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We obtain an estimate for the convergence rate of the Fourier series of a continuous periodic function in terms of the modulus of continuity of the function and the value of its -variation. We prove that the leading term of the estimate is sharp. [ABSTRACT FROM AUTHOR]

An estimate for the convergence rate of the Fourier series of a continuous periodic function of bounded variation is refined. [ABSTRACT FROM AUTHOR]

General Rogosinsky–Bernstein linear polynomial means of Fourier series are introduced and three convergence criteria as are obtained: for convergence in the space of continuous periodic functions and for convergence almost everywhere with two guaranteed sets (Lebesgue points and -points). For smooth functions, the rate of convergence in norm of , as well as of their interpolation analogues, is also studied. For approximation of functions in , the asymptotics is found along with the rate of decrease of the remainder term. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The paper deals with the spherical Fourier sums $$S_r (f,x) = \sum\nolimits_{\left\| k \right\| \leqslant r} {\hat f} (k)e^{ik \cdot x}$$ of a periodic function f in m variables and the strong integral means of these sums ((∫ | S( f, x)| dr)/ R for p ≥ 1. We establish the exact growth order as R → +∞ of the corresponding operators, i.e., the growth order of the quantities sup((∫| S( f, 0)| dr)/ R). The upper and lower bounds differ by their coefficients, which depend only on the dimension m. A sufficient condition on the function ensuring the uniform strong p-summability of its Fourier series is given. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We find sharp inequalities between the best approximations of periodic differentiable functions by trigonometric polynomials and moduli of continuity of mth order in the space L as well as present their applications. For some classes of functions defined by these moduli of continuity, we calculate the exact values of n-widths in L. [ABSTRACT FROM AUTHOR]

We prove the sharpness of Zygmund’s theorem, which asserts that if a 2 π-periodic function f belongs to L ln+ L, then its Fourier series is convergent in mean. [ABSTRACT FROM AUTHOR]

We obtain order-sharp estimates of best approximations for the classes B of periodic functions of several variables by trigonometric polynomials whose spectra are generated by the level surfaces of the function Ω( t). [ABSTRACT FROM AUTHOR]

The article presents a numerical solution on the obtainment of order-sharp estimates of Fourier widths of classes of Nikol'skii-Besov and Lizorkin-Triebel periodic functions of various variables. It discusses formulas and definitions of Fourier widths and function spaces. It provides theorems on the Fourier widths with a comprehensive derivation of the equation.

The article focuses on exact constants in Jackson-type inequalities and exact values of widths. Application of other characteristics of smoothness of 2ω-periodic functions, for example, of the trigonometric modulus of continuity, allows one to prove new meaningful results related to further studies of Jackson-type inequalities.

The article reports that the Turan problem was stated by mathematician P. Turán in 1970 in connection with possible applications in the number theory. Further progress in the Turán problem was made in the papers of mathematicians D.V. Gorbachev and A.S. Manoshina. They've proved the Popov's conjecture and reduced the Turán problem for a rational "h" to a finite dimensional external problem and evaluated the result for it. In this article the Turán problem is solved for all rational "h" by considering several linear programming problems.

The article focuses on the deviation of partial sums of Fourier-Haar systems with arbitrary dilation factor. It states that Fourier series converges uniformly for any continuous 1-periodic function. It adds that at the points of discontinuity, the values of the functions are taken as the half-sum of the left and right limits.

The article deals with the Jackson theorem in the space L2 on the interval [-1,1] with power-law weight. It provides equations for Bessel function and Hilbert space. Further, this analysis is intended to compare inequality with the well-known exact Jackson inequality in the space L2 of periodic functions, which was proved by Cherhykh.

The article presents an estimate of the approximation of periodic continuous functions by Fourier series. The uniform convergence of the Fourier series of a 2π-periodic continuous function generalizes the Dini-Lipshitz criterion. The estimate of integral A is taken into account to prove the continuous function theorem.