Higher Order Modulus Of Continuity Research Articles

On a cosine operator function framework of approximation processes in Banach space

We introduce the cosine-type approximation processes in abstract Banach space setting. The historical roots of these processes go back to W. W. Rogosinski in 1926. The given new definitions use a cosine operator functions concept. We proved that in presented setting the cosine-type operators possess the order of approximation, which coincide with results known in trigonometric approximation. Moreover, a general method for factorization of certain linear combinations of cosine operator functions is presented. The given method allows to find the order of approximation using the higher order modulus of continuity. Also applications for the different type of approximations are given.

Estimates of the norm of a function orthogonal to the piecewise-constant functions in terms of higher-order moduli of continuity

The uniform norm of a function that is defined on the real line and has zero integrals between integer points is estimated in terms of its modulus of continuity of arbitrary even order. Sharp bounds of this kind are known for periodic functions. The passage to nonperiodic functions significantly complicates the problem. In general, the constant for nonperiodic functions is greater than that for periodic functions. The constants in the bound are improved compared with those known earlier. The proof is based on a representation of the error of the polynomial interpolation as the product of the influence polynomial and an integrated difference of higher order.

Linear Methods in the Proof of Jackson Type Inequalities and Applications to Estimates of Functionals with Two Known Moments

We propose a new method for estimating functionals in terms of higher order moduli of continuity with explicit constants. Using this method, we estimate deviations of linear methods of approximation by entire functions of finite degree and, in particular, by trigonometric polynomials. For illustration of the results, we derive estimates for the Riesz and Akhiezer–Krein–Favard averages. Bibliography: 14 titles.

Simultaneous Approximation for Generalized Baskakov-Durrmeyer-Type Operators

The present paper deals with the study of a Durrmeyer-type integral modification of certain modified Baskakov operators. Here we study simultaneous approximation properties for these operators by using the iterative combinations. We obtain an asymptotic formula and an error estimation in terms of higher order modulus of continuity for these operators.

Direct estimates in simultaneous approximation for Durrmeyer type operators

In the present paper, we study a Durrmeyer type integral modification of the wellknown Baskakov operators with the weight function of Beta basis function. Some approximation properties of these operators were recently studied by Finta [2]. Here we study simultaneous approximation properties for these operators. We estimate local direct result in terms of modulus of continuity. The operators considered in this paper reproduce not only the constant functions but also the linear ones, due to this property we can improve the order of approximation for these operators by applying the iterative combinations, which were first studied by Micchelli [7]. We establish an asymptotic formula and error estimation in terms of higher order modulus of continuity in simultaneous approximation for the Micchelli combinations of these operators. Mathematics subject classification (2000): 41A30, 41A36.

Jackson and Bernstein theorems for the weight exp(−|x|) on ℝ

In 1978, Freud, Giroux and Rahman established a weightedL 1 Jackson theorem for the weight exp(−|x|) on the real line, using methods that work only inL 1. This weight is somewhat exceptional, for it sits on the boundary between weights like exp(-|x|α), α≥1, where weighted polynomials are dense, and the case α<1, where weighted polynomials are not dense. We obtain the firstL p Jackson theorem for exp(−|x|), valid in allL p , 0<p≤∞, as well as for higher order moduli of continuity. We also establish a converse Bernstein type theorem, characterizing rates of approximation in terms of smoothness of the approximated function.

Error estimation for mixed summation-integral type operators

Srivastava and Gupta [H.M. Srivastava, V. Gupta, A certain family of summation integral type operators, Math. Comput. Modelling 37 (2003) 1307–1315] proposed a certain family of summation integral type operators and estimated the rate of convergence for bounded variation functions. Very recently Gupta et al. [V. Gupta, R.N. Mohapatra, Z. Finta, On certain family of mixed summation-integral type operators, Math. Comput. Modelling, in press] defined the mixed summation integral type operators and obtained some direct results in simultaneous approximation. In the present paper, we consider the linear combinations of the mixed summation integral type operators and establish the error estimation for simultaneous approximation in terms of higher order modulus of continuity. To prove the main result we use the technique of linear approximating method viz. Steklov mean.

Some Approximation Properties for Modified Baskakov Type Operators

Abstract We study some direct results for the recently introduced family of modified Baskakov type operators. In particular, we obtain local direct results on ordinary and simultaneous approximation and an estimation of error for linear combinations in terms of higher order modulus of continuity. We have applied the Steklov mean as a tool for the linear approximating method.

Analyticity of Higher-Order Moduli of Continuity of Real-Analytic Functions

The Perel'man's result according to which the first modulus of continuity of any real-analytic function f is a function analytic in a certain neighborhood of the origin is generalized to the case of arbitrary moduli of continuity of higher order.

Widths in L2 of classes of differentiable functions, defined by higher-order moduli of continuity

Sharp inequalities are obtained in the space L2, connecting the best approximations of differentiable 2π-periodic functions by trigonometric polynomials with integrals containing the higher-order moduli of continuity of the derivatives of these functions. The Kolmogorov widths of certain classes of functions, defined by these moduli of continuity, are computed.