УДК 517.5 Встановлено, що задача Боянова–Найдьонова k = 0,1 , … , r - 1 , на класах функцій де q ≥ 1 , якщо k ≥ 1 , і q ≥ p , якщо k = 0 , еквівалентна задачі про точну константу C = C ( λ ) в нерівності колмогоровського типу де α = r - k + 1 / q r + 1 / p , δ > 0 , Ω p , λ r : = ⋃ { Ω p r ( A 0 , A r ) : A 0 = A r L ( φ λ , r ) p } , λ > 0 , φ λ , r --- стиск ідеального сплайна Ейлера порядку r , Зокрема, отримано точну на класах Ω p , λ r , λ > 0 , нерівність вигляду (1). Теореми про взаємозв'язок і наслідки з них (точні нерівності бернштейнівського типу) доведено також для задачі Боянова–Найдьонова на просторах тригонометричних поліномів та сплайнів.

Владислав Федорович Бабенко (до 75-річчя від дня народження)

It is shown that the Bojanov–Naidenov problem ‖x^(k)‖_q, δ → sup, k = 0, 1, . . . , r − 1, on the classes of functions Ω_p^r(A_0, A_r) := {x ∈L_∞^r: ‖x^(r)‖_∞≤A_r, L(x)_p≤A_0}, where q ≥ 1 for k ≥ 1 and q ≥ p for k = 0, is equivalent to the problem of finding the sharp constant C = C(λ) in the Kolmogorov-type inequality ‖x^(r)‖_q,δ≤ CL(x)_p^α‖x^(r)‖_∞^1-α, x∈Ω_p,λ^r, (1) where α =r-k+1/q/r+1/p, ‖ x‖_p,δ := sup ‖ x‖_L_p[a,b] :a, b, ∈ R, 0 < b – a ≤ δ δ > 0, Ω_p,λ^r := ⋃{Ω_p^r(A_0, A_r):A_0=A_rL(φλ ,r)p}, ⋋ > 0, φ⋋,r is a contraction of the ideal Euler spline of order r, and L(x)p : = sup ‖ x‖_L_p[a,b]: a, b, ∈ R |x(t)| > 0, t ∈ (a,b). In particular, we obtain a sharp inequality of the form (1) in the classes Ω_p,λ^r, ⋋ > 0. We also prove the theorems on relationships for the Bojanov–Naidenov problems in the spaces of trigonometric polynomials and splines and establish the corresponding sharp Bernstein-type inequalities.

УДК 517.5 Розв'язано екстремальну задачу на класi пар ( x , I ) функцiй x ∈ S φ k , похiднi яких x ( i ) , i = 0,1 , … , k , мають функцiями порiвняння вiдповiднi похiднi φ ( i ) , та інтервалів I = [ a , b ] , які задовольняють умови де Як наслiдок розв'язано такi ж задачі на класах i на обмежених множинах просторів тригонометричних полiномiв i сплайнiв та задачу Ердьоша для додатних (вiд'ємних) частин полiномiв і сплайнiв.

We solve an extremal problem ‖x_±^(k)‖_L_p[a,b]→sup, k=0,1,… ,r-1,p>0, in a class of pairs ( x, I ) of functions x ∈S_φ^k such that φ^(i) are the comparison functions for x^(i), i = 0 , 1 , … ,k, and the intervals I = [ a, b ] satisfy the conditions L(x)_p≤ A, μ{supp_[a,b]x_±^(k)}≤μ , where L(x)_p:=sup{(ab∫|x(t)|^pdt)^1/p:a,b∈𝐑,|x(t)|>0,t∈(a,b)}. In particular, we solve the same problems on the classes W_∞^r(𝐑) and on bounded sets of spaces of trigonometric polynomials and splines, as well as the Erdős problem for the positive (negative) parts of polynomials and splines.

The research is devoted to the study of long-term dynamics of fluctuations in average monthly flow rates during the summer-autumn and winter low-water periods in the rivers of Belarus. The assessment of extreme water flows of rare frequency in the summer-autumn period for the two time intervals (1961‒1990, 1991‒2020) demonstrates an increase in their frequency at most gauging stations over the last 30-year period. The minimum average monthly flow rates in the summer-autumn period are most often observed in August or September. The research shows that all months covering the winter low-water period are characterized by a tendency to increase runoff, more pronounced after 1988.

