Abstract
The theory of approximation is a very extensive field and the study of the theory of trigonometric approximation is of great mathematical interest and of great practical importance. Positive approximation processes play an important role in approximation theory and appear in a very natural way dealing with approximation of continuous functions, especially one, which requires further qualitative properties, such as monotonicity, convexity, and shape preservation and so on. The theory of summability arises from the process of summation of series and the significance of the concept of summability has been strikingly demonstrated in various contexts, e.g., in Fourier analysis, fractional calculus, analytic continuation, quantum mechanics, probability theory, approximation theory, nonlinear analysis, and fixed point theory. The methods of almost summability and statistical summability have become an active area of research in recent years. This chapter contains two sections. In the first section, an attempt is made to obtain a theorem on the degree of approximation of functions belonging to the \(Lip\, (\alpha , r)\)-class, using almost Riesz summability method of its infinite Fourier series, so that some theorems become particular case of our main theorem. In the second section, a theorem concerning the degree of approximation of the conjugate of a function f belonging to \(Lip\, (\xi (t), r)\)-class by Euler (E, q) summability of conjugate series of its Fourier series has been established which in turn generalizes the results of Shukla [Certain Investigations in the theory of Summability and that of Approximation, Ph.D. Thesis, 2010, V.B.S. Purvanchal University, Jaunpur (Uttar Pradesh)] and others.
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References
Chandra, P.: On the degree of approximation of continuous functions. Commun. Fac. Sci. Uni. Ankara 30(A), 7–16 (1981)
Chandra, P.: Trigonometric approximation of functions in \(L_{p} \)-norm. J. Math. Anal. Appl. 275, 13–26 (2002)
Das, G., Kuttner, B., Nanda, S.: Some sequence spaces and absolute almost convergence. Trans. Am. Math. Soc. 283(2), 729–739 (1984)
Hardy, G.H.: Divergent Series, 1st edn. Oxford University Press (1949)
Khan, H.H.: On the degree of approximation to a function belonging to weighted \((L^{p} ,\, \xi (t))\) class. Aligarh Bull. Math. 3–4, 83–88 (1973–1974)
Khan, H.H.: On degree of approximation to a functions belonging to the class \(Lip\, (\alpha, \, p)\). Indian J. Pure Appl. Math. 5, 132–136 (1974)
Khan, H.H.: On the degree of approximation to a function by triangular matrix of its Fourier series I. Indian J. Pure Appl. Math. 6, 849–855 (1975)
Khan, H.H.: On the degree of approximation to a function by triangular matrix of its conjugate Fourier series II. Indian J. Pure Appl. Math. 6, 1473–1478 (1975)
Khan, H.H.: A note on a theorem Izumi, Communications. De La Facult Des Sciences Mathmatiques Ankara (TURKEY) 31, 123-127 (1982)
Khan, H.H., Ram, G.: On the degree of approximation. Facta Universitatis Series Mathematics and Informatics (TURKEY) 18, 47–57 (2003)
King, J.P.: Almost summable sequences. Proc. Am. Math. Soc. 17, 1219–1225 (1966)
Kuttner, B., Sahney, B.N.: On nonuniqueness of the degree of saturation. Math. Proc. Camb. Philos. Soc. 84, 113–116 (1978)
Leindler, L.: Trigonometric approximation in \(L_{p}\)-norm. J. Math. Anal. Appl. 302, 129–136 (2005)
Lorentz, G.G.: A contribution to the theory of divergent series. Acta Math. 80, 167–190 (1948)
Maddox, I.J.: Elements of Functional Analysis. Cambridge University Press (1970)
Maddox, I.J.: A new type of convergence. Math. Proc. Camb. Philos. Soc. 83, 61–64 (1978)
McFadden, L.: Absolute Nörlund summability. Duke Math. J. 9, 168–207 (1942)
Mishra, V.N.: Some problems on approximations of functions in banach spaces. Ph.D. thesis, Indian Institute of Technology, Roorkee-247 667, Uttarakhand, India (2007)
Mishra, V.N.: On the degree of approximation of signals (functions) belonging to the weighted \(W \left(L_{p}, \xi \, (t)\right),\, (p \ge 1)\)-class by almost matrix summability method of its conjugate Fourier series. Int. J. Appl. Math. Mech. 5(7), 16–27 (2009)
Mishra, V.N., Mishra, L.N.: Trigonometric approximation of signals (functions) in \(L_{p} (p\ge 1)\)-norm. Int. J. Contemp. Math. Sci. 7(19), 909–918 (2012)
