 1} researches on the binomial series in 1826. However, it was observed that there were certain convergent series which particularly in dynamical Astronomy furnished nearly correct result. A theory of divergent series was formulated explicitly for the first time in 1890, when Cesaro ^{ 2} published a paper on the multiplication of series. Since then the theory of series, whose equation of partial sums oscillates, has been the center of attraction and fascination for most of the pioneering mathematical analysis. These process of associating generalized sums known as method of summability Szasz ^{ 4, 5} and Hardy ^{ 3, 6} provide a natural generalization of the classical notion of convergence. Hobson ^{ 9} and are thus responsible for bringing within the field of applicability a wider class of erstwhile rejected series that used to be tabooed as divergent.

Let ∑ be an infinite series with as the sequence of its n^{th }Partial sums.

Let {p_{n}} and {q_{n}} be two non-negative sequence, with

(1.1) |

The n^{th }(N,p,q) mean ) of the sequence at point x in a domain E is defined by the sequence to sequence transformations.

(1.2) |

If t_{n}(x) s(x) as n

Then the series or the sequence of its its partial sum is said to be summable (N,p,q) to the sum s(*x*) at the point x in a domain E

If

(1.3) |

uniformly in set E then we say that series is summable (N,p,q) uniformly in E to the sum s (*x*) where q_{n} = 1 n. (N,p,q) summability reduces to the summability (N,) for = summability reduces to (N, ) such summability called as harmonic summability.

Let F(t) be a 2- periodic and Lebesgue integrable function of t in (-). Then the Fourier series ^{ 7, 8} of the function F(t) is given by

(1.4) |

The conjugate series of Fourier series is given by

(1.5) |

We write a point at t = *x*

= the integral part of

and

**Theorem 1:**

Let λ(t) and be two positive function of t such that λ(t), and t. Increase monotonically with t. Let and be two non- negative monotonic non- increasing sequence of constant with there non- vanishing partial sum and tending to infinity as m, n respectively.

If

(2.1) |

(2.2) |

Where 0 and

(2.3) |

as t uniformly in a domain E in which F(x) is bounded then the Fourier series (1.4) is summable (N,p,q) uniformly in E to the sum F(x).

**Theorem 2:**

Let and be non-negative monotonic, non-increasing sequence of constant with their non-varnishing partial sums and tending to infinity as m, n resp if = 0 as n and also λ(t) and be the function of t such that λ(t), and t. increase monotonically with t and which satisfy the following condition λ(n). R_{n }= 0 as n 0

If

(2.4) |

as t 0 uniformly in E, then the conjugate series of Fourier series (1.5) is summable (N,p,q) uniformly in the domain E to the sum

(2.5) |

Whenever the integral exists uniformly in E.

The following lemmas are required in order to prove our theorem.

**Lemma (3.1)** for 0 Where c is an absolute constant

**Lemma (3.2)** If as and the condition (2.1) is satisfy then n. as

**Proof of Lemma (3.2)**

It may be easily noted that if = 0 Then,

Now,

Therefore,

Implies that n = 0 as n

**Lemma (3.3) **If 0 then

**Proof: **

We have

**Proof of lemma (3.1)**

Denoting the n^{th }partial sum of the series (1.4) by

Therefore following (1.2), the (N,p,q)^{th} transform of is given by

(3.1) |

Now in order to prove the **theorem 1** we have to show that under our assumption,

(3.2) |

as n uniformly in E.

Now for I_{1},

= 0(n) (using if 0 then )

= 0 (n) 0,** **(using 2.3)

= 0 ( using lemma 3.2 )

(3.3) |

Uniformly in E,

Also for I_{2},

(3.4) |

Now

(3.5) |

Further

But

Thus

(3.6) |

Again,

(3.7) |

From (3.4) to (3.7) it follows that

(3.8) |

Again,

(3.9) |

Uniformly in E by virtue of a known result due to Hardy and Rogosinski and regularity of the method of summation.

Now combing (3.3), (3.8) and (3.9), we get the required result in (3.2). This completes the proof of the **theorem (2).**

**Proof of theorem (2)**

The n^{th }partial sum of the series (1.5) at the point t = *x* in E, is given by

Denoting by the (N,p,q)^{th} transform of

We have the following (1.2),

Where,

Now we have to show that

(3.10) |

Let us write with 0

(3.11) |

For

(3.12) |

(3.13) |

Again,

(3.14) |

Using the result due to Hardy and Rogosinski and regulation of the method of summation.

