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We study the order (velocity) of the approximation of functions on the axis by entire functions of exponential type not higher than σ as σ → ∞ (the linear and best approximations). The exact order of approximation of individual functions on R d by the classical summation methods of Fourier integrals (Gauss-Weierstrass, Bochner-Riesz, Marcinkiewicz) and the nonclassical Bernstein-Stechkin method is found. For functions on a torus, similar theorems of approximation by polynomials were obtained previously.Keywords. Entire functions of exponential type, Bernstein-Riesz inequality, algebra of absolutely convergent Fourier integrals, summation methods, modulus of smoothness, K-functional, Fourier multiplier, total variation of a measure.

511.72We study the equation ν 1 ( ) = x x, where ν 1 ( ) x is the function of frequency of the digit 1 in the ternary expansion of x. We prove that this equation has a unique rational root and a continuum set of irrational solutions. An algorithm for the construction of solutions is proposed. We also describe the topological and metric properties of the set of all solutions. Some additional facts about the equations ν i x x ( ) = , i = 0 2 , , are given.