A New Theorem on Absolute Matrix Summability of Fourier Series. Şebnem Yildiz

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1 PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouelle série, tome 0??)) 20?), Prliminary ersion; to be edited DOI: Not assigned yet A New Theorem on Absolute Matrix Summability of Fourier Series Şebnem Yildiz Abstract. We generalize a main theorem dealing with absolute weighted mean summability of Fourier series to the A, k summability factors of Fourier series under weaker conditions. Also some new and known results are obtained.. Introduction Let a n be a gien infinite series with artial sums s n ). By u α n and t α n we denote the nth Cesàro means of order α, with α >, of the sequence s n ) and na n ), resectiely, that is see [6]).) where.2) A α n = u α n = A α n n =0 A α n s and t α n = A α n α + )α + 2)...α + n) n! n =0 A α n a, = On α ), A α n = 0 for n > 0. The series a n is said to be summable C, α k, k, if see [8],[0]) n u α n u α n k.3) = n tα n k <. If we take α =, then C, α k summability reduces to C, k summability. Let ) be a sequence of ositie real numbers such that n.4) = as n, P i = i = 0, i ). =0 200 Mathematics Subject Classification: 26D5; 42A24; 40F05; 40G99. Key words and hrases: Summability factors, absolute matrix summability, Fourier series, infinite series, Hölder inequality, Minkowski inequality. Communicated by Gradimir Miloanoić.

2 2 YILDIZ The sequence-to-sequence transformation.5) t n = n s defines the sequence t n ) of the Riesz mean or simly the N, ) mean of the sequence s n ) generated by the sequence of coefficients ) see [9]). The series a n is said to be summable N, n k, k, if see []) ).6) t n t n k <. In the secial case when = for all alues of n res. k = ), N, n k summability is the same as C, k res. N, n ) summability. =0 2. Known Results Following theorems are dealing with N, n k summability factors of infinite series. 2.) Theorem 2.. [2]) Let ) be a sequence of ositie numbers such that = On ) as n. Let X n ) be a ositie monotonic nondecreasing sequence. If the sequences X n ), λ n ) and ) satisfy the conditions 2.2) λ m X m as m, 2.3) m nx n 2 λ n as m, 2.4) m t n k = OX m ) as m, then the series a n λ n is summable N, n k, k. Theorem 2.2. [5]) Let X n ) be a ositie monotonic nondecreasing sequence. If the sequences X n ), λ n ), and ) satisfy the conditions 2.)-2.3) and m t n k 2.5) = OX m ) as m, Xn then the series a n λ n is summable N, n k, k. REMARK It should be noted that condition 2.5) is reduced to the condition 2.4), when k =. When k >, condition 2.5) is weaker than condition 2.4) but the conerse is not true see [5] for details).

3 A NEW THEOREM ON ABSOLUTE MATRIX SUMMABILITY OF FOURIER SERIES 3 3. An alication of absolute matrix summability to Fourier series Let A = a n ) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, maing the sequence s = s n ) to As = A n s)), where 3.) A n s) = n a n s, n = 0,,... =0 The series a n is said to be summable A k, k, if see [3]) 3.2) n An s) k <, and it is said to be summable A, k, k, if see [2]) ) 3.3) An s) k <. where 3.4) A n s) = A n s) A n s). If we take = for all n, A, k summability is the same as A k summability. Also, if we take a n =, then A, k summability is the same as N, n k summability. For any sequence λ n ) we write that 2 λ n = λ n λ n+ and λ n = λ n λ n+. A sequence λ n ) is said to be of bounded ariation, denoted by λ n ) BV, if λ n <. Let ft) be a eriodic function with eriod 2π, and Lebesgue integrable oer π, π). Write 3.5) fx) 2 a 0 + a n cosnx + b n sinnx) = C n x), ϕt) = 2 {fx + t) + fx t)}, and ϕ αt) = α t α t 0 t u)α ϕu) du, α > 0). It is well known that if ϕt) BV0, π), then t n x), where t n x) is the C, ) mean of the sequence nc n x)) see [7]). Many works hae been done dealing with absolute summability factors of Fourier series see [3]-[5],[]). Among them, in [5], Bor has roed the following theorem dealing with the Fourier series. n=0

4 4 YILDIZ Theorem 3.. If ϕ t) BV0, π), X n ) is a ositie monotonic nondecreasing sequence, the sequences ), λ n ) satisfy the conditions 2.)-2.3) and m t n x) k 3.6) = OX m ) as m, Xn then the series C n x)λ n is summable N, n k, k. If we take = for all alues of n, then we obtain a new result dealing with C, k summability factors of Fourier series. 4. Main Results The aim of this aer is to generalize Theorem 3. for A, k summability factors of Fourier series. Before stating the main theorem, we must first introduce some further notations. Gien a normal matrix A = a n ), we associate two lower semimatrices Ā = ā n) and  = â n) as follows: n 4.) ā n = a ni, n, = 0,,... and 4.2) i= â 00 = ā 00 = a 00, â n = ā n ā n,, n =, 2,... It may be noted that Ā and  are the well-known matrices of series-to-sequence and series-to-series transformations, resectiely. Then, we hae n n 4.3) A n s) = a n s = ā n a and 4.4) 4.5) 4.6) 4.7) 4.8) =0 A n s) = =0 n â n a. =0 Theorem 4.. Let k and A = a n ) be a ositie normal matrix such that a no =, n = 0,,..., a n, a n, for n +, a nn = O ), â n,+ = O â n ). If all the conditions of Theorem 3. are satisfied, then the series C n x)λ n is summable A, k, k. It should be noted that if we take a n =, then we get Theorem 3.. We need the following lemma for the roof of our theorem.

