International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2009Β (2009), Article IDΒ 989865, 8 pages
doi:10.1155/2009/989865
Research Article

On ( 𝑁 , 𝑝 , π‘ž ) ( 𝐸 , 1 ) Summability of Fourier Series

H. K. NigamΒ and Ajay Sharma

Department of Mathematics, Faculty of Engineering & Technology, Mody Institute of Technology and Science (Deemed University), Laxmangarh, Sikar (Rajasthan) 332311, India

Received 7 November 2008; Revised 17 March 2009; Accepted 30 March 2009

Academic Editor: HüseyinΒ Bor

Copyright Β© 2009 H. K. Nigam and Ajay Sharma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A new theorem on ( 𝑁 , 𝑝 , π‘ž ) ( 𝐸 , 1 ) summability of Fourier series has been established.

1. Introduction

Let { 𝑝 𝑛 } and { π‘ž 𝑛 } be the sequences of constants, real or complex, such that

𝑃 𝑛 = 𝑝 0 + 𝑝 1 + 𝑝 2 + β‹― + 𝑝 𝑛 = 𝑛  𝜈 = 0 𝑝 𝜈 ⟢ ∞ , a s 𝑄 𝑛 ⟢ ∞ , 𝑛 = π‘ž 0 + π‘ž 1 + π‘ž 2 + β‹― + π‘ž 𝑛 = 𝑛  𝜈 = 0 π‘ž 𝜈 ⟢ ∞ , a s 𝑅 𝑛 ⟢ ∞ , 𝑛 = 𝑝 0 π‘ž 𝑛 + 𝑝 1 π‘ž 𝑛 βˆ’ 1 + β‹― + 𝑝 𝑛 π‘ž 0 = 𝑛  𝜈 = 0 𝑝 𝜈 π‘ž 𝑛 βˆ’ 𝜈 ⟢ ∞ , a s 𝑛 ⟢ ∞ . ( 1 . 1 ) Given two sequences { 𝑝 𝑛 } and { π‘ž 𝑛 } convolution ( 𝑝 βˆ— π‘ž ) is defined as 𝑅 𝑛 = ( 𝑝 βˆ— π‘ž ) 𝑛 = 𝑛  π‘˜ = 0 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ . ( 1 . 2 ) Let βˆ‘ ∞ 𝑛 = 0 𝑒 𝑛 be an infinite series with the sequence of its 𝑛 th partial sums { 𝑠 𝑛 } .

We write 𝑑 𝑛 𝑝 , π‘ž = 1 𝑅 𝑛 𝑛  π‘˜ = 0 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ 𝑠 π‘˜ . ( 1 . 3 )

If 𝑅 𝑛 β‰  0 , for all 𝑛 , the generalized Nörlund transform of the sequence { 𝑠 𝑛 } is the sequence { 𝑑 𝑛 𝑝 , π‘ž } .

If 𝑑 𝑛 𝑝 , π‘ž β†’ 𝑆 , a s 𝑛 β†’ ∞ , then the series βˆ‘ ∞ 𝑛 = 0 𝑒 𝑛 or sequence { 𝑠 𝑛 } is summable to 𝑆 by generalized Nörlund method (Borwein [1]) and is denoted by

𝑆 𝑛 ⟢ S ( 𝑁 , 𝑝 , π‘ž ) . ( 1 . 4 )

The necessary and sufficient conditions for ( 𝑁 , 𝑝 , π‘ž ) method to be regular are 𝑛  π‘˜ = 0 | | 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ | | ξ€· | | 𝑅 = 𝑂 𝑛 | | ξ€Έ , ( 1 . 5 )

and 𝑝 𝑛 βˆ’ π‘˜ = π‘œ ( | 𝑅 𝑛 | ) , a s 𝑛 β†’ ∞ for every fixed π‘˜ β‰₯ 0 , for which π‘ž π‘˜ β‰  0 .

