Periodic Functions Of Many Variables Research Articles

About Absolute Convergence of Fourier Series of Almost Periodic Functions

The current stage of development of the theory of almost periodic functions is characterized by a desire for analysis and processing of a huge amount of accumulated scientific and practical material. The theory of almost periodic functions arose in the 20-30 s of the twentieth century; currently, extensive literature has accumulated on various issues of this theory. Long before the creation of the general theory of almost periodic functions, the outstanding Riga mathematician P. Bol drew attention to such functions. For functions of many variables f(x<sub>1</sub>, x<sub>2,</sub>...x<sub>p</sub>), Bol considered the corresponding multiple Fourier series and, in p-dimensional Euclidean space, a straight line passing through the origin: x<sub>1</sub>=a<sub>1</sub> t, x<sub>2</sub>=a<sub>2</sub> t,..., x<sub>p</sub>=a<sub>p</sub>t, where a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> - some real, non-zero numbers. Considering the value of the function f(x<sub>1</sub>, x<sub>2,</sub>...x<sub>p</sub>) on this line, he obtains a function of one variable φ(t) = f(a<sub>1</sub> t, a<sub>2</sub> t,...a<sub>p</sub> t) and proves that this function is almost periodic. With some choice of numbers a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> - it may happen that this function is periodic. However, if the numbers a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> are linearly independent, then you can easily make sure that the function will not be a periodic function. Further development of the problem was carried out by the French mathematician E. Escalangon. However, the main significant drawback of the results of Bol and Escalangon was that from the very beginning, starting with the definition of almost-periodic functions, they introduced into consideration a fixed system of numbers a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> associated with the almost-period (τ). This drawback was eliminated by the Danish mathematician G. Bohr, who developed in general terms the theory of continuous almost-periodic functions. Bohr's research in its methods was closely related to Bohl's research. However, Bohr did not impose restrictions such as Bohl’s inequality in advance for the almost period. The results obtained by Bol and Bohr were based on the deep connection between almost periodic functions and periodic functions of many variables. The article examines the question of sufficient conditions for the absolute and uniform convergence of Fourier series of uniform almost periodic functions in the case when the Fourier exponents have a single limit point at zero, i.e. λ<sub>k</sub>→0 (k→∞). In this case, the Laplace transform is used for the first time as a structural characteristic.

Kolmogorov Widths of the Nikol’skii–Besov Classes of Periodic Functions of Many Variables in the Space of Quasicontinuous Functions

Kolmogorov Widths of the Nikol’skii–Besov Classes of Periodic Functions of Many Variables in the Space of Quasicontinuous Functions

A deterministic algorithm for constructing multiple rank-1 lattices of near-optimal size

In this paper we present the first known deterministic algorithm for the construction of multiple rank-1 lattices for the approximation of periodic functions of many variables. The algorithm works by converting a potentially large reconstructing single rank-1 lattice for some d-dimensional frequency set I ⊂{0,…,N − 1}d into a collection of much smaller rank-1 lattices which allow for accurate and efficient reconstruction of trigonometric polynomials with coefficients in I (and, therefore, for the approximation of multivariate periodic functions). The total number of sampling points in the resulting multiple rank-1 lattices is theoretically shown to be less than \( \mathcal {O}\left (|I| \log ^{2 }(N |I|) \right ) \) with constants independent of d, and by performing one-dimensional fast Fourier transforms on samples of trigonometric polynomials with Fourier support in I at these points, we obtain exact reconstruction of all Fourier coefficients in fewer than \( \mathcal {O}\left (d |I|\log ^{4} (N|I|)\right ) \) total operations. Additionally, we present a second multiple rank-1 lattice construction algorithm which constructs lattices with even fewer sampling points at the cost of only being able to reconstruct exact trigonometric polynomials rather than having additional theoretical approximation guarantees. Both algorithms are tested numerically and surpass the theoretical bounds. Notably, we observe that the oversampling factors #samples/|I| appear to grow only logarithmically in |I| for the first algorithm and appear near-optimally bounded by four in the second algorithm.