For odd $r\in \mathbb{N}$; $\alpha, \beta >0$; $p\in [1, \infty]$; $\delta \in (0, 2 \pi)$, any $2\pi$-periodic function $x\in L^r_{\infty}(I_{2\pi})$, $I_{2\pi}:=[0, 2\pi]$, and arbitrary measurable set $B \subset I_{2\pi},$ $\mu B \leqslant \delta/\lambda,$ where $\lambda=$ $\left({\left\|\varphi_{r}^{\alpha, \beta}\right\|_{\infty} \left\| {\alpha^{-1}}{x_+^{(r)}} + {\beta^{-1}}{x_-^{(r)}}\right\|_\infty}{E^{-1}_0(x)_\infty}\right)^{1/r}$, we obtain sharp Remez type inequality $$E_0(x)_\infty \leqslant \frac{\|\varphi_r^{\alpha, \beta}\|_\infty}{E_0(\varphi_r^{\alpha, \beta})^{\gamma}_{L_p(I_{2\pi} \setminus B_\delta)}} \left\|x \right\|^{\gamma}_{{L_p} \left(I_{2\pi} \setminus B \right)}\left\| {\alpha^{-1}}{x_+^{(r)}} + {\beta^{-1}}{x_-^{(r)}}\right\|_\infty^{1-\gamma},$$ where $\gamma=\frac{r}{r+1/p},$ $\varphi_r^{\alpha, \beta}$ is non-symmetric ideal Euler spline of order $r$, $B_\delta:= \left[M- \delta_2, M+ \delta_1 \right]$, $M$ is the point of local maximum of spline $\varphi_r^{\alpha, \beta}$ and $\delta_1 > 0$, $\delta_2 > 0$ are such that $\varphi_r^{\alpha, \beta}(M+ \delta_1) = \varphi_r^{\alpha, \beta}(M- \delta_2), \;\; \delta_1 + \delta_2 = \delta .$In particular, we prove the sharp inequality of Hörmander-Remez type for the norms of intermediate derivatives of the functions $x\in L^r_{\infty}(I_{2\pi})$.

For any q ≥ p > 0, 𝛼 = (r + 1/q)/(r + 1/p), fp ∈ [0,∞], and β ∈ [0, 2𝜋), we prove a sharp Remez-type inequality ‖ x‖_q≤‖φ_r+c‖_q/‖φ_r+c‖_L_p([0,2]/B_y(β))^α‖x^(r)‖_L_p([0,2]/B)^α‖x^(r)‖_∞^1-α for 2𝜋-periodic functions x ∈ Lr∞, which have zeros and satisfy the condition ‖x_+‖_p0.5em ‖x_-‖_p^-1=f_p,10em (1) where 𝜑r is Euler’s perfect spline of order r, the number c is such that the function x = 𝜑r +c satisfies condition (1), B is an arbitrary Lebesgue-measurable set such that μ B≤β(‖φ_r+c‖_p‖x^(r)‖_∞‖ x‖_p^-1)^-1/(r+1/p), the set By(β) is defined by By(β) := t ∈ [0, 2𝜋] : |𝜑r(t) + c| > y(β), and moreover, μBy(β) = β. We also establish sharp Remez-type inequalities of various metrics for trigonometric polynomials and polynomial splines satisfying relation (1).

We obtain the strengthened Kolmogorov comparison theorem in asymmetric case.In particular, it gives us the opportunity to obtain the following strengthened Kolmogorov inequality in the asymmetric case:$$\|x^{(k)}_{\pm }\|_{\infty}\le \frac{\|\varphi _{r-k}( \cdot \;;\alpha ,\beta )_\pm \|_{\infty }}{E_0(\varphi _r( \cdot \;;\alpha ,\beta ))^{1-k/r}_{\infty }}|||x|||^{1-k/r}_{\infty}\|\alpha^{-1}x_+^{(r)}+\beta^{-1}x_-^{(r)}\|_\infty^{k/r}$$for functions $x \in L^r_{\infty }(\mathbb{R})$, where$$|||x|||_\infty:=\frac12 \sup_{\alpha ,\beta}\{ |x(\beta)-x(\alpha)|:x'(t)\neq 0 \;\;\forallt\in (\alpha ,\beta) \}$$$k,r \in \mathbb{N}$, $k 0$, $\varphi_r( \cdot \;;\alpha ,\beta )_r$ is the asymmetric perfect spline of Euler of order $r$ and $E_0(x)_\infty $ is the best uniform approximation of the function $x$ by constants.

