On quasilinear Beltrami-type equations with degeneration

We consider the solvability problem for the equation $f_{\bar z} $ = v(z, f(z))f _z , where the function v(z,w) of two variables may be close to unity. Such equations are called quasilinear Beltrami-type equations with ellipticity degeneration. We prove that, under some rather general conditions on v(z,w), the above equation has a regular homeomorphic solution in the Sobolev classW _loc ^1,1 . Moreover, such solutions f satisfy the inclusion f ^?1 ∈ W _loc ^1,2 .     

A method of constructing integrable linear equations and its application to Hill's equation

Starting with a given equation of the form $$\ddot x + [\lambda + \varepsilon f(t)] x = 0$$ , where λ > 0 and ? ? l is a small parameter [heref(t) may be periodic, and so Hill's equation is included], we construct an equation of the form y + [λ + ?f (t) + ?² _g (t)]y = 0, integrable by quadratures, close in a certain sense to the original equation. For x₀ = y₀ and x ₀ ^′ = y ₀ ^′ , an upper bound is obtained for ¦y—x¦ on an interval of length Δt.     

Formulas for setting a location of a wavefront propagating in a medium with power dependence of velocity on a coordinate

The 2D eikonal equation f _x ² + f _y ² = (ky)^2α is considered. If f is its solution, then the relation f(x, y) = C gives a wavefront location. However, obtaining solutions is associated with some difficulties. We develop the earlier proposed technique, which enables us to track a parametric curve to set a wavefront without solving the above equation.     

On copositive approximation by algebraic polynomials

Для функцииf ∈C[?1, 1] с ог раниченным числом пе ремен знака строится последовательность многочленовр _п , коположительных сf (т.е.f(x)p _n (x)≥0, ?1≤х<1) и таких, что $$\left\| {f - p_n } \right\|_\infty \leqslant C\omega _\varphi ^3 (f,n^{ - 1} ),$$ гдеω _? ³ (f, δ) — модуль непр ерывности Дитциана-Т отика третьего порядка. Изв естно, чтоω _? ³ нельзя заменить ни наω _? ⁴ , ни на ω⁴. Таким образом, приведенная оценка точна в некотором смы сле. В качестве следст вия установлена эквивал ентность соотношений $$E_n (f) = O(n^{ - \alpha } )\user2{}E_n^{(0)} (f,r) = O(n^{ - \alpha } )\user2{}0< \alpha< 3.$$      

A new modification of the Hermite-Fejér interpolation

пУсть жАДАНы Ужлы $$ - \infty< x_1< x_2< ...< x_k< x_{k + 1}< ...< x_n< + \infty ,$$ , И пУстьx ₁ ^* <x ₂ ^* <...<x _n-1 ^* — кОРНИ МНОгО ЧлЕНА Ω′(х). гДЕ $$\omega (x) = \prod\limits_{k = 1}^n {(x - x_k ).} $$ В РАБОтЕ ИсслЕДУЕтсь жАДАЧА: кАк ОпРЕДЕлИт ь МНОгОЧлЕНР(х) МИНИМАльНОИ стЕп ЕНИ, Дль кОтОРОгО ВыпОлНь Утсь слЕДУУЩИЕ ИНтЕР пОльцИОННыЕ УслОВИь гДЕ {y _k И {y _k′}-жАДАННы Е сИстЕМы жНАЧЕНИИ.     

On detecting a wavefront described by a 2D eikonal equation when the velocity in the medium depends on one spatial variable

In this paper, the 2D eikonal equation f _x ² + f _y ² = (ky + b)^2α is considered. If a solution is found, the relation f (x, y) = C determines the location of the wavefront. However, finding the solutions is still an open question. In this paper, we propose to directly find a curve in parametric form, which determines the wavefront without solving the equation.     

Inversion complexity of self-correcting circuits for a certain sequence of boolean functions

The asymptotics L _k ^? (f ₂ ⁿ ) ?? n min{k+1, p} is obtained for the sequence of Boolean functions $f_2^n \left( {x_1 , \ldots ,x_n } \right) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n}$ for any fixed k, p ?? 1 and growing n, here L _k ^? (f ₂ ⁿ ) is the inversion complexity of realization of the function f ₂ ⁿ by k-self-correcting circuits of functional elements in the basis B = {&, ?}, p is the weight of a reliable invertor.     

Best trigonometric and bilinear approximations of classes of functions of several variables

Order-sharp estimates of the best orthogonal trigonometric approximations of the Nikol’skii-Besov classes B _p,θ ^r of periodic functions of several variables in the space L _q are obtained. Also the orders of the best approximations of functions of 2d variables of the form g(x, y) = f(x?y), x, y ∈ $\mathbb{T}$ ^d = Π _j=1 ^d [?π, π], f(x) ∈ B _p,θ ^r , by linear combinations of products of functions of d variables are established.     