Combing (3.11), (3.12), (3.13) and (3.14) we get the required result in (3.10). This completes the proof of the theorem (2).

When and be non-negative monotonic non-increasing sequence of constant with their non-varnishing partial sums and tending to infinity as m, n resp if = 0 as n and also λ(t) and be the function of t such that λ(t), and . Increase monotonically wit t. and which satisfy the following condition λ(n). R_{n}= 0 as n 0

If = = 0

As t 0 uniformly in E, then the conjugate series of** **Fourier series (1.5) is summable (N,p,q) uniformly in the domain E to the sum

Whenever the integral exists uniformly in E.

[1] | Abel, N.H, Utersuchumgen ber die reine…, Journal r die reine and angewandte Mathematik, (Crelles), L, 1826, 311-339. | ||

In article | View Article | ||

[2] | Cesaro, E. surla multiplication des series, Bull. sci.Math.(2) vol. 14, 1890, pp. 144-120 | ||

In article | |||

[3] | Hardy, G.H, Divergent series. The clarlndon press, oxford. 1949, | ||

In article | |||

[4] | Szasz, O, Introduction to the theory of divergent Series University of Cincinnati, 1946. | ||

In article | |||

[5] | Szasz, O. Introduction to the theory of divergent Series. University of Cincinnati, Revised edition, 1952. | ||

In article | |||

[6] | Hardy, G.H, Theorems relating to the summability and Convergence of slowly oscillating series, proc. London Math. Soc. (2) vol. 8, 1910, 301-320. Esp. P 309. | ||

In article | View Article | ||

[7] | Prasad, B.N. The summability of Fourier series. Presidential address (section of Mathematics and statistics), 32^{nd}^{ }Indian sciences congress Association Nagpur. | ||

In article | |||

[8] | Prasad, B.N and Bhatta, S.N. The summability factor of Fourier series. Duke Math. 1957, 103-120. | ||

In article | View Article | ||

[9] | Hobson, E.W, The theory of the function of a real variable and the theory of Fourier series, Cambridge, 1950. | ||

In article | |||

Published with license by Science and Education Publishing, Copyright © 2020 Suresh Kumar Sahani

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Suresh Kumar Sahani. Analysis on Uniform [N, p, q] Summability of Fourier Series and Its Conjugate Series. *American Journal of Mathematical Analysis*. Vol. 8, No. 1, 2020, pp 9-13. http://pubs.sciepub.com/ajma/8/1/2

Sahani, Suresh Kumar. "Analysis on Uniform [N, p, q] Summability of Fourier Series and Its Conjugate Series." *American Journal of Mathematical Analysis* 8.1 (2020): 9-13.

Sahani, S. K. (2020). Analysis on Uniform [N, p, q] Summability of Fourier Series and Its Conjugate Series. *American Journal of Mathematical Analysis*, *8*(1), 9-13.

Sahani, Suresh Kumar. "Analysis on Uniform [N, p, q] Summability of Fourier Series and Its Conjugate Series." *American Journal of Mathematical Analysis* 8, no. 1 (2020): 9-13.

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[1] | Abel, N.H, Utersuchumgen ber die reine…, Journal r die reine and angewandte Mathematik, (Crelles), L, 1826, 311-339. | ||

In article | View Article | ||

[2] | Cesaro, E. surla multiplication des series, Bull. sci.Math.(2) vol. 14, 1890, pp. 144-120 | ||

In article | |||

[3] | Hardy, G.H, Divergent series. The clarlndon press, oxford. 1949, | ||

In article | |||

[4] | Szasz, O, Introduction to the theory of divergent Series University of Cincinnati, 1946. | ||

In article | |||

[5] | Szasz, O. Introduction to the theory of divergent Series. University of Cincinnati, Revised edition, 1952. | ||

In article | |||

[6] | Hardy, G.H, Theorems relating to the summability and Convergence of slowly oscillating series, proc. London Math. Soc. (2) vol. 8, 1910, 301-320. Esp. P 309. | ||

In article | View Article | ||

[7] | Prasad, B.N. The summability of Fourier series. Presidential address (section of Mathematics and statistics), 32^{nd}^{ }Indian sciences congress Association Nagpur. | ||

In article | |||

[8] | Prasad, B.N and Bhatta, S.N. The summability factor of Fourier series. Duke Math. 1957, 103-120. | ||

In article | View Article | ||

[9] | Hobson, E.W, The theory of the function of a real variable and the theory of Fourier series, Cambridge, 1950. | ||

In article | |||