Now 𝐸 1 𝑛 = 1 2 𝑛 𝑛  π‘˜ = 0 ξ‚΅ 𝑛 π‘˜ ξ‚Ά 𝑠 π‘˜ . ( 1 . 6 )

If 𝐸 1 𝑛 β†’ 𝑠 , a s 𝑛 β†’ ∞ , then the series βˆ‘ ∞ 𝑛 = 0 𝑒 𝑛 is said to be ( 𝐸 , 1 ) summable to 𝑠 (Hardy [2]): 𝑑 𝑛 𝑝 , π‘ž , 𝐸 = 1 𝑅 𝑛 𝑛  π‘˜ = 0 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ 𝐸 1 π‘˜ = 1 𝑅 𝑛 𝑛  π‘˜ = 0 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ β‹… 1 2 π‘˜ π‘˜  𝜈 = 0 ξ‚΅ π‘˜ 𝜈 ξ‚Ά 𝑠 𝜈 . ( 1 . 7 )

If 𝑑 𝑛 𝑝 , π‘ž , 𝐸 β†’ ∞ , a s 𝑛 β†’ ∞ , then we say that the series βˆ‘ ∞ 𝑛 = 0 𝑒 𝑛 or the sequence { 𝑠 𝑛 } is summable to 𝑆 by ( 𝑁 , 𝑝 , π‘ž ) ( 𝐸 , 1 ) summability method.

Particular Cases
(1) ( 𝑁 , 𝑝 , π‘ž ) ( 𝐸 , 1 ) mean reduces to ( 𝑁 , 𝑝 𝑛 ) ( 𝐸 , 1 ) summability mean if π‘ž 𝑛 = 1 , βˆ€ 𝑛 .(2) ( 𝑁 , 𝑝 , π‘ž ) ( 𝐸 , 1 ) mean reduces to ( 𝑁 , 1 / ( 𝑛 + 1 ) ) ( 𝐸 , 1 ) mean if 𝑝 𝑛 = 1 / ( 𝑛 + 1 ) a n d π‘ž 𝑛 = 1 , βˆ€ 𝑛 . (3) ( 𝑁 , 𝑝 , π‘ž ) ( 𝐸 , 1 ) method reduces to ( 𝑁 , π‘ž 𝑛 ) ( 𝐸 , 1 ) if 𝑝 𝑛 = 1 , βˆ€ 𝑛 . (4) ( 𝑁 , 𝑝 , π‘ž ) ( 𝐸 , 1 ) method reduces to ( 𝐢 , 𝛼 ) ( 𝐸 , 1 ) if 𝑝 𝑛 = ( 𝑛 + 𝛼 βˆ’ 1 𝛼 βˆ’ 1 ) , 𝛼 > 0 , a n d π‘ž 𝑛 = 1 , βˆ€ 𝑛 .Let 𝑓 ( 𝑑 ) be a periodic function with period 2 πœ‹ and integrable in the sense of Lebesgue over the interval ( βˆ’ πœ‹ , πœ‹ ) .
Let its Fourier series be given by
1 𝑓 ( 𝑑 ) ∼ 2 π‘Ž 0 + ∞  𝑛 = 1 ξ€· π‘Ž 𝑛 c o s 𝑛 𝑑 + 𝑏 𝑛 s i n ξ€Έ . 𝑛 𝑑 ( 1 . 8 ) We use the following notations: ξ€œ πœ™ ( 𝑑 ) = 𝑓 ( π‘₯ + 𝑑 ) βˆ’ 𝑓 ( π‘₯ βˆ’ 𝑑 ) βˆ’ 2 𝑓 ( π‘₯ ) , Ξ¦ ( 𝑑 ) = 𝑑 0 | | | |  1 πœ™ ( 𝑒 ) 𝑑 𝑒 , 𝜏 = 𝑑 ξ‚„ = t h e i n t e g r a l p a r t o f 1 𝑑 , 𝑅 ξ‚€ 1 𝑑  = 𝑅 𝜏 , 𝑅 𝑛 𝐾 = 𝑅 ( 𝑛 ) , 𝑛 1 ( 𝑑 ) = 2 πœ‹ 𝑅 𝑛 𝑛  π‘˜ = 0 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ c o s π‘˜ ( 𝑑 / 2 ) c o s ( π‘˜ + 1 ) ( 𝑑 / 2 ) s i n . ( 𝑑 / 2 ) ( 1 . 9 )

2. Theorem

A quite good amount of work is known for Fourier series by ordinary summability method. The purpose of this paper is to study Fourier series by ( 𝑁 , 𝑝 , π‘ž ) ( 𝐸 , 1 ) summability method in the following form.