УДК 517.5 Для довiльних q ≥ p > 0 , α = ( r + 1 / q ) / ( r + 1 / p ) , f p ∈ [ 0 , ∞ ] , β ∈ [ 0,2 π ) , доведено точну нерiвнiсть типу Ремеза < / ⅆ i v > < ⅆ i v > ‖ x ‖ q ≤ < / ⅆ i v > < ⅆ i v > ‖ φ r + c ‖ q ‖ φ r + c ‖ L p ( [ 0 , < / ⅆ i v > < ⅆ i v > 2 π ] ∖ B y ( β ) ) α ‖ x ‖ L p ( [ 0 , < / ⅆ i v > < ⅆ i v > 2 π ] ∖ B ) α ‖ x ( r ) ‖ ∞ 1 - α < / ⅆ i v > < ⅆ i v > для 2 π -перiодичних функцiй x ∈ L ∞ r , що мають нулі, і задовольняють умову < / ⅆ i v > < ⅆ i v > ‖ x + ‖ p ⋅ ‖ x - ‖ p -1 = f p ( 1 ) < / ⅆ i v > < ⅆ i v > де φ r - ідеальний сплайн Ейлера порядку r , а число c обрано так, що функція x = φ r + c задовольняє рівність (1), B - довільна вимірна за Лебегом множина, така що μ B ≤ β ( ‖ φ r + c ‖ p ⋅ ‖ x ( r ) ‖ ∞ ⋅ ‖ x ‖ p -1 ) - 1 / ( r + 1 / p ) , а множина B y ( β ) означена рівністю B y ( β ) : = { t ∈ [ 0,2 π ] : | φ r ( t ) + c | > y ( β ) } , причому μ B y ( β ) = β . Також отримано точні нерівності різних метрик типу Ремеза для тригонометричних поліномів і поліноміальних сплайнів, що задовольняють умову (1).

We obtain generalization of the known A.A. Ligun's inequality to non-normed $L_q$-spaces for derivatives of periodic functions.

For any $q > p > 0$, $\omega > 0,$ $d \ge 2 \omega,$ we obtain the following sharp inequality of various metrics$$\|x\|_{L_q(I_{d})} \le \frac{\|\varphi +c\|_{L_q(I_{2\omega})}}{\|\varphi + c \|_{L_p(I_{2\omega})}}\|x\|_{L_p(I_{d})}$$on the set $S_{\varphi}(\omega)$ of $d$-periodic functions $x$ having zeros with given the sine-shaped $2\omega$-periodic comparison function $\varphi$, where $c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]$ is such that$$\|x_{\pm}\|_{L_p(I_{d})} = \|(\varphi +c)_{\pm}\|_{L_p(I_{2\omega})}\,.$$In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient of the norms $\|x_{+}\|_{L_p(I_{d})} / \|x_-\|_{L_p(I_{d})}$.

УДК 517.5 Доведено теорему про взаємозв'язок точних констант у нерівностях типу Колмогороваі Колмогорова–Ремеза для диференційовних періодичних функцій. Як наслідок встановлено новi точнi нерiвностi типу Колмогорова–Ремеза на класах таких функцій. Крім того, отримано новi точнi нерiвностi типу Бернштейна–Ремеза для тригонометричних поліномів і поліноміальних сплайнів.

We obtain new sharp inequality of Kolmogorov type for differentiable periodic functions $x \in L_1^3$.

We obtain the estimates of the seminorms of Weil of the functions on the real line and their derivatives with the help of local $L_p$-norms of the functions and uniform norms of their highest derivatives.

We prove the inequality that estimates seminorm of Weil of the derivatives of the functions on the real line with the help of uniform norm of the functions and their derivatives. We also solve the corresponding problem of Kolmogorov.

In the paper, we have found the supremum of the best mean approximations by algebraic polynomials of differentiable functions from $W^r_L$ classes for $r=1,2$.

We establish sharp estimates of the $L_q$-norms on any finite interval for the polynomials and splines and their derivatives with the help of local $L_p$-norms of these polynomials and splines.

We establish a new theorem on the relationship between sharp constants in Kolmogorov-type inequalities and sharp constants in Kolmogorov–Remez-type inequalities for differentiable periodic functions. As a consequence, we obtain new sharp Kolmogorov–Remez-type inequalities for these functions. We also derive new sharp Bernstein–Remez-type inequalities for trigonometric polynomials and polynomial splines.