Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions Ratio Mathematica Volume 38, 2020, pp. 341-348 341 Approximation of functions by (C,2)(E,1) product summability method of Fourier series Jitendra Kumar Kushwaha* Abstract Various investigators such as Leindler [10], Chandra [1], Mishra et al. [7], Khan [11], Kushwaha [6] have determined the degree of approximation of 2 -periodic functions belonging to classes Lip , ),( rLip  , )),(( rtLip  of functions through trigonometric Fourier approximation using different summability means. Recently Nigam [12] has determined that the Fourier series is summable under the summability means (C,2)(E,1) but he did not find the degree of approximation of function belonging to various classes. In this paper a theorem concerning the degree of approximation of function f belonging to )),(( rtLip  class by (C,2)(E,1) product summability method of Fourier series has been established which in turn generalizes the result of H. K. Nigam [12]. Keywords: Degree of approximation; Fourier series; Pruduct summability methods. 2010 AMS subject classification: 42B05; 42B08.† *Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur, India. jitendra.mathstat@ddugu.ac.in. † Received on March 29th, 2020. Accepted on June 19th, 2020. Published on June 30th, 2020. doi: 10.23755/rm.v38i0.504. ISSN: 1592-7415. eISSN: 2282-8214. ©Jitendra Kumar Kushwaha. Jitendra Kumar Kushwaha 342 1. Introduction The study of the theory of trigonometric approximation is of great mathematical interest and of great practical importance. Broadly speaking, signals are treated as function of single variable and images are represented by function of two variables. The study of these concepts is directly related to the emerging area of information technology. Studies on trigonometric approximation of functions in pL -norm using different linear operators such Hölder,Nörlund, Euler, Riesz, Borel etc. were made by several researchers like Mohapatra & Chandra [9], Holland, Mohapatra & sahney [8], Chandra [2]. The degree o approximation of a function belonging to different class of functions by product summability methods were made by Lal & Singh [5], Lal & Kushwaha [6]. The aim of this paper is to study Fourier series and conjugate series by product operators. The advantage of considering product operators over linear operators can be understood with the observation that the infinite series, which is neither summable by left linear operators nor by right linear operators individually, is summable to some number by the product operators obtained from the same linear operators placed in the same sequential order. Moreover , in studies of error estimates )( fEn through Trigonometric Fourier Approximation, product operators give better approximation than individual linear operators. Generalizing the result of Nigam [12], the degree of approximation of function f belonging to )),(( rtLip  class by (C,2)(E,1) product summability method of Fourier series has been established. Therefore, in this paper, (C,2)(E,1) product summability method is introduced and a theorem on the approximation of functions belonging to ( )rtL ),( class has been established. Let   =0n n u be given infinite series with ns for its th n partial sum. Let  1Ent denote the sequence of (E,1) mean of the sequence  ns . If the (E,1) transform of ns is defined as →→      =  = nassxfs k n xft k n k n E n );( 2 1 );( 0 1 (1.1) The series   =0n n u is said to summable to the number s by the (E,1) method (Hardy [14]). Approximation of functions by (C,2)(E,1) product summability method of Fourier series 343 Let  2C n t denote the sequence of (C, 2) mean of the sequence   n s . If the (C, 2) transform of ns is defined as ( ) →→+− ++ =  = nassxfskn nn xft k n k C n );(1 )2)(1( 2 );( 0 2 (1.2) the series   =0n n u is said to be summable to the number s by (C, 2) method (Cesàro method). Thus if ( ) →→      +− ++ =  == nassxfs v n kn nn xft v n v k n k EC n );( 2 1 1 )2)(1( 2 );( 00 . 12 ( 1.3) Where 12 . EC n t denotes the sequence of (C,2)(E,1) product mean of the sequence ns . The series   =0n n u is said to summable to the number s by (C,2)(E,1) method. We observe that (C,2)(E,1) method is regular. Let f be 2 -periodic and Lebesgue integrable function. The Fourier series associated with f at a point x is defined by ( ) )(sincos 2 ~)( 01 0 xAnxbnxa a xf n n n nn   =  = ++ (1.4) with partial sum );( xfs n . Throughout this paper, we use following notations: )()()(),()( xftxftxftxt −−++==   = =               +      +− ++ = n k k v kn t tv v kkn nn tM 0 0 )2/sin( )2/1sin( 2 1 )2)(1( 1 )(  . 2. Main Theorem We prove the following theorem Theorem . If RRf →: is 2 -periodic, Lebesgue integrable on ],[ − and belonging to ( )rtLip ),( class then the degree of approximation of f by the (C,2)(E,1) product means ( ) );( 2 1 1 )2)(1( 2 );( 00 . 