Madan Lal Mittal

Approximation of signals (functions) belonging to the weighted W(Lp,ξ(t))-class by linear operators

Mittal and Rhoades (1999–2001) and Mittal et al. (2006) have initiated the studies of error estimates En(f) through trigonometric Fourier approximations (TFA) for the situations in which the summability matrix T does not have monotone rows. In this paper, we determine the degree of approximation of a function f˜, conjugate to a periodic function f belonging to the weighted W(Lp,ξ(t))-class (p≥1), where ξ(t) is nonnegative and increasing function of t by matrix operators T (without monotone rows) on a conjugate series of Fourier series associated with f. Our theorem extends a recent result of Mittal et al. (2005) and a theorem of Lal and Nigam (2001) on general matrix summability. Our theorem also generalizes the results of Mittal, Singh, and Mishra (2005) and Qureshi (1981-1982) for Norlund (Np)-matrices.

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] .

Approximation of Signals (Functions) by Trigonometric Polynomials in -Norm

Mittal and Rhoades (1999, 2000) and Mittal et al. (2011) have initiated a study of error estimates through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix does not have monotone rows. In this paper, the first author continues the work in the direction for to be a -matrix. We extend two theorems on summability matrix of Deger et al. (2012) where they have extended two theorems of Chandra (2002) using -method obtained by deleting a set of rows from Cesaro matrix . Our theorems also generalize two theorems of Leindler (2005) to -matrix which in turn generalize the result of Chandra (2002) and Quade (1937).

Applications of Cesàro Submethod to Trigonometric Approximation of Signals (Functions) Belonging to Class in -Norm

We prove two Theorems on approximation of functions belonging to Lipschitz class in -norm using Cesaro submethod. Further we discuss few corollaries of our Theorems and compare them with the existing results. We also note that our results give sharper estimates than the estimates in some of the known results.

Degree of approximation of signals (functions) in Besov space using linear operators

Mittal, Rhoades (1999–2001), Mittal et al. (2005, 2006, 2011) have initiated a study of error estimates through trigonometric Fourier approximation (tfa) for the situation in which the summability matrix T ≡ (an,k) does not have monotone rows. Recently Mohanty et al. (2011) have obtained a theorem on the degree of approximation of functions in Besov space Bqα(L p) by choosing T to be a Norlund (Np)-matrix with non-increasing weights {pn}. In this paper, we continue the work of Mittal et al. and extend the result of Mohanty et al. (2011) to the general matrix T.

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

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).

On the strong Nörlund summability of conjugate Fourier series

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].

Approximation of functions in the generalized Zygmund class using Hausdorff means

In this paper we investigate the degree of approximation of a function belonging to the generalized Zygmund class Zp(ω)

Approximation of Functions of Class \mathrm {Lip} (\alpha , { p}) in L_{p}-Norm

Z_{p}^{(\omega)}

Error Estimation of Functions by Fourier-Laguerre Polynomials Using Matrix-Euler Operators

(p≥1