Theorem 2.1. Let { 𝑝 𝑛 } a n d { π‘ž 𝑛 } be positive monotonic, nonincreasing sequences of real numbers such that 𝑅 𝑛 = 𝑛  π‘˜ = 0 𝑝 π‘˜ π‘ž 𝑛 βˆ’ π‘˜ ⟢ ∞ , a s 𝑛 ⟢ ∞ . ( 2 . 1 ) Let 𝛼 ( 𝑑 ) be a positive, nondecreasing function of 𝑑 . If ξ€œ Ξ¦ ( 𝑑 ) = 𝑑 0 | | | | ξ‚΅ 𝑑 πœ™ ( 𝑒 ) 𝑑 𝑒 = π‘œ ξ‚Ά , 𝛼 ( 1 / 𝑑 ) a s 𝑑 ⟢ + 0 , ( 2 . 2 ) 𝛼 ( 𝑛 ) ⟢ ∞ , a s 𝑛 ⟢ ∞ , ( 2 . 3 ) then a sufficient condition that the Fourier Series (1.8) be summable ( 𝑁 , 𝑝 , π‘ž ) ( 𝐸 , 1 ) to 𝑓 ( π‘₯ ) at the point 𝑑 = π‘₯ is ξ€œ 𝑛 1 𝑅 ( 𝑒 ) ξ€· 𝑅 𝑒 𝛼 ( 𝑒 ) 𝑑 𝑒 = 𝑂 𝑛 ξ€Έ , a s 𝑛 ⟢ ∞ . ( 2 . 4 )

3. Lemmas

Proof of the theorem needs some lemmas.

Lemma 3.1. For 0 ≀ 𝑑 ≀ 1 / 𝑛 , | | 𝐾 𝑛 | | ( 𝑑 ) = 𝑂 ( 𝑛 ) . ( 3 . 1 )

Proof. | | 𝐾 𝑛 | | = 1 ( 𝑑 ) 2 πœ‹ 𝑅 𝑛 | | | | | 𝑛  π‘˜ = 0 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ c o s π‘˜ ( 𝑑 / 2 ) s i n ( π‘˜ + 1 ) ( 𝑑 / 2 ) s i n | | | | | ≀ 1 ( 𝑑 / 2 ) 2 πœ‹ 𝑅 𝑛 𝑛  π‘˜ = 0 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ | | ( π‘˜ + 1 ) s i n | | ( 𝑑 / 2 ) | | s i n | | 1 ( 𝑑 / 2 ) = 𝑂 ( 𝑛 ) 𝑅 𝑛 𝑛  π‘˜ = 0 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ = 𝑂 ( 𝑛 ) . ( 3 . 2 )

Lemma 3.2. If { 𝑝 𝑛 } and { π‘ž 𝑛 } are nonnegative and nonincreasing, then for 0 ≀ π‘Ž ≀ 𝑏 < ∞ , 0 ≀ 𝑑 ≀ πœ‹ , and any 𝑛 we have 1 2 πœ‹ 𝑅 𝑛 | | | | | 𝑏  π‘˜ = π‘Ž 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ c o s π‘˜ ( 𝑑 / 2 ) s i n ( π‘˜ + 1 ) ( 𝑑 / 2 ) | | | | | ξ‚΅ 𝑅 s i n ( 𝑑 / 2 ) = 𝑂 𝜏 𝑑 𝑅 𝑛 ξ‚Ά . ( 3 . 3 )