5 A NEW THEOREM ON ABSOLUTE MATRIX SUMMABILITY OF FOURIER SERIES 5 Lemma 4.. [2]) Under the conditions of Theorem 2.2 we hae that 4.9) 4.0) nx n λ n as n, X n λ n <. Proof of Theorem 4. Let I n x)) denotes the A-transform of the series C nx)λ n. Then, by 4.3) and 4.4), we hae I n x) = n â n C x)λ. Alying Abel s transformation to this sum, we get that n n I n x) = â n C x)λ = ânλ ) n = ânλ n = â n )λ t x) + r= rc r x) + ânnλ n n n + ) + )t x) + â nn λ n n t nx) n + = I n, x) + I n,2 x) + I n,3 x) + I n,4 x). â n,+ λ t x) + n rc r x) r= n + â n,+ λ + t x) To comlete the roof of Theorem 4., by Minkowski s inequality, it is sufficient to show that ) 4.) I n,rx) k <, for r =, 2, 3, 4. First, by alying Hölder s inequality with indices k and k, where k > and =, we hae that k + k ) I n,x) k ) { n ) n n + â n ) λ t x) ) a nn â n ) λ k t x) k { n { n â n ) } â n ) λ k t x) k + a nn λ n t n x) n + n }

6 6 YILDIZ m m λ λ t x) k X λ λ t x) k P r= Xr n=+ λ X + O) λ m X m as m, â n ) r t r x) k m t x) k + O) λ m P r P X by irtue of the hyotheses of Theorem 4. and Lemma 4.. Now, using Hölder s inequality we hae that ) I n,2x) k ) { n + â n,+ λ t x) ) { n â n,+ λ t x) n ) n n ) n a nn λ ) k â n ) t x) k λ ) k â n ) t x) k m λ ) λ ) t x) k â n ) m P X λ ) t x) k λ ) r= r P r Xr n=+ λ ) X + O)m λ m X m { n t r x) k + O)m λ m â n ) m X 2 λ + O) X λ + O)m λ m X m as m, P X } t x) k

7 A NEW THEOREM ON ABSOLUTE MATRIX SUMMABILITY OF FOURIER SERIES 7 by irtue of the hyotheses of Theorem 4. and Lemma 4.. Again, we hae that ) I n,3x) k = n ) n ) { n n ) n n m m m â n,+ λ + t x) â n,+ λ + t x) ) { n â n ) λ + t x) ) n a nn k â n ) λ + k t x) k λ + k t x) k n=+ â n ) λ + k t x) k â n ) P t x) k λ + λ + X as m, λ + t x) k P { n â n ) } by irtue of the hyotheses of Theorem 4. and Lemma 4.. Finally, as in T n,, we hae that m ) I n,4x) k m m X n ) a k nn λ n k t n x) k m λ n t n x) k as m, λ n λ n t n x) k by irtue of hyotheses of the Theorem 4. and Lemma 4.. This comletes the roof of Theorem 4.. If we take a n = in Theorem 4., then we get Theorem 3. and if we take = for all alues of n in Theorem 4., then we get a new result dealing with the A k summability method. Also, if we take a n = and = for all alues of n in Theorem 4., then we get a result concerning the C, k summability methods.

8 8 YILDIZ References. H. Bor, On two summability methods, Math. Proc. Cambridge Philos Soc ), H. Bor, On absolute summability factors, Proc. Amer. Math. Soc ), H. Bor, D. S. Yu and P. Zhou, On local roerty of absolute summability of factored Fourier series, Filomat ), H. Bor, Some new results on infinite series and Fourier series, Positiity 9 205), H. Bor, On absolute weighted mean summability of infinite series and Fourier series, Filomat in ress). 6. E. Cesàro, Sur la multilication des sèries, Bull. Sci. Math ), K. K. Chen, Functions of bounded ariation and the cesaro means of Fourier series, Acad. Sin. Sci. Record 945), T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. Lond. Math. Soc ), G. H. Hardy, Diergent Series, Oxford Uni. Press, Oxford 949). 0. E. Kogbetliantz, Sur lès series absolument sommables ar la methode des moyennes arithmetiques, Bull. Sci. Math ), H. S. Özarslan, Ş. Yıldız, A new study on the absolute summability factors of Fourier series, J. Math. Anal ), W. T. Sulaiman, Inclusion theorems for absolute matrix summability methods of an infinite series, IV. Indian J. Pure Al. Math ), N. Tanoi c-miller, On strong summability, Glas. Mat. Ser III 4 34) 979), Deartment of Mathematics Receied ) Ahi Eran Uniersity Kırşehir, Turkey sebnemyildiz@ahieran.edu.tr; sebnem.yildiz82@gmail.com