Estimates of the best orthogonal trigonometric approximations and orthoprojective widths of the classes of periodic functions of many variables in a uniform metric

Some approximative characteristics of classes of periodic functions of many variables $$ {L}_{\beta, p}^{\psi }, $$ 1 < p < 1, in a uniform metric are investigated. The first part of the paper is devoted to the construction of estimates of the best orthogonal trigonometric approximations of the mentioned classes in the space L∞. In the second part, we have established the ordinal estimates of the orthoprojective widths of the classes $$ {L}_{\beta, p}^{\psi }, $$ 1 < p < 1, in the same space, as well as the estimates of another approximative characteristic which is close, in a definite meaning, to the orthoprojective width.

Estimates of the best orthogonal trigonometric approximations and orthoprojective widths of the classes of periodic functions of many variables in a uniform metric

Some approximative characteristics of classes of periodic functions of many variables L^{\psi}_{\beta, p}, 1 &lt; p &lt; \infty, in a uniform metric are investigated. The first part of the paper is devoted to the construction of estimates of the best orthogonal trigonometric approximations of the mentioned classes in the space L_\infty. In the second part, we have established the ordinal estimates of the orthoprojective widths of the classes L^{\psi}_{\beta, p}, 1 &lt; p &lt; \infty, in the same space, as well as the estimates of another approximative characteristic which is close, in a definite meaning, to the orthoprojective width.

Entropy Numbers of the Nikol'skii—Besov-type Classes of Periodic Functions of many Variables

The order estimates of entropy numbers of the Nikol’skii–Besov–type classes of periodic functions of many variables in the Lebesgue space are obtained. For the corresponding choice of a majorant function for the mixed modulus of continuity, those classes coincide with the Nikol’skii–Besov classes.

Integral presentation of deviations of rectangular linear means of Fourier series on classes of periodic differentiable functions

The paper deals with the problems of approximation in a uniform metric of periodic functions of many variables by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. Questions of asymptotic behavior of the upper bounds of deviations of linear operators generated by the use of linear methods of summation of Fourier series on the classes of periodic differentiable functions are studied in many works. Methods of investigation of integral representations of deviations of polynomials on the classes of periodic differentiable functions of real variable originated and received its development through the works of S.M. Nikol'skii, S.B. Stechkin, N.P.Korneichuk, V.K. Dzadik, A.I. Stepanets, etc. Along with the study of approximation by linear methods of classes of functions of one variable, are studied similar problems of approximation by linear methods of classes of functions of many variables. In addition to the approximative properties of rectangular Fourier sums, are studied approximative properties of other approximation methods: the rectangular sums of Valle Poussin, Zigmund, Rogozinsky, Favar. In this paper we consider the classes of \(\overline{\psi}\)-differentiable periodic functions of many variables, allowing separately to take into account the properties of partial and mixed \(\overline{\psi}\)-derivatives, and given by analogy with the classes of \(\overline{\psi}\)-differentiable periodic functions of one variable. Integral representations of rectangular linear means of Fourier series on classes of \(\overline{\psi}\)-differentiable periodic functions of many variables are obtained. The obtained formulas can be useful for further investigation of the approximative properties of various linear rectangular methods on the classes \(\overline{\psi}\)-differentiable periodic functions of many variables in order to obtain a solution to the corresponding Kolmogorov-Nikolsky problems.

Approximation of Periodic Functions of Many Variables by Functions of a Smaller Number of Variables in Orlicz Metric Spaces

For periodic functions of many variables, we propose a method for their approximation in the Orlicz spaces L(𝕋m). According to this method, the functions are approximated by the sums of functions of smaller number of variables each of which is piecewise-constant in one variable for fixed values of the other variables. A Jackson-type inequality is analyzed for these approximations in terms of the mixed module of continuity.