Proof. 1 2 πœ‹ 𝑅 𝑛 | | | | | 𝑏  π‘˜ = π‘Ž 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ c o s π‘˜ ( 𝑑 / 2 ) s i n ( π‘˜ + 1 ) ( 𝑑 / 2 ) s i n | | | | | ≀ 1 ( 𝑑 / 2 ) 𝑑 πœ‹ 𝑅 𝑛 | | | | | 𝑏  π‘˜ = π‘Ž 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ c o s π‘˜ ξ‚€ 𝑑 2  s i n 𝑑 ( π‘˜ + 1 ) 2 | | | | | = 1 𝑑 πœ‹ 𝑅 𝑛 | | | | | I m ξƒ― 𝑏  π‘˜ = π‘Ž 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ c o s π‘˜ ξ‚€ 𝑑 2  𝑒 𝑖 ( π‘˜ + 1 ) ( 𝑑 / 2 ) ξƒ° | | | | | ≀ 1 𝑑 πœ‹ 𝑅 𝑛 | | | | | 𝑏  π‘˜ = π‘Ž 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ c o s π‘˜ ξ‚€ 𝑑 2  𝑒 𝑖 π‘˜ 𝑑 / 2 | | | | | | | 𝑒 𝑖 𝑑 / 2 | | ≀ 1 𝑑 πœ‹ 𝑅 𝑛 | | | | | 𝑏  π‘˜ = π‘Ž 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ c o s π‘˜ ξ‚€ 𝑑 2  𝑒 𝑖 π‘˜ 𝑑 / 2 | | | | | ≀ 1 𝑑 πœ‹ 𝑅 𝑛 ⎧ βŽͺ ⎨ βŽͺ ⎩ | | | | | 𝜏 βˆ’ 1  π‘˜ = π‘Ž 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ c o s π‘˜ ξ‚€ 𝑑 2  𝑒 𝑖 π‘˜ 𝑑 2 | | | | | + | | | | | | 𝑏  π‘˜ = 𝜏 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ c o s π‘˜ ξ‚€ 𝑑 2  𝑒  𝑖 π‘˜ 𝑑 2 ξƒͺ | | | | | | ⎫ βŽͺ ⎬ βŽͺ ⎭ . ( 3 . 4 ) Now considering first term of (3.4), we have 1 𝑑 πœ‹ 𝑅 𝑛 | | | | | 𝜏 βˆ’ 1  π‘˜ = π‘Ž 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ c o s π‘˜ ξ‚€ 𝑑 2  𝑒 𝑖 π‘˜ ( 𝑑 / 2 ) | | | | | ≀ 1 𝑑 πœ‹ 𝑅 𝑛 𝜏 βˆ’ 1  π‘˜ = π‘Ž 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ | | 𝑒 𝑖 π‘˜ ( 𝑑 / 2 ) | | ≀ 1 𝑑 πœ‹ 𝑅 𝑛 𝜏 βˆ’ 1  π‘˜ = π‘Ž 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ ≀ 1 𝑑 πœ‹ 𝑅 𝑛 𝜏 βˆ’ 1  π‘˜ = π‘Ž 𝑝 𝜏 βˆ’ π‘˜ π‘ž π‘˜ ≀ 1 𝑑 πœ‹ 𝑅 𝑛 ξ€· 𝑅 𝜏 ξ€Έ ξ‚΅ 𝑅 = 𝑂 𝜏 𝑑 𝑅 𝑛 ξ‚Ά . ( 3 . 5 ) Now considering second term of (3.4) and using Abel’s lemma, we have 1 𝑑 πœ‹ 𝑅 𝑛 | | | | | 𝑏  π‘˜ = 𝜏 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ c o s π‘˜ ξ‚€ 𝑑 2  𝑒 𝑖 π‘˜ ( 𝑑 / 2 ) | | | | | ≀ 1 𝑑 πœ‹ 𝑅 𝑛 | | | | | 𝑏  π‘˜ = 𝜏 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ 𝑒 𝑖 π‘˜ ( 𝑑 / 2 ) | | | | | ≀ 2 𝑝 𝑛 βˆ’ 𝑏 π‘ž 𝜏 𝑑 πœ‹ 𝑅 𝑛 m a x 𝜏 + 1 ≀ π‘˜ ≀ 𝑏 | | | | 1 βˆ’ 𝑒 𝑖 ( π‘˜ + 1 ) ( 𝑑 / 2 ) 1 βˆ’ 𝑒 𝑖 𝑑 / 2 | | | | ≀ 4 𝑝 𝑛 βˆ’ 𝑏 π‘ž 𝜏 𝑑 πœ‹ 𝑅 𝑛 | | | | 𝑒 βˆ’ 𝑖 𝑑 / 4 𝑒 𝑖 𝑑 / 4 βˆ’ 𝑒 βˆ’ 𝑖 𝑑 / 4 | | | | ≀ 2 π‘ž 𝜏 𝑑 πœ‹ 𝑅 𝑛 ξ‚΅ 𝑝 𝑛 βˆ’ 𝑏 𝑃 𝜏 ξ‚Ά 𝑃 𝜏 | | | | 1 s i n | | | |  ( 𝑑 / 4 ) w h e r e 𝑃 𝜏 = 𝜏  π‘˜ = 0 𝑝 𝜏 βˆ’ π‘˜ ξƒͺ ≀ 8 π‘ž 𝜏 𝑑 πœ‹ 𝑅 𝑛 ξ‚΅ 𝑝 𝑛 βˆ’ 𝑏 𝑃 𝜏 ξ‚Ά 𝑃 𝜏 | | | 1 𝑑 | | | ≀ 8 π‘ž 𝜏 𝑃 𝜏 𝑑 πœ‹ 𝑅 𝑛 ≀ 8 𝑅 𝜏 𝑑 πœ‹ 𝑅 𝑛  s i n c e 𝑅 𝜏 = 𝜏  π‘˜ = 0 𝑝 𝜏 βˆ’ π‘˜ π‘ž π‘˜ β‰₯ 𝑃 𝜏 π‘ž 𝜏 ξƒͺ ξ‚΅ 𝑅 = 𝑂 𝜏 𝑑 𝑅 𝑛 ξ‚Ά . ( 3 . 6 ) Using (3.5) and (3.6), we get the required result of Lemma 3.2.

