Maximal functions and ergodic averages related to Waring’s problem

We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of [13], [14] and [16]. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring’s problem to an approximation formula for the Fourier transform of the solution set of lattice points on hypersurfaces arising in Waring’s problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation formula, ? ²-maximal theorems and ergodic theorems were previously known. Our contribution is in reducing the dimensional constraint in the approximation formula using recent bounds of Wooley, and improving the range of ? ^p spaces in the maximal and ergodic theorems. We also conjecture the expected range of spaces.     

Interpolation of Lorentz martingale spaces

In this paper,we apply function parameters,introduced by Persson,to real interpolation of Lorentz martingale spaces.Some new interpolation theorems concerning Lorentz martingale spaces are formulated.The results that we obtain generalize some fundamental interpolation theorems in classical martingale Hp theory.     

Application of Interpolational Methods to the Study of Properties of Functions of Several Variables

In this paper, interpolation theorems for spaces of functions of several variables are used to generalize and refine Hörmander's theorem on the multipliers of the Fourier transform from L _p to L _q and the Hardy--Littlewood--Paley inequality for a class of multiple Fourier series in the multidimensional case.     

Singular operators and Fourier multipliers in weighted Lebesgue spaces with variable index

Mikhlin’s ideas and results related to the theory of spaces L _ρ ^p(·) with nonstandard growth are developed. These spaces are called Lebesgue spaces with variable index; they are used in mechanics, the theory of differential equations, and variational problems. The boundedness of Fourier multipliers and singular operators on the spaces L _ρ ^p(·) are considered. All theorems are derived from an extrapolation theorem due to Rubio de Francia. The considerations essentially use theorems on the boundedness of operators and maximal Hardy-Littlewood functions on Lebesgue spaces with constant index.     

Existence and multiplicity of periodic solutions for some second order differential systems with p(t)-Laplacian

In this paper, we deal with the existence and multiplicity of periodic solutions for the p(t)-Laplacian Hamiltonian system. Some new existence theorems are obtained by using the least action principle and minmax methods in critical point theory, and our results generalize and improve some existence theorems.     

Periodic solutions for some nonautonomous p(t)-Laplacian Hamiltonian systems

In this paper, we deal with the existence of periodic solutions of the p(t)-Laplacian Hamiltonian system . Some new existence theorems are obtained by using the least action principle and minimax methods in critical point theory, and our results generalize and improve some existence theorems.     

Generalized Grötzsch ring function and generalized elliptic integrals

In this paper, we study the quotient of hypergeometric functions μ_a(r) in the theory of Ramanujan's generalized modular equation for a ∈(0, 1/2]. Several new inequalities are given for this and related functions. Our main results complement and generalize some known results in the literature.     

On the convergence of double Fourier series of functions in L P[0, 1]2, where p=(p 1, p 2)

Pointwise convergence of double trigonometric Fourier series of functions in the Lebesgue space L _p[0,2]²was studied by M. I. Dyachenko. In this paper, we also consider the problems of the convergence of double Fourier series in Pringsheim"s sense with respect to the trigonometric as well as the Walsh systems of functions in the Lebesgue space L _p[0,1]², p=(p₁,p₂), endowed with a mixed norm, in the particular case when the coefficients of the series in question are monotone with respect to each of the indices. We shall obtain theorems which generalize those of M. I. Dyachenko to the case when p is a vector. We shall also show that our theorems in the case of trigonometric Fourier series are best possible.     

The problem of approximation in mean on arcs in the complex plane

Classical theorems on the approximation of curves in the complex domain are studied; in particular, direct and inverse theorems on the arcs Γ in the complex plane in the metric of L_p(Γ) are obtained. The results obtained are new in the case of a closed interval [?1, 1] as well.     

The average errors for Hermite-Fejr interpolation on the Wiener space

For 1≤ p ∞, firstly we prove that for an arbitrary set of distinct nodes in [-1, 1], it is impossible that the errors of the Hermite-Fejr interpolation approximation in L p -norm are weakly equivalent to the corresponding errors of the best polynomial approximation for all continuous functions on [-1, 1]. Secondly, on the ground of probability theory, we discuss the p-average errors of Hermite-Fejr interpolation sequence based on the extended Chebyshev nodes of the second kind on the Wiener space. By our results we know that for 1≤ p ∞ and 2≤ q ∞, the p-average errors of Hermite-Fejr interpolation approximation sequence based on the extended Chebyshev nodes of the second kind are weakly equivalent to the p-average errors of the corresponding best polynomial approximation sequence for L q -norm approximation. In comparison with these results, we discuss the p-average errors of Hermite-Fejr interpolation approximation sequence based on the Chebyshev nodes of the second kind and the p-average errors of the well-known Bernstein polynomial approximation sequence on the Wiener space.     