The exponential diophantine equation x y + y x = y 2 with xy odd

For any fixed positive integer D which is not a square, let (u, υ) = (u ₁, υ ₁) be the fundamental solution of the Pell equation u ² ? Dυ ² = 1. Further let $\mathbb{D}$ be the set of all positive integers D such that D is odd, D is not a square and gcd(D, υ ₁) > max(1, √D/8). In this paper we prove that if (x, y, z) is a positive integer solution of the equation x ^y + y ^x = z ² satisfying gcd(x, y) = 1 and xy is odd, then either $x \in \mathbb{D}$ or $y \in \mathbb{D}$ .     

Bounds for singular integrals associated with a class of hypersurfaces

For the hypersurface Γ=(y,γ(y)), the singular integral operator along Γ is defined by. $$Tf(x,x_n ) = P.V.\int_{\mathbb{R}^n } {, f(x - y,x_n ) - } \gamma (y))_{\left| y \right|^{n - 1} }^{\Omega (v)} dy$$ where Σ is homogeneous of order 0, $ \int_{\Sigma _{n \lambda } } {\Omega (y')dy'} = 0 $ . For a certain class of hypersurfaces, T is shown to be bounded on L^p(Rⁿ) provided Ω∈L _α ¹ (Σ _n?2),P>1.     

Theory of Maxwell fluids

The solvability in-the-large is proved of the initial-boundary-value problem for the stationary quasilinear system describing the flow of a linear viscoelastic fluid with a finite number of discretely distributed relaxation times and with delay times {? ₀ ^?4 ? _m }, m=1, ..., L?1 (generalized Maxwell fluid).     

Walsh-Lebesgue points of multi-dimensional functions

Walsh-Lebesgue points are introduced for higher dimensions and it is proved that a.e. point is a Walsh-Lebesgue point of a function f from the Hardy space H ₁ ⁱ [0, 1) ^d , where $$ H_1^i [0,1]^d \supset L(\log L)^{d - 1} [0,1)^d for all i = 1,...,d $$ . Every function f ∈ H ₁ ⁱ [0, 1) ^d is Fejér summable at each Walsh-Lebesgue point. Similar theorem is verified for ?-summability.     

A system of functions

A system of functions $$f_k (x) = \sum\nolimits_{i = 1}^r a _i \varphi _\iota (x)^k + b_i \overline {\varphi _\iota } (x)^k , k = 1,2,...$$ is considered on the interval [0,l]. Under certain conditions on the? _i(x), it is proved that the system 1 ∪ {f_k(x)} _k=1 ^∞ is complete in the space L_p(0,l). In the case r=1 it is proved, under certain additional assumptions, that the system {f_k(x)} _k=0 ^∞ is minimal.     

Solution of the Szili type conjugate problem on the roots of the laguerre polynomials

Let ?1<α≤0 and let $$L_n^{(\alpha )} (x) = \frac{1}{{n!}}x^{ - \alpha } e^x \frac{{d^n }}{{dx^n }}(x^{\alpha + n} e^{ - x} )$$ be the generalizednth Laguerre polynomial,n=1,2,… Letx ₁,x ₂,…,x _n andx*₁,x*₂,…,x* _n?1 denote the roots ofL _n ^(α) (x) andL _n ^(α)′ (x) respectively and putx*₀=0. In this paper we prove the following theorem: Ify ₀,y ₁,…,y _n ?1 andy ₁ ^′ ,…,y _n ^′ are two systems of arbitrary real numbers, then there exists a unique polynomialP(x) of degree 2n?1 satisfying the conditions $$\begin{gathered} P\left( {x_k^* } \right) = y_k (k = 0,...,n - 1) \hfill \\ P'\left( {x_k } \right) = y_k^\prime (k = 1,...,n). \hfill \\ \end{gathered} $$ .     

Limit ultraspherical series and their approximation properties

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x ₂)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S _n ^?1 (f) = S _n ^?1 (f, x) is the projection onto the subspace of algebraic polynomials p _n = p _n (x) of degree at most n, i.e., S _n (p _n ) = p _n ; in addition, S _n ^?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S _n ^?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ _n (x) of the partial sums S _n ^?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

A remark concerning Pincherle bases

In this note we find sufficient conditions for uniqueness of expansion of any two functionsf(z) and g(z) which are analytic in the circle ¦ z ¦ < R (0 < R <∞) in series $$f(z) = \sum\nolimits_{n = 0}^\infty {(a_n f_2 (z) + b_n g_n (z))}$$ and $$g_i (z) = \sum\nolimits_{n = 0}^\infty {a_n \lambda _n f_n (z)} + b_n \mu _n f_n (x)),$$ which are convergent in the compact topology, where (f _n {z} _n=0 ^∞ and {g} _n=0 ^∞ are given sequences of functions which are analytic in the same circle while {λ _n } _n=0 ^∞ and {μ _n } _n=0 ^∞ are fixed sequences of complex numbers. The assertion obtained here complements a previously known result of M. G. Khaplanov and Kh. R. Rakhmatov.     

