Pointwise convergence of multiple Fourier series using different convergence notions

, . , L ^p (²) p>1 . , C(T³), - , . - .

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This work was completed while the first named author was a visiting professor at Indiana University, Bloomington, Indiana; and the second named author was a visiting professor at Ohio State University, Columbus, Ohio, U.S.A.     

Tangential Markov Inequalities on Transcendental Curves

Abstract. We show that on the curves γ:=(x,e ^t(x) ) , x∈ [a,b] , where t(x) is a fixed polynomial, there holds a tangential Markov inequality of exponent four for algebraic polynomials P _N (x,y) of degree at most N in each variable x,y: ||(P _N (x,e ^t(x) ))'|| _[a,b] ≤ CN ⁴ ||P _N || _γ , and the exponent four is sharp. On the other hand, the corresponding tangential Markov factors on curves y=x ^α with irrational α grow exponentially in the degree of the polynomials.     

Trigonometric series with a generalized monotonicity condition

In this paper,we consider numerical and trigonometric series with a very general monotonicity condition.First,a fundamental decomposition is established from which the sufficient parts of many classical results in Fourier analysis can be derived in this general setting.In the second part of the paper a necessary and sufficient condition for the uniform convergence of sine series is proved generalizing a classical theorem of Chaundy and Jolliffe.     

Riesz-type inequalities on general sets

Sharp Riesz–Bernstein-type inequalities are proven for the derivatives of algebraic polynomials on general subsets of unit circle. The sharp Riesz–Bernstein constant involves the equilibrium density of the set in question.