Uaday Singh

Approximation of signals of class Lip(α, p) by linear operators

Abstract Mittal, Rhoades [5] , [6] , [7] , [8] and Mittal et al. [9] , [10] have initiated a study of error estimates E n ( f ) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper we continue the work. Here we extend two theorems of Leindler [4] , where he has weakened the conditions on { p n } given by Chandra [2] , to more general classes of triangular matrix methods. Our Theorem also partially generalizes Theorem 4 of Mittal et al. [11] by dropping the monotonicity on the elements of matrix rows, which in turn generalize the results of Quade [15] .

Degree of Approximation of Function

In this paper, we shall compute the degree of approximation of function \( f\in H_p^{(w)}\), a new Banach space introduced by Das, Nath and Ray ([2], [3]) through matrix means of Fourier series of f , which in turn generalizes most of the results of Liendler [5]. We use rows of the matrix T ≡ (an, k) without monotonicity and derive also some analogous results for the cases when rows of T are monotonic.

f\in H_p^{(w)}

Analysis of signals or time functions are of great importance, because it convey information or attributes of some phenomenon. In this paper, we consider the functions/signals defined in terms of their integral modulus of continuity and approximate them by matrix means of their trigonometric Fourier series. The error of approximation obtained in this paper depends on the integral modulus of continuity of the signal. We also discuss an example to show the application of the result in the situation when the Fourier series of the signal has Gibbs phenomenon. We also discuss some corollaries of our theorems.

Class in Generalized Hölder Metric by Matrix Means

In this paper, we determine the degree of approximation of the conjugate of 2π-periodic signals (functions) belonging to Lip(α,r) (0<α≤1, r≥1)-class by using Cesàro-Euler (C,1)(E,q) means of their conjugate trigonometric Fourier series. Our result generalizes the result of Lal and Singh (Tamkang J. Math. 33(3):269-274, 2002).MSC:41A10.

Trigonometric approximation of periodic functions belonging to Lip(ω(t),p)-class

We estimate the pointwise approximation of periodic functions belonging to L(ω)β-class, where ω is an integral modulus of continuity type function associated with f , using product means of the Fourier series of f generated by the product of two general linear operators. We also discuss the case p = 1 separately. This case has not been mentioned in the earlier results given by various authors. The deviations obtained in our theorems are free from p and more sharper than the earlier results.

Degree of approximation of the conjugate of signals (functions) belonging to Lip(α,r)-class by (C,1)(E,q) means of conjugate trigonometric Fourier series

Various investigators such as Khan (1974), Chandra (2002), and Liendler (2005) have determined the degree of approximation of 2π-periodic signals (functions) belonging to Lip(𝛼,𝑟) class of functions through trigonometric Fourier approximation using different summability matrices with monotone rows. Recently, Mittal et al. (2007 and 2011) have obtained the degree of approximation of signals belonging to Lip(𝛼,𝑟)- class by general summability matrix, which generalize some of the results of Chandra (2002) and results of Leindler (2005), respectively. In this paper, we determine the degree of approximation of functions belonging to Lip α and 𝑊(𝐿𝑟, 𝜉(𝑡)) classes by using Cesaro-Norlund (𝐶1⋅𝑁𝑝) summability without monotonicity condition on {𝑝𝑛}, which in turn generalizes the results of Lal (2009). We also note some errors appearing in the paper of Lal (2009) and rectify them in the light of observations of Rhoades et al. (2011).

Approximation of periodic integrable functions in terms of modulus of continuity

In continuation of a recent work by Mittal [M.L. Mittal, On strong Norlund summability of Fourier series, J. Math. Anal. Appl. 314 (2006),75-84], the present authors obtain a sufficient condition for the summability [N, p n (1) , 2] of the conjugate Fourier series. In conjunction with the known Tauberian theorem on the strong Norlund summability, which was also considered earlier by Mittal [M.L. Mittal, A Tauberian theorem on strong Norlund summability, J. Indian Math. Soc. (N.S.) 44 (1980), 369-377], our result gives a sufficient condition for the summability [C, 1,2] of the conjugate Fourier series. Our main theorem generalizes the results given earlier by Prasad [G. Prasad, On Norlund Summability of Fourier Series, Ph.D. thesis, University of Roorkee, Roorkee, 1967] and Singh [U.N. Singh, On the strong summability of a Fourier series and its conjugate series, Proc. Nat. Inst. Sci. India Part A 13 (1947), 319-325].

Trigonometric Approximation of Signals (Functions) Belonging to (,()) Class by Matrix (1⋅) Operator

In this paper, we study the degree of approximation of 2π-periodic functions of two variables, defined on T2=[−π,π]×[−π,π]

On the strong Nörlund summability of conjugate Fourier series

T^{2}=[-\pi,\pi]\times[-\pi,\pi]

Approximation of certain bivariate functions by almost Euler means of double Fourier series

and belonging to certain Lipschitz classes, by means of almost Euler summability of their Fourier series. The degree of approximation obtained in this way depends on the modulus of continuity associated with the functions. We also derive some corollaries from our theorems.