Boundedness of trigonometric integrals and polynomials in the L metric

Conditions are derived for the uniform boundedness of special trigonometric integrals and sums in the L metric.Translated from Matematicheskie Zametki, Vol. 8, No. 6, pp. 811–822, December, 1970.     

Convergence in the mean of the Fourier series in orthogonal polynomials

For weights p(t) and q(t) with a finite number of power-law-type singularities we obtain necessary and sufficient conditions for the inequality $$\parallel s_n^{(p)} (f)q\parallel _{L^\eta - (1,1)} \leqslant C\parallel fq\parallel _{L^\eta ( - 1,1)'}$$ to hold, where s_n ^(p)(f) is a partial sum of the Fourier series of the function f in terms of polynomials orthogonal on [?1, 1] with weight p(t). This inequality is used to solve the problem concerning convergence in the mean and also convergence almost everywhere of the partial sums s_n ^(p)(f).     

Some integral mean estimates for polynomials

In this paper we establish Lq inequalities for polynomials, which in particular yields interesting generalizations of some Zygmund-type inequalities.     

Approximation of periodic functions by trigonometric polynomials in the mean

For arbitrary summation methods we obtain inequalities between upper bounds of deviations in the L metric and corresponding upper bounds in the C metric with respect to a certain class of functions. These inequalities constitute a generalization of known relationships due to S. M. Nikol'skii. We consider the cases wherein these inequalities become exact or asymptotic equalities.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 15–26, July, 1974     

New integral mean estimates for polynomials

In this paper we prove someL ^P inequalities for polynomials, wherep is any positive number. They are related to earlier inequalities due to A Zygmund, N G De Bruijn, V V Arestov, etc. A generalization of a polynomial inequality concerning self-inversive polynomials, is also obtained.     

One-sided approximation of a step by algebraic polynomials in the mean

An asymptotically sharp estimate is obtained for the best one-sided approximation of a step by algebraic polynomials in the space L ₁:     

Regular simplices, symmetric polynomials and the mean value property

LetP be ann-dimensional regular simplex in ℝⁿ centered at the origin, and let P(k) be thek-skeleton ofP fork = 0, 1,…,n. Then the set of all continuous functions in ℝⁿ satisfying the mean value property with respect to P(k) forms a finite-dimensional linear space of harmonic polynomials. In this paper the function space is explicitly determined by group theoretic and combinatorial arguments for symmetric polynomials.     

On mean approximation by polynomials with gaps on non-caratheodory domains

The paper presents some new results on approximation by polynomials with gaps in the norm of the space L _p , 1 ≤ p < ∞, on non-Caratheodory domains in the complex plane. Lacunary versions of several results due to A. L. Shahinian, M. M. Djrbashian, S. M. Mergelian and J. Brennan are proved.     

A bound for Smale's mean value conjecture for complex polynomials

Smale's mean value conjecture is an inequality that relatesthe locations of critical points and critical values of a polynomialp to the value and derivative of p at some given non-criticalpoint. Using known estimates for the logarithmic capacity ofa connected set in the plane containing three given points,we give a new bound for the constant in Smale's inequality interms of the degree d of p. The bound improves previous resultswhen d 8.     

On mean approximation by polynomials with gaps on Caratheodory sets

The paper presents some new results on the possibility of approximation by polynomials with gaps. The approximations are done in the norm of the space L _p , 1 ≤ p < + ∞, on the Caratheodory sets in the complex plane. The lacunary versions of some results by Farell—Markushevich, S. Sinanian, A. L. Shahinian are obtained (Theorems 1, 3, 5). Similar approximations by the real parts of lacunary polynomials are given (Theorems 2, 4, 6). Dedicated to the memory of academician S. N. Mergelyan     

Uniform and mean approximation by certain weighted polynomials,with applications

LetW(x):= exp(-{tiQ(x})), where, for example, Q(x) is even and convex onR, and Q(x)/logx → ∞ asx → ∞. A result of Mhaskar and Saff asserts that ifa _n =a _n (W) is the positive root of the equation $$n = ({2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi })\int_0^1 {{{a_n xQ'(a_n x)} \mathord{\left/ {\vphantom {{a_n xQ'(a_n x)} {\sqrt {1 - x^2 } }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 - x^2 } }}dx,}$$ then, given any polynomialP _n(x) of degree at mostn, the sup norm ofP _n(x)W(a _n x) overR is attained on [-1, 1]. In addition, any sequence of weighted polynomials {p _n (x)W(a _n x)} ₁ ^∞ that is uniformly bounded onR will converge to 0, for ¦x¦>1. In this paper we show that under certain conditions onW, a function g(x) continuous inR can be approximated in the uniform norm by such a sequence {p _n (x)W(a _n x)} ₁ ^∞ if and only if g(x)=0 for ¦x¦? 1. We also prove anL _p analogue for 0W(x)=exp(?|x| ^α ), when α >1. Further applications of our results are upper bounds for Christoffel functions, and asymptotic behavior of the largest zeros of orthogonal polynomials. A final application is an approximation theorem that will be used in a forthcoming proof of Freud's conjecture for |x| ^p exp(?|x| ^α ),α > 0,p > ?1.     

On the best approximation in the mean and overconvergence of a sequence of polynomials of the best approximation

We investigate one property of a sequence of polynomials of the best approximation in the mean related to the convergence in a neighborhood of every point of regularity of a function on the level line ∂ G _R.