4. Proof of Theorem

Following Zygmund [3], the 𝑛 th partial sum 𝑠 𝑛 ( π‘₯ ) of the series (1.8) at 𝑑 = π‘₯ is given by 𝑠 𝑛 1 ( π‘₯ ) = 𝑓 ( π‘₯ ) + ξ€œ 2 πœ‹ πœ‹ 0 πœ™ π‘₯ ( 𝑑 ) s i n ( 𝑛 + 1 / 2 ) 𝑑 s i n ( 𝑑 / 2 ) 𝑑 𝑑 . ( 4 . 1 )

So the ( 𝐸 , 1 ) mean of the series (1.8) at 𝑑 = π‘₯ is given by 𝐸 1 𝑛 1 ( π‘₯ ) = 2 𝑛 𝑛  π‘˜ = 0 ξ‚΅ 𝑛 π‘˜ ξ‚Ά 𝑠 π‘˜ 1 ( π‘₯ ) = 𝑓 ( π‘₯ ) + 2 𝑛 + 1 πœ‹ ξ€œ πœ‹ 0 πœ™ π‘₯ ( 𝑑 ) s i n ξƒ― ( 𝑑 / 2 ) 𝑛  π‘˜ = 0 ξ‚΅ 𝑛 π‘˜ ξ‚Ά s i n ξ‚€ 1 π‘˜ + 2  𝑑 ξƒ° 1 𝑑 𝑑 = 𝑓 ( π‘₯ ) + 2 𝑛 + 1 πœ‹ ξ€œ πœ‹ 0 πœ™ π‘₯ ( 𝑑 ) s i n ( 𝑑 / 2 ) I m ξ€½ 𝑒 𝑖 𝑑 / 2 ξ€· 1 + 𝑒 𝑖 𝑑 ξ€Έ 𝑛 ξ€Ύ 1 𝑑 𝑑 = 𝑓 ( π‘₯ ) + 2 𝑛 + 1 πœ‹ ξ€œ πœ‹ 0 πœ™ π‘₯ ( 𝑑 ) s i n ( 𝑑 / 2 ) I m ξ€½ 𝑒 𝑖 𝑑 / 2 ( 1 + c o s 𝑑 + 𝑖 s i n 𝑑 ) 𝑛 ξ€Ύ 1 𝑑 𝑑 = 𝑓 ( π‘₯ ) + 2 𝑛 + 1 πœ‹ ξ€œ πœ‹ 0 πœ™ π‘₯ ( 𝑑 ) s i n ( 𝑑 / 2 ) I m  𝑒 𝑖 𝑑 / 2 2 𝑛 c o s 𝑛 ξ‚€ 𝑑 2 𝑑  ξ‚€ c o s 2 + 𝑖 s i n 𝑑 2  𝑛  1 𝑑 𝑑 = 𝑓 ( π‘₯ ) + 2 𝑛 + 1 πœ‹ ξ€œ πœ‹ 0 πœ™ π‘₯ ( 𝑑 ) s i n ( 𝑑 / 2 ) I m  𝑒 𝑖 𝑑 / 2 2 𝑛 c o s 𝑛 ξ‚€ 𝑑 2   ξ‚€ c o s 𝑛 𝑑 2 + 𝑖 s i n 𝑛 𝑑 2 1    𝑑 𝑑 = 𝑓 ( π‘₯ ) + ξ€œ 2 πœ‹ πœ‹ 0 πœ™ π‘₯ ( 𝑑 ) c o s 𝑛 ( 𝑑 / 2 ) s i n ( 𝑛 + 1 ) ( 𝑑 / 2 ) s i n ( 𝑑 / 2 ) 𝑑 𝑑 . ( 4 . 2 )