12 xfs v n kn nn xft v n v k n k EC n  ==       +− ++ = of its Fourier series (1.4) is given by Jitendra Kumar Kushwaha 344             + =− 1 1 12 . n Oft r EC n  . 3. Lemmas 3.1 Lemma 1 For )1/(10 + nt , )1()( += nOtK n . Proof For )1/(10 + nt , tnnt sinsin    = =       +      +− ++  n k k kn t kkn nn tM 0 0 )2/sin( )2/1sin( 2 )1( )2)(1( 1 )(      = =       +      +− ++  n k k kn t tkkn nn tM 0 0 )2/sin( )2/sin()12( 2 )1( )2)(1( 1 )(      = =             + +− ++  n k k k k k kn nn 0 0 )12( 2 )1( )2)(1( 1   =  = ++− ++ n k kkn nn 0 )]12)(1[( )2)(1( 1   == + ++ −+ ++ + = n k n k kk nn k nn n 00 )]12([ )2)(1( 1 )12( )2)(1( 1        + ++ −+ + =  === n k n k n k kk nn k n 00 2 0 2 )2)(1( 1 )12( )2( 1        + + ++ ++ − + + = 2 )1( 3 )12)(1( )2)(1( 1 )2( )1( 2 nnnnn nnn n  ).1( += nO 3.2 Lemma 2 For + tn )1/(1 , )./1()( tOtK n = Proof For + tn )1/(1 , applying Jordan’s lemma, /)2/sin( tt  and 1sin nt .   = =             ++ +  n k k k t k nn n 0 0 )/( 1 2 1 )2)(1( )1(     = =       +      +− ++  n k k kn t kkn nn tM 0 0 )2/sin( )2/1sin( 2 )1( )2)(1( 1 )(    Approximation of functions by (C,2)(E,1) product summability method of Fourier series 345 -   = =             ++ n k k k t kk nn 0 0 )/( 1 2)2)(1( 1    == ++ − + = n k n k k nntnt 00 )2)(1( 1 1 )2( 1 )./1( tO= 4. Proof of the Theorem Following Titchmarsh [13] and using Riemann Lebesgue theorem, );( xfs n of the series (1.4) is given by dt t tn txfxfs n )2/sin( )2/1sin( )( 2 1 )();( 0 + =−     Using (1), the (E,1) transforms of );( xfs n is given by dt t tn k n txft n k n E n         +       =−  = + )2/sin( )2/1sin( )( 2 1 )( 00 1 )1,(    The (C,2) (E,1) transform of );( xfs n is given by   = =               +     +− ++ =− n k k k EC n dtt k t tkn nn xft 0 00 . )2/1sin( )2/sin( )( 2 )1( )2)(1( 1 )(12       dttKt n )()( 0 =   dttKtdttKt n n n n )()()()( )1/(1 )1/(1 0  + + +=   21 II += (4.1) Now , dttKtI n n )()( )1/(1 0 1  + =  =         + + dtntO n )1()( )1/(1 0  , by Lemma (1)         +=  + )1/(1 0 )()1( n dttnO  Jitendra Kumar Kushwaha 346  +             + += )1/(1 1 1 )1( n dt n nO   , where )1/(10 + n by first mean value theorem of calculus             + = 1 1 n O  . (4.2) Lastly,         =  + dttKtOI n n )()( )1/(1 2           + =  + dt tn t O n   )1/(1 )1( )( , by lemma ( 2)             + = 1 1 n O  . (4.3) Combining (4.1)-(4.3), we get             + =− 1 1 12 . n Oft r EC n  . This completes the proof of the theorem. 5. Conclusions The result of main theorem is . 1 1 12 .             + =− n Oft r EC n  from which the results of H.K. Nigam [12] can be derived directly. Acknowledgement Author is highly thankful to Professor Shyam Lal, Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, India for his encouragement and support to this work. Approximation of functions by (C,2)(E,1) product summability method of Fourier series 347 References [1] P. Chandra, Approximation by Nörlund operators, Mat. Vestnik, Vol. 8, 263-269, 1986. [2] P. Chandra, functions of classes p L and ),( pLip  and their Riesz means, Riv. Mat. Univ. Parm, Vol. 4, No. 12, 275-282, 1986. [3] ] P. Chandra , On the degree of approximation of a class of function by means of Fourier series , Acta Mathematica Hungarica, Vol. 52 , No. 3-4, 199- 205, 1988. [4] A.B.S. Hollend, R.N. Mohapatra and B.N. Sahney , pL approximation of function by Euler means, Rendiconti di Mathematica (Rome)(2), Vol. 3, 341- 355, 1983. [5] S. Lal and P.N. Singh, Degree o approximation of conjugate of function by (C,1)(E,1) means of conjugate series of Fourier series, Tamkang Journal of Mathematics, Vol. 33, No. 3, 269-274, 2002. [6] S. Lal and J. K. Kushwaha, Degree of approximation of Lipschitz Function by (C,1)(E,q) product Summability Method, Int. Math. Forum Vol.4 (no. 43), 2009, pp. 2101-2107. [7] V.N. Mishra and L. N. Mishra, Trigonometric Approximation by signals(Functions) in pL -norm, International Journal of Contemporary Mathematical Sciences , Vol. 7, No. 19, 2012, pp. 909-918. [8] R.N. Mohapatra and B.N. Sahney, Approximation of continuous function by their Fourier series, Mathemtica: Journal L’ Anlyse Numerique la Theorie de l’approxiamtion, Vol.10, 81-87, 1981. [9] R. N. Mohapatra and P. Chandra, Holder continuous function and their Euler, Borel and Taylor meas, Math. Cronicle (New Zealand), Vol. 11 81-96, 1988. [10] L. Leindler, Trigonometric Approximation in pL norm, Journal of Mathematical Analysis and Applicaiton, Vol. 302, No. 1, 2005, pp. 129-136. [11] H.H. Khan, On Degree of Approximation of a functions Belonging to class ),( pLip  , Indian Journal of Pure and Applied Mathematics, Vol. 5, No. 2, 1974,pp. 132-136. [12] H.K. Nigam, On (C,2)(E,1) Product means of Fourier series, Electronic Journal of Mathematical Analysis an Application, Vol.1(2) July 2013, pp. 334- 344. Jitendra Kumar Kushwaha 348 [13] E.C. Titchmarsh, The Theory of functions, Oxford University Press, 402- 403, 1939. [14] G.H. Hardy, Divergent Series, Oxford University Press, Oxford, 1949.