Entropy Numbers and Widths for the Nikol’skii–Besov Classes of Functions of Many Variables in the Space L∞

We establish order estimates for the entropy numbers of the Nikol’skii–Besov classes $$B_{p,\theta}^r$$ of periodic functions of many variables in the metric of the space L∞. In the two-dimensional case the results obtained are used for establishing exact-order estimates of the Kolmogorov widths of the classes $$B_{p,\theta}^r$$ , 2 ≤ p ≤ ∞, and of the linear and the trigonometric widths of the classes $$B_{\infty,\theta}^r$$ in the metric of the space L∞.

Widths of the classes B p , θ Ω $$ {B}_{p,\theta}^{\varOmega } $$ of periodic functions of many variables in the space B1,1

We have obtained the exact-by-order estimates of Kolmogorov, linear, and trigonometric widths of the classes \( {B}_{p,\theta}^{\varOmega } \) of periodic functions of many variables in the space B1,1 the norm in which is stronger than the L1-norm.

Estimates of approximative characteristics of the classes B p , θ Ω $$ {B}_{p,\theta}^{\Omega} $$ of periodic functions of many variables with given majorant of mixed continuity moduli in the space L ∞

We obtain exact-order estimates of the approximation of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic functions of several variables in the space L ∞ , by using operators of orthogonal projection, as well as linear operators subjected to some conditions.

Entropy Numbers and Widths for the Classes B p , θ r $$ {B}_{p,\theta}^r $$ of Periodic Functions of Many Variables

We establish the exact-order estimates for the entropy numbers of the Nikol’skii–Besov classes of periodic functions of two variables in the space L ∞ and use the obtained results to estimate the lower bounds for the Kolmogorov, linear, and trigonometric widths. We also study the behavior of similar asymptotic characteristics of the classes $$ {B}_{p,\theta}^r $$ of periodic functions of many variables in the spaces L 1 and B 1,1 .

Best m-Term Trigonometric Approximation for Periodic Functions with Low Mixed Smoothness from the Nikol’skii–Besov-Type Classes

We establish the exact-order estimates of the best m-term trigonometric approximation for periodic functions of many variables (with low mixed smoothness) from the Nikol’skii–Besov-type classes.

Kolmogorov Widths of the Anisotropic Besov Classes of Periodic Functions of Many Variables

We establish exact-order estimates for the Kolmogorov widths of the anisotropic Besov classes of periodic functions of many variables in the spaces L q , 1 ≤ q ≤ ∞.

Estimation of the Entropy Numbers and Kolmogorov Widths for the Nikol’skii–Besov Classes of Periodic Functions of Many Variables

We establish order estimates for the entropy numbers of the Nikol’skii–Besov classes B , of periodic functions of many variables in the space L q with certain relations between the parameters p and q. By using the obtained lower estimates of the entropy numbers, we establish the exact-order estimates for the Kolmogorov widths of the same classes of functions in the space L 1.

On M-term approximations of the Nikolskii - Besov class

In this paper, we consider a Lebesgue space with a mixed norm of periodic functions of many variables. We obtain the exact estimation of the best M-term approximations of Nikol’skii’s and Besov’s classes in the Lebesgue space with the mixed norm.

Estimations of Linear Widths of the Classes B p,θ Ω of Periodic Functions of Many Variables in the Space L q

The exact-order estimates of linear widths are established for the classes B , Ω of periodic functions of many variables in the space L q for certain relations between on the parameters p and q.

On the Problem of Linear Widths of the Classes B p,θ r of Periodic Functions of Many Variables

We establish order estimates for the linear widths of the classes B , of periodic functions of many variables in the space L q for some relationships between the parameters p, q, and θ.

Trigonometric Widths of the Nikol’skii–Besov Classes in the Lebesgue Space with Mixed Norm

We establish exact-order estimates for the trigonometric widths of the Nikol’skii–Besov classes of periodic functions of many variables in the Lebesgue space with mixed norm.

Approximation of the Classes H p Ω of Periodic Functions of Many Variables in the Space L p

We establish upper estimates for the approximation of the classes H Ω of periodic functions of many variables by polynomials constructed by using the system obtained as the tensor product of the systems of functions of one variable. These results are then used to establish the exact-order estimates of the orthoprojective widths for the classes H Ω in the space L p with p ∈ {1, ∞}.