Separate asymptotics of two series of eigenvalues for a single elliptic boundary-value problem

The spectral problem in a bounded domain Ω?Rⁿ is considered for the equation Δu= λu in Ω, ?u=λ?υ/?ν on the boundary of Ω (ν the interior normal to the boundary, Δ, the Laplace operator). It is proved that for the operator generated by this problem, the spectrum is discrete and consists of two series of eigenvalues {λ _j ⁰ } _j=1 ^∞ and {λ _j ^∞ } _j=1 ^∞ , converging respectively to 0 and +∞. It is also established that $$N^0 (\lambda ) = \sum\nolimits_{\operatorname{Re} \lambda _j^0 \geqslant 1/\lambda } {1 \approx const} \lambda ^{n - 1} , N^\infty (\lambda ) \equiv \sum\nolimits_{\operatorname{Re} \lambda _j^\infty \leqslant \lambda } {1 \approx const} \lambda ^{n/1} .$$ The constants are explicitly calculated.     

Singular integrals along Lipschitz curves with holomorphic kernels

If γ(x)=x+iA(x),tan ^?1‖A′‖_∞<ω<π/2,S _ω ⁰ ={z∈C}| |argz|<ω, or, |arg(-z)|<ω} We have proved that if φ is a holomorphic function in S _ω ⁰ and \(\left| {\varphi (z)} \right| \leqslant \frac{C}{{\left| z \right|}}\) , denotingT f (z)= ∫?(z-ζ)f(ζ)dζ, ?f∈C ₀(γ), ?z∈suppf, where C_c(γ) denotes the class of continuous functions with compact supports, then the following two conditions are equivalent:

1. T can be extended to be a bounded operator on L²(γ);
2. there exists a function ?₁ ∈H ^∞(S _ω ⁰ ) such that ?′₁(z)=?(z)+?(-z), ?z∈S _ω ⁰ ?z∈S _w ⁰ .

     

Greedy expansions in Hilbert spaces

We study the rate of convergence of expansions of elements in a Hilbert space H into series with regard to a given dictionary D. The primary goal of this paper is to study representations of an element f ∈ H by a series f ～ ∑ _j=1 ^∞ c _j (f)g _j (f), $g_j \left( f \right) \in \mathcal{D}$ . Such a representation involves two sequences: {g _j (f)} _j=1 ^∞ and {c _j (f) _j=1 ^∞ . In this paper the construction of {g _j (f)} _j=1 ^∞ is based on ideas used in greedy-type nonlinear approximation, hence the use of the term greedy expansion. An interesting open problem questions, “What is the best possible rate of convergence of greedy expansions for f ∈ A ₁(D)?” Previously it was believed that the rate of convergence was slower than $m^{ - \tfrac{1} {4}}$ . The qualitative result of this paper is that the best possible rate of convergence of greedy expansions for $f \in A_1 \left( \mathcal{D} \right)$ is faster than $m^{ - \tfrac{1} {4}}$ . In fact, we prove it is faster than $m^{ - \tfrac{2} {7}}$ .     

Local properties of functions and approximation by trigonometric polynomials

Suppose Φ_{p, E} (p>0 an integer, E ?[0, 2π]) is a family of positive nondecreasing functions? _x(t) (t>0, x∈ E) such that? _x(nt)≤n^P ? _x(t) (n=0,1,...), t_n is a trigonometric polynomial of order at most n, and Δ _h ^l (f, x) (l>0 an integer) is the finite difference of orderl with step h of the functionf.THEOREM. Supposef (x) is a function which is measurable, finite almost everywhere on [0, 2π], and integrable in some neighborhood of each point xε E,? _X εΦ_p,E and $$\overline {\mathop {\lim }\limits_{\delta \to \infty } } |(2\delta )^{ - 1} \smallint _{ - \delta }^\delta \Delta _u^l (f,x)du|\varphi _x^{ - 1} (\delta ) \leqslant C(x)< \infty (x \in E).$$ . Then there exists a sequence {t _n } _n=1 ^∞ which converges tof (x) almost everywhere, such that for x ε E $$\overline {\mathop {\lim }\limits_{n \to \infty } } |f(x) - l_n (x)|\varphi _x^{ - 1} (l/n) \leqslant AC(x),$$ where A depends on p andl.