Mishra, V.N., Khan, H.H., Khatri, K.: Degree of approximation of conjugate of signals (functions) by lower triangular matrix operator. Appl. Math. 2(12), 1448–1452 (2011)
Mishra, V.N., Khan, H.H., Khan, I.A., Mishra, L.N.: On the degree of approximation of signals of \(Lip (\alpha, r)\), \((r \ge 1)\)-class by almost Riesz means of its Fourier series. J. Class. Anal. 4(1), 79–87 (2014). doi:10.7153/jca-04-05
Mishra, V.N., Khan, H.H., Khan, I.A., Khatri, K., Mishra, L.N.: Approximation of signals belonging to the Lip (\(\xi \)(t), p), (p> 1)-class by (E, q) (q > 0)—means, of the conjugate series of its Fourier series. Adv. Pure Math. 3, 353–358 (2013). doi:10.4236/apm.2013.33050
Mittal, M.L., Mishra, V.N.: Approximation of signals (functions) belonging to the weighted \(W \left(L_{p}, \xi (t)\right), (p \ge 1)\)-class by almost matrix summability method of its Fourier series. Int. J. Math. Sci. Eng. Appl. (IJMSEA) 2(IV), 285–294 (2008)
Mittal, M.L., Rhoades, B.E.: Approximations by matrix means of double Fourier series to continuous functions in two variables functions. Radovi Mat. 9, 77–99 (1999)
Mittal, M.L., Rhoades, B.E.: On the degree of approximation of continuous functions by using linear operators on their Fourier series. Int. J. Math. Game Theor. Algebra 9, 259–267 (1999)
Mittal, M.L., Rhoades, B.E.: Degree of approximation to functions in a normed space. J. Comput. Anal. Appl. 2, 1–10 (2000)
Mittal, M.L., Rhoades, B.E.: Degree of approximation of functions in the Hölder metric. Radovi Mat. 10, 61–75 (2001)
Mittal, M.L., Rhoades, B.E., Mishra, V.N.: Approximation of signals (functions) belonging to the weighted \(W(L_{P} ,\xi (t),(p\ge 1)\)-class by linear operators. Int. J. Math. Math. Sci. ID 53538, 1–10 (2006)
Mittal, M.L., Rhoades, B.E., Mishra, V.N., Singh, U.: Using infinite matrices to approximate functions of class \(Lip\, (\alpha \,\, \, p)\) using trigonometric polynomials. J. Math. Anal. Appl. 326, 667–676 (2007)
Mittal, M.L., Singh, U., Mishra, V.N., Priti, S., Mittal, S.S.: Approximation of functions belonging to \(Lip(\xi (t), p),(p\ge 1)\)-class by means of conjugate Fourier series using linear operators. Indian J. Math. 47(2–3), 217–229 (2005)
Mohapatra, R.N., Sahney, B.N.: Approximation by a class of linear operators involving a lower triangular matrix. Stud. Sci. Math. Hung. 14, 87–94 (1979)
Nanda, S.: Some sequence spaces and almost convergence. J. Aust. Math. Soc. 22 (Series A), 446–455 (1976)
Petersen, G.M.: Regular Matrix Transformations. Mc Graw Hill Book Co. Inc. London (1966)
Proakis, J.G.: Digital Communications. McGraw-Hill, New York (1995)
Psariks, E.Z., Moustakids, G.V.: An \(L_2\)-based method for the design of 1 D zero phase FIR digital filters. IEEE Trans. Circuit Syst. Fundam. Theory Appl. 4(7), 591–601 (1997)
Rhoades, B.E., Ozkoklu, K., Albayrak, I.: On degree of approximation to a functions belonging to the class Lipschitz class by Hausdroff means of its Fourier series. Appl. Math. Comput. 217, 6868–6871 (2011)
Schaefer, P.: Almost convergent and almost summable sequences. Proc. Am. Math. Soc. 20, 51–54 (1969)
Sharma, P.L., Qureshi, K.: On the degree of approximation of a periodic function by almost Riesz means. Ranchi Univ. Math. J. 11 (1980)
Shukla, R.K.: Certain Investigations in the theory of summability and that of approximation. Ph.D. thesis, V.B.S. Purvanchal University, Jaunpur, Uttar Pradesh (2010)
Sunouchi, G.: On the class of saturation in the theory of approximation II, III. Tôhoku Math. J. 13(112–118), 320–328 (1961)
Sunouchi, G.: Saturation in the local approximation. Tôhoku Math. J. 17, 16–28 (1965)
Sunouchi, G., Watari, C.: On determination of the class of saturation in the theory of approximation of functions II. Tôhoku Math. J. 11, 480–488 (1959)
Zamansky, M.: Classes de saturation de certaines procedes d’approximation des sries de Fourier des fonctions continues. Ann. Sci Ecole Normale Sup. 66, 19–93 (1949)
Zygmund, A.: Trigonometric Series, vol. I. Cambridge University Press, Cambridge (1959)
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Mishra, V.N. (2016). Degree of Approximation of Functions Through Summability Methods. In: Dutta, H., E. Rhoades, B. (eds) Current Topics in Summability Theory and Applications. Springer, Singapore. https://doi.org/10.1007/978-981-10-0913-6_9
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