Therefore ( 𝑁 , 𝑝 , π‘ž ) transform of { 𝐸 1 𝑛 ( π‘₯ ) } is given by 𝑑 𝑛 𝑝 , π‘ž , 𝐸 1 ( π‘₯ ) = 𝑅 𝑛 𝑛  π‘˜ = 0 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ 𝐸 1 π‘˜ 1 ( π‘₯ ) = 𝑓 ( π‘₯ ) + ξ€œ 2 πœ‹ πœ‹ 0 1 𝑅 𝑛 𝑛  π‘˜ = 0 𝑝 𝑛 βˆ’ π‘˜ π‘ž π‘˜ πœ™ π‘₯ ( 𝑑 ) c o s π‘˜ ( 𝑑 ) s i n ( π‘˜ + 1 ) ( 𝑑 / 2 ) s i n ξ€œ ( 𝑑 / 2 ) = 𝑓 ( π‘₯ ) + πœ‹ 0 𝐾 𝑛 ( 𝑑 ) πœ™ π‘₯ 𝑑 ( 𝑑 ) 𝑑 𝑑 , 𝑛 𝑝 , π‘ž , 𝐸 ξ‚Έ ξ€œ ( π‘₯ ) βˆ’ 𝑓 ( π‘₯ ) = 0 1 / 𝑛 + ξ€œ 𝛿 1 / 𝑛 + ξ€œ πœ‹ 𝛿 ξ‚Ή 𝐾 0 π‘₯ 0 2 0 0 𝑑 𝑛 ( 𝑑 ) πœ™ π‘₯ ( 𝑑 ) 𝑑 𝑑 = 𝐼 1 + 𝐼 2 + 𝐼 3 ( s a y ) . ( 4 . 3 ) We have | | 𝐼 1 | | ≀ ξ€œ 0 1 / 𝑛 | | 𝐾 𝑛 | | | | πœ™ ( 𝑑 ) π‘₯ | | ξ€œ ( 𝑑 ) 𝑑 𝑑 = 𝑂 ( 𝑛 ) 0 1 / 𝑛 | | πœ™ π‘₯ | | ( ( 𝑑 ) 𝑑 𝑑 u s i n g L e m m a ) ξ‚΅ 1 3 . 1 = 𝑂 ( 𝑛 ) π‘œ ξ‚Ά ( 𝑛 𝛼 ( 𝑛 ) b y ξ‚΅ 1 ( 2 . 2 ) ) = π‘œ ξ‚Ά 𝛼 ( 𝑛 ) = π‘œ ( 1 ) a s 𝑛 ⟢ ∞ ( b y ( 2 . 3 ) ) . ( 4 . 4 )

