We prove that for any (varepsilon>0) and any trigonometric polynomial (f) with frequencies in the set ({n^3: N leq nleq N+N^{2/3-varepsilon}}) one has
$$|f|_4 ll varepsilon^{-1/4}|f|_2$$with implied constant being absolute. We also show that the set ({n^3: Nleq nleq N+(0.5N)^{1/2}}) is a Sidon set.
We generalize the Khinchin singularity phenomenon for the problem in which, for a given irrational linear subspace, rational subspaces forming the least angle with the given subspace are sought.
In this paper, conditions are found for the boundedness of the fractional maximal operator, the Riesz potential, and their commutators in Orlicz spaces.
We study the convergence of Fourier series in the polynomial system ({m_{n,N}^{alpha,r}(x)}) orthonormal in the sense of Sobolev and generated by the system of modified Meixner polynomials. In particular, we show that the Fourier series of (fin W^r_{l^p_{rho_N}(Omega_delta)}) in this system converges to (f) pointwise on the grid (Omega_delta) as (pge2). In addition, we study the approximation properties of partial sums of Fourier series in the system ({m_{n,N}^{0,r}(x)}).
We construct a Parseval wavelet frame with compact support for an arbitrary continuous (2pi)-periodic function (f), (f(0)=1), satisfying the inequality (|f(x)|^2+|f(x+pi)|^2le 1). The frame refinement mask uniformly approximates (f). The refining function has stable integer shifts.
It is proved that every connected boundedly compact locally Chebyshev set in a normed space is a Chebyshev set.
This paper presents some generalizations of a Thébault conjecture, provides an analog of the Thébault conjecture for the (n)-simplex, and also solves a conjecture in a 2022 paper by the authors by using linear algebra.
Upper and lower bounds are obtained for the approximation numbers of the two-dimensional rectangular Hardy operator on weighted Lebesgue spaces on (mathbb{R}_+^2).
Vanishing theorems for the kernels of Lichnerowicz and Hodge Laplacians on a complete Riemannian manifold are proved, and the eigenvalues of a Lichnerowicz Laplacian on a closed Riemannian manifold are estimated.
This paper is the first to introduce a fixed point problem of integral type in a (b)-metric space. We study sufficient conditions for the existence and uniqueness of a common fixed point of contractive mappings of integral type. We also give two examples to support our results.
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