On the Convergence of Rational Functions of Best Lp-Approximation

In the present paper, for sequences of rational functions, analytic in some domain, a theorem of Montel’s type is proved. As an application, sequences of rational functions of best Lp — approximation of a function ?(z) ∈ L_p(Γ), with Γ being a closed rectifiable analytic curve, are considered. The case ?(z) ∈ Hp is discussed, too.     

Simpson's formula for functions whose derivatives belong to Lp spaces

In [1], Dragomir gave an inequality of Simpson's type for functions whose derivatives belong to L_p spaces. Here, we generalize his results using functions whose n^th derivatives, n ∈ {2, 3, 4}, belong to L_p spaces.     

Fixed point theorems for better admissible multimaps on almost convex sets

We obtain new fixed point theorems on multimaps in the class Bp defined on almost convex subsets of topological vector spaces. Our main results are applied to deduce various fixed point theorems, coincidence theorems, almost fixed point theorems, intersection theorems, and minimax theorems. Consequently, our new results generalize well-known works of Kakutani, Fan, Browder, Himmelberg, Lassonde, and others.     

On the partial sums of Vilenkin-Fourier series

The aim of this paper is to investigate weighted maximal operators of partial sums of Vilenkin-Fourier series. Also, the obtained results we use to prove approximation and strong convergence theorems on the martingale Hardy spaces H _p , when 0 < p ≤ 1.     

Approximation of signals of class Lip(α, p) by linear operators

Mittal, Rhoades [5], [6], [7] and [8] and Mittal et al. [9] and [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].     

Conformal invariants associated with quadratic differentials

We associate a functional of pairs of simply-connected regions D₂ ? D₁ to any quadratic differential on D₁ with specified singularities. This functional is conformally invariant, monotonic, and negative. Equality holds if and only if the inner domain is the outer domain minus trajectories of the quadratic differential. This generalizes the simply-connected case of results of Z. Nehari [20], who developed a general technique for obtaining inequalities for conformal maps and domain functions from contour integrals and the Dirichlet principle for harmonic functions. Nehari’s method corresponds to the special case that the quadratic differential is of the form (?q)² for a singular harmonic function q on D₁.As an application we give a one-parameter family of monotonic, conformally invariant functionals which correspond to growth theorems for bounded univalent functions. These generalize and interpolate the Pick growth theorems, which appear in a conformally invariant form equivalent to a two-point distortion theorem of W. Ma and D. Minda [16].     

Computation of k1 Via mennicke symbols

For any ring A, the group K₁A is filtered by the Whitehead determinants of invertible matrices over A of different sizes. We want to compute the corresponding graded group (especially the highest degree non-zero term) in terms of symbols which generalize Mennicke’s symbol. In particular, we generalize the Bass-Milnor-Serre result which presents SK₁A of a Dedekind ring A via the Mennicke symbol, to an arbitrary commutative ring A satisfying the Bass second stable range condition. As an application, SK₁ is computed for some rings of continuous functions. Some of our theorems are partially known, but we have often weakened hypotheses, using stable range conditions rather than Krull dimension (having in mind applications to rings of continuous functions).     

A result analogous to G. Szego’s theorem

The paper proves a new result similar to the well known theorems of G. Szego and A. Kolmogorov on the best approximation by analytic polynomials in weighted L ^p spaces. Such results are essential in prediction theory for stationary processes. It is well known, that for one step prediction, the size of the best approximation is the same for all p. The paper proves that for two step prediction the best approximation sizes are different for p = 2 and p = ∞.     

Inhomogeneous Diophantine approximation on curves and Hausdorff dimension

The goal of this paper is to develop a coherent theory for inhomogeneous Diophantine approximation on curves in Rn akin to the well established homogeneous theory. More specifically, the measure theoretic results obtained generalize the fundamental homogeneous theorems of R.C. Baker (1978) [2], Dodson, Dickinson (2000) [18] and Beresnevich, Bernik, Kleinbock, Margulis (2002) [8]. In the case of planar curves, the complete Hausdorff dimension theory is developed.