Now we consider | | 𝐼 2 | | ≀ ξ€œ 𝛿 1 / 𝑛 | | 𝐾 𝑛 | | | | πœ™ ( 𝑑 ) π‘₯ | | ( 𝑑 ) 𝑑 𝑑 ( w h e r e = ξ€œ 0 < 𝛿 < 1 ) 𝛿 1 / 𝑛 𝑂 ξ‚΅ 𝑅 ( 1 / 𝑑 ) 𝑑 𝑅 𝑛 ξ‚Ά | | πœ™ π‘₯ | | ( ( 𝑑 ) 𝑑 𝑑 u s i n g L e m m a ) ξ‚΅ 1 3 . 2 = 𝑂 𝑅 𝑛 ξ‚Ά ξ€œ 𝛿 1 / 𝑛 ξ‚΅ 𝑅 ( 1 / 𝑑 ) 𝑑 ξ‚Ά | | πœ™ π‘₯ | | ξ‚΅ 1 ( 𝑑 ) 𝑑 𝑑 = 𝑂 𝑅 𝑛 ξ‚Ά  ξ‚» ξ‚΅ 𝑅 ( 1 / 𝑑 ) 𝑑 ξ‚Ά πœ™ π‘₯ ξ‚Ό ( 𝑑 ) 𝛿 1 / 𝑛 βˆ’ ξ€œ 𝛿 1 / 𝑛 𝑑 ξ‚΅ 𝑅 ( 1 / 𝑑 ) 𝑑 ξ‚Ά πœ™ π‘₯ ξƒ­ ξ‚΅ 1 ( 𝑑 ) = 𝑂 ξ‚Ά  ξ‚» π‘œ ξ‚΅ 𝑅 ( 𝑛 ) 𝑅 ( 1 / 𝑑 ) 𝛼 ( 1 / 𝑑 ) ξ‚Ά ξ‚Ό 𝛿 1 / 𝑛 βˆ’ ξ€œ 𝛿 1 / 𝑛 πœ™ π‘₯ ξ‚΅ ( 𝑑 ) 𝑑 𝑅 ( 1 / 𝑑 ) 𝑑 ξ‚Ά ξƒ­ ( b y ξ‚΅ 1 ( 2 . 1 ) ) = π‘œ ξ‚Ά ξ‚΅ 1 𝑅 ( 𝑛 ) + π‘œ ξ‚Ά ξ‚΅ 1 𝛼 ( 𝑛 ) + π‘œ ξ€œ 𝑅 ( 𝑛 ) ξ‚Ά ξ‚Έ 𝛿 1 / 𝑛 πœ™ π‘₯ ( ξ‚» 𝑑 ξ‚΅ 𝑑 ) 𝑅 ( 1 / 𝑑 ) 𝛼 ( 1 / 𝑑 ) ξ‚΅ 1 𝑑 𝛼 ( 1 / 𝑑 ) ξ‚Ά ξ‚Ό ξ‚Ή = π‘œ ξ‚Ά ξ‚΅ 1 𝑅 ( 𝑛 ) + π‘œ ξ‚Ά ξ‚΅ 1 𝛼 ( 𝑛 ) + π‘œ ξ‚Ά Γ— ξ‚Έ ξ€œ 𝑅 ( 𝑛 ) 𝛿 1 / 𝑛 π‘œ ξ‚΅ 𝑑 ξ‚΅ 𝛼 ( 1 / 𝑑 ) ξ‚Ά ξ‚» 𝑑 𝛼 ( 1 / 𝑑 ) 𝑅 ( 1 / 𝑑 ) ξ‚€ 1 𝑑 𝛼 ( 1 / 𝑑 ) ξ‚Ά ξ‚Ό + 𝛼 𝑑  𝑑 ξ‚΅ 𝑅 ( 1 / 𝑑 ) ξ‚΅ 1 𝑑 𝛼 ( 1 / 𝑑 ) ξ‚Ά ξ‚Ή = π‘œ 𝑅 ξ‚Ά ξ‚΅ 1 ( 𝑛 ) + π‘œ 𝛼 ξ‚Ά ξ‚Έ ξ€œ ( 𝑛 ) + π‘œ ( 1 ) 𝛿 1 / 𝑛 𝑑 𝛼 ( 1 / 𝑑 ) { 𝛼 ( 1 / 𝑑 ) } 2 ξ‚΅ 1 + π‘œ 𝑅 ξ‚Ά ξ€œ ( 𝑛 ) 𝛿 1 / 𝑛 ξ‚΅ 𝑑 𝑑 𝑅 ( 1 / 𝑑 ) ξ‚΅ 1 𝑑 𝛼 ( 1 / 𝑑 ) ξ‚Ά ξ‚Ή = π‘œ ξ‚Ά ξ‚΅ 1 𝑅 ( 𝑛 ) + π‘œ ξ‚Ά ξ‚» 1 𝛼 ( 𝑛 ) + π‘œ ( 1 ) ξ‚Ό 𝛼 ( 1 / 𝑑 ) 𝛿 1 / 𝑛 ξ‚΅ 1 + π‘œ ξ‚Ά  ξ‚» 𝑅 ( 𝑛 ) 𝑑 𝑅 ( 1 / 𝑑 ) ξ‚Ό 𝑑 𝛼 ( 1 / 𝑑 ) 𝛿 1 / 𝑛 βˆ’ ξ€œ 𝛿 1 / 𝑛 ξ‚΅ 𝑅 ( 1 / 𝑑 ) ξ‚Ά ξƒ­ ξ‚΅ 1 𝑑 𝛼 ( 1 / 𝑑 ) 𝑑 𝑑 = π‘œ ξ‚Ά ξ‚΅ 1 𝑅 ( 𝑛 ) + π‘œ ξ‚Ά ξ‚΅ 1 𝛼 ( 𝑛 ) + π‘œ ξ‚Ά ξ‚΅ 1 𝛼 ( 𝑛 ) + π‘œ ξ‚Ά ξ€œ 𝑅 ( 𝑛 ) 1 1 / 𝑛 ξ‚΅ 𝑅 ( 1 / 𝑑 ) ξ‚Ά ξ‚΅ 1 𝑑 𝛼 ( 1 / 𝑑 ) 𝑑 𝑑 = π‘œ ξ‚Ά ξ‚΅ 1 𝑅 ( 𝑛 ) + π‘œ ξ‚Ά ξ‚΅ 1 𝛼 ( 𝑛 ) + π‘œ ξ‚Ά ξ€œ 𝑅 ( 𝑛 ) 𝑛 1 ξ‚΅ 𝑅 ( 𝑒 ) ξ‚Ά  ∡ 1 𝑒 𝛼 ( 𝑒 ) 𝑑 𝑒 𝑑  ξ‚΅ 1 = 𝑒 = π‘œ ξ‚Ά ξ‚΅ 1 𝑅 ( 𝑛 ) + π‘œ ξ‚Ά ξ‚΅ 1 𝛼 ( 𝑛 ) + π‘œ ξ‚Ά 𝑂 ξ€· 𝑅 𝑅 ( 𝑛 ) 𝑛 ξ€Έ ( b y ξ‚΅ 1 ( 2 . 4 ) ) = π‘œ ξ‚Ά ξ‚΅ 1 𝑅 ( 𝑛 ) + π‘œ ξ‚Ά 𝛼 ( 𝑛 ) + π‘œ ( 1 ) = π‘œ ( 1 ) , a s 𝑛 β†’ ∞ ( b y v i r t u e o f ( 2 . 1 ) a n d ( 2 . 2 ) ) . ( 4 . 5 )

