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General uniform approximation theory by multivariate singular integral operators

George A. Anastassiou (2012)

Annales Polonici Mathematici

We study the uniform approximation properties of general multivariate singular integral operators on ℝ N , N ≥ 1. We establish their convergence to the unit operator with rates. The estimates are pointwise and uniform. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss-Weierstrass, Poisson-Cauchy and trigonometric singular integral operators to which this theory can be applied directly.

Generalizations of Jensen-Steffensen and related integral inequalities for superquadratic functions

Shoshana Abramovich, Slavica Ivelić, Josip Pečarić (2010)

Open Mathematics

We present integral versions of some recently proved results which improve the Jensen-Steffensen and related inequalities for superquadratic functions. For superquadratic functions which are not convex we get inequalities analogous to the integral Jensen-Steffensen inequality for convex functions. Therefore, we get refinements of all the results which use only the convexity of these functions. One of the inequalities that we obtain for a superquadratic function φ is y ¯ ⩾ φ x ¯ + 1 λ β - λ α ∫ α β φ f t - x ¯ d λ t , where x ¯ = 1 λ β - λ α ∫ α β f t d λ t and y ¯ = 1 λ β - λ α ∫ α β φ f t d λ t which under...

Generalizations of the Jensen-Steffensen and related inequalities

Milica Bakula, Marko Matić, Josip Pečarić (2009)

Open Mathematics

We present a couple of general inequalities related to the Jensen-Steffensen inequality in its discrete and integral form. The Jensen-Steffensen inequality, Slater’s inequality and a generalization of the counterpart to the Jensen-Steffensen inequality are deduced as special cases from these general inequalities.

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