Search

Series in multiplicative systems χ with generalized monotone coefficients are studied. Necessary and sufficient Hardy–Littlewood type conditions for the sums of such series to belong to the Lorentz space are proved. As corollaries, we establish estimates of best approximation in the system χ and Konyushkov-type theorems on the equivalence of O- and ≍-relations for the weighted sums of the Fourier coefficients in the system χ and for the best approximations.

In this paper, we obtain five tests (three of which are symmetric) of pointwise convergence of Fourier series with respect to generalized Haar systems; the tests are similar to the Dini convergence tests. It is shown that the Dini convergence tests for Price systems are also valid for generalized Haar systems. It is also shown that the classicalDini convergence test does not apply, in general, even to generalized Haar systems, although the classical symmetric Dini test for generalized Haar systems is valid. Also upper bounds for the Dirichlet kernels for generalized Haar systems are obtained.

We study Fourier series of Jacobi polynomials P k α−r,−r (x), k = r, r +1,..., orthogonal with respect to the Sobolev-type inner product of the following form: $$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{\left( v \right)}}} \left( { - 1} \right){g^{\left( v \right)}}\left( { - 1} \right) + \int_{ - 1}^1 {{f^{\left( r \right)}}} {g^{\left( r \right)}}\left( t \right){\left( {1 - t} \right)^\alpha }dt$$ . It is shown that such series are a particular case of mixed series of Jacobi polynomials P k α,β(x), k = 0, 1,..., considered earlier by the author. We study the convergence of mixed series of general Jacobi polynomials and their approximation properties. The results obtained are applied to the study of the approximation properties of Fourier series of Sobolev orthogonal Jacobi polynomials P k α−r,−r (x).

We prove that the Zygmund space L(lnL)1/2 is the largest among symmetric spaces X in which any uniformly bounded orthonormal system of functions contains a sequence such that the corresponding space of Fourier coefficients F(X) coincides with l2. Moreover, we obtain a description of spaces of Fourier coefficients corresponding to appropriate subsequences of arbitrary uniformly bounded orthonormal systems in symmetric spaces located between the spaces L(lnL)1/2 and L1.

The notion of general mixed modulus of smoothness of periodic functions of several variables in the spaces Lp is introduced. The proposed construction is, on the one hand, a natural generalization of the general modulus of smoothness in the one-dimensional case, which was introduced in a paper of the first author and in which the coefficients of the values of a given function at the nodes of a uniform lattice are the Fourier coefficients of a 2π-periodic function called the generator of the modulus; while, on the other hand, this construction is a generalization of classical mixed moduli of smoothness and of mixed moduli of arbitrary positive order. For the modulus introduced in the paper, in the case 1 ≤ p ≤ +∞, the direct and inverse theorems on the approximation by the “angle” of trigonometric polynomials are proved. The previous estimates of such type are obtained as direct consequences of general results, new mixed moduli are constructed, and a universal structural description of classes of functions whose best approximation by “angle” have a certain order of convergence to zero is given.

It is shown that the exact order of decrease of the norm in L of the remainder of a Fourier sine series with monotone coefficients can be expressed in terms of the coefficients of the series just as for a series with convex coefficients. But the numerical multipliers in the estimates are different.

We study the relationship between the strongly regular growth of an entire function f of order zero, the existence of the angular density of its zeros, the behavior of the Fourier coefficients of the logarithm of f, and the regular growth of the logarithm of the modulus and the argument of f in the Lp[0, 2π]-metric, p ≥ 1.

We obtain asymptotic estimates for best approximations by trigonometric polynomials in the metric of the space C(Lp) for classes of periodic functions expressible as convolutions of kernels Ψβ with Fourier coefficients decreasing to zero faster than any power sequence, and with functions ϕ ∈ C (ϕ ∈ Lp) whose moduli of continuity do not exceed the given majorant of ω(t). It is proved that, in the spaces C and L1, for convex moduli of continuity ω(t), the obtained estimates are asymptotically sharp.

The paper deals with the question of the divergence of Fourier series in function spaces wider than L = L[−π, π], but narrower than Lp = Lp[−π, π] for all p ∈ (0, 1). It is proved that the recent results of Filippov on the generalization to the space ϕ(L) of Kolmogorov’s theorem on the convergence of Fourier series in Lp, p ∈ (0, 1), cannot be improved.