Now by Riemann-Lebesgue theorem and by regularity of the method of summability we have

𝐼 3 = ξ€œ πœ‹ 𝛿 | | π‘˜ 𝑛 | | | | πœ™ ( 𝑑 ) π‘₯ | | ( 𝑑 ) 𝑑 𝑑 = π‘œ ( 1 ) , a s 𝑛 ⟢ ∞ . ( 4 . 6 )

This completes the proof of the theorem.

5. Corollaries

Following corollaries can be derived from our main theorem.

Corollary 5.1. If ξ‚Έ 𝑑 Ξ¦ ( 𝑑 ) = π‘œ ξ‚Ή , l o g ( 1 / 𝑑 ) a s 𝑑 ⟢ + 0 , ( 5 . 1 ) then the Fourier series (1.8) is ( 𝐢 , 1 ) ( 𝐸 , 1 ) summable to 𝑓 ( π‘₯ ) at the point 𝑑 = π‘₯ .

Corollary 5.2. If Ξ¦ ( 𝑑 ) = π‘œ ( 𝑑 ) , a s 𝑑 ⟢ + 0 , ( 5 . 2 ) then the Fourier series (1.8) is ( 𝑁 , 𝑝 𝑛 ) ( 𝐸 , 1 ) summable to 𝑓 ( π‘₯ ) at the point 𝑑 = π‘₯ , provided that { 𝑝 𝑛 } be a positive, monotonic, and nonincreasing sequence of real numbers such that 𝑝 𝑛 = 𝑝 0 + 𝑝 1 + β‹― + 𝑝 𝑛 ⟢ ∞ , a s 𝑛 ⟢ ∞ . ( 5 . 3 )

Acknowledgments

The authors are grateful to Professor Shyam Lal, Department of Mathematics, Harcourt Butler Technological Institute, Kanpur, India, for his valuable suggestions and guidence in preparation of this paper. The authors are also thankful to Professor M. P. Jain, Vice Chancellor, Mody Institute of Technology and Science (Deemed University), Lakshmangarh, Sikar, Rajasthan, India, and to Professor S. N. Puri, Former Dean, Faculty of Engineering, Technology, Mody Institute of Technology and Science (Deemed University), Laxmangarh, Sikar, Rajasthan.

References

  1. D. Borwein, β€œOn product of sequences,” Journal of the London Mathematical Society, vol. 33, pp. 352–357, 1958. View at Zentralblatt MATH Β· View at MathSciNet
  2. G. H. Hardy, Divergent Series, Oxford University Press, Oxford, UK, 1st edition, 1949. View at Zentralblatt MATH Β· View at MathSciNet
  3. A. Zygmund, Trigonometric Series. Vol. I, Cambridge University Press, Cambridge, UK, 2nd edition, 1959. View at Zentralblatt MATH Β· View at MathSciNet