The properties of the Fourier coefficients for some classes of functions with respect to both the Haar system and general orthonormal systems are studied. It is established that the results obtained in the paper cannot be strengthened.

We investigate the equiconvergence on T N = [−π, π) N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L p (T N ) and g ∈ L p (R N ), p > 1, N ≥ 3, g(x) = f(x) on T N , in the case where the “partial sums” of these expansions, i.e., S n (x; f) and J α(x; g), respectively, have “numbers” n ∈ Z N and α ∈ R N (n j = [α j ], j = 1,..., N, [t] is the integral part of t ∈ R1) containing N − 1 components which are elements of “lacunary sequences.”

It is established that the formulas determining the jump of a periodic function from the derivatives of the partial sums of its Fourier series and valid for functions of harmonic bounded variation (the HBV class) possibly will not hold for functions of Φ-bounded variation (in the sense of Schramm) if this class is wider than the HBV class.

For multiplicative Fourier transforms, analogs of results obtained by R. P. Boas, F. Moricz, M. I. D’yachenko, I. P. Liflyand, and S. Yu. Tikhonov and dealing with conditions for the uniform convergence and weighted integrability with power weight of classical Fourier transforms as well as conditions for these transforms to belong to Lipschitz classes are proved. Certain results of C. W. Onneweer concerning conditions for multiplicative Fourier transforms to belong to Lipschitz—Besov and Herz spaces are also generalized.

Lower and upper bounds for the mixed norms of derivatives and mixed moduli of smoothness of functions having monotone Fourier coefficients are proved.

For the number of solutions of the congruence xy ≡ 1 (modp) under the graph of a linear function, an asymptotic formula with square root cancellation in the error term is proved.

For the two-dimensional nonstationary Euler equations of an ideal incompressible fluid, the flow problem with a given vortex is considered. The uniqueness of the solution is proved.

Questions dealing with the approximation of functions from the classes CβψHα by Poisson integrals are studied. The Kolmogorov-Nikol’skii problem for Poisson integrals for the classes CβψHα is solved in the uniform metric.

We introduce the notion of general modulus of smoothness in the spaces Lp of 2π-periodic pth-power integrable functions; in these spaces, the coefficients multiplying the values of a given function at the nodes of the uniform lattice are the Fourier coefficients of some 2π-periodic function called the generator of the modulus. It is shown that all known moduli of smoothness are special cases of this general construction. For the introduced modulus, in the case 1 ≤ p ≤ +∞ we prove a direct theorem of approximation theory (a Jackson-type estimate). It is shown that the known Jackson-type estimates for the classical moduli, the modulus of positive fractional order, and the modulus of smoothness related to the Riesz derivative are its direct consequences. We also obtain a universal structural description of classes of functions whose best approximations have a certain order of convergence to zero.

A new lower bound for the growth of the Dirichlet kernel for the Walsh-Kaczmarz system is obtained and an example of an almost everywhere divergent Fourier series with respect to this system from a class narrower than that examined earlier is constructed.

We present necessary and sufficient conditions for the absolute convergence of the Fourier series of almost-periodic (in the sense of Besicovitch) functions when the Fourier exponents have limit points at infinity or at zero. The structural properties of the functions are described by the modulus of continuity or the modulus of averaging of high order, depending on the behavior of the Fourier exponents. The case of uniform almost-periodic functions of bounded variation is considered.

We obtain sufficient conditions for β-absolute convergence (0 < β ≤ 1) of multiple Fourier series of functions of the classes $$L^2 ([0,2\pi ]^N ),(\Lambda ^1 ,\Lambda ^2 ,...,\Lambda ^N )BV^{(p)} ([0,2\pi ]^N ),r - BV([0,2\pi ]^N )$$ .

The problem of the stability of wave propagation in anisotropic inhomogeneous media is considered. The class of approximate solutions possessing the stability property with respect to the small deviations of the input data in the form regularizing the operators R(φ, ψ, x, t, α) is constructed. Here an important role is played by the choice of the smoothing function and by the conditions for matching the regularization parameter with the error.