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 .

     

Interpolation in certain Hilbert spaces of analytic functions

In this article we study, for a Hilbert spaceB of analytic functions in the open unit disk, the dependence of the structure of the space of sequencesB(Z)={{f(z_k)} _k=1 ^∞ :f∈B} on the choice of the sequence Z={z_k} _k=1 ^∞ of distinct points of the unit disk [6].     

Best trigonometric approximations for some classes of periodic functions of several variables in the uniform metric

We study best M-term trigonometric approximations and best orthogonal trigonometric approximations for the classes B _pθ ^r and W _pα ^r of periodic functions of several variables in the uniform metric.     

Wavelet estimations for density derivatives

Donoho et al. in 1996 have made almost perfect achievements in wavelet estimation for a density function f in Besov spaces Bsr,q(R). Motivated by their work, we define new linear and nonlinear wavelet estimators flin,nm, fnonn,m for density derivatives f(m). It turns out that the linear estimation E(‖flinn,m-f(m)‖p) for f(m) ∈ Bsr,q(R) attains the optimal when r≥ p, and the nonlinear one E(‖fnonn,m-f(m)‖p) does the same if r≤p/2(s+m)+1 . In addition, our method is applied to Sobolev spaces with non-negative integer exponents as well.     

On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces

We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = ?y″ + q(x)y with potentials from the Sobolev space W ₂ ^θ?1 , θ ≥ 0, including the nonclassical case θ ∈ [0, 1) in which the potential is a distribution. The results are obtained in new terms. Let s _2k (q) = λ _k ^1/2 (q) ? k, s _2k?1(q) = μ _k ^1/2 (q) ? k ? 1/2, where {λ _k } ₁ ^∞ and {μ _k } ₁ ^∞ are the sequences of eigenvalues of the operator L generated by the Dirichlet and Dirichlet-Neumann boundary conditions, respectively,. We construct special Hilbert spaces t ₂ ^θ such that the mapping F:W ₂ ^θ?1 →t ₂ ^θ defined by the equality F(q) = {s _n } ₁ ^∞ is well defined for all θ ≥ 0. The main result is as follows: for θ > 0, the mapping F is weakly nonlinear, i.e., can be expressed as F(q) = Uq + Φ(q), where U is the isomorphism of the spaces W ₂ ^θ?1 and t ₂ ^θ , and Φ(q) is a compact mapping. Moreover, we prove the estimate ∥Ф(q)∥_τ ≤ C∥q∥_θ?1, where the exact value of τ = τ(θ) > θ ? 1 is given and the constant C depends only on the radius of the ball ∥q∥_θ? ≤ R, but is independent of the function q varying in this ball.     

Uniform stability of the inverse Sturm-Liouville problem with respect to the spectral function in the scale of Sobolev spaces

We consider the inverse problem of recovering the potential for the Sturm-Liouville operator Ly = ?y″ + q(x)y on the interval [0, π] from the spectrum of the Dirichlet problem and norming constants (from the spectral function). For a fixed θ ≥ 0, with this problem we associate a map F: W ₂ ^θ → l _D ^θ , F(σ) = {s _k } ₁ ^∞ , where W ₂ ^θ = W ₂ ^θ [0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential q ∈ W ₂ ^{θ ? 1} , and l _D ^θ is a specially constructed finite-dimensional extension of the weighted space l ₂ ^θ ; this extension contains the regularized spectral data s = {s _k } ₁ ^∞ for the problem of recovering the potential from the spectral function. The main result consists in proving both lower and upper uniform estimates for the norm of the difference ‖σ ? σ ₁‖ _θ in terms of the l _D ^θ norm of the difference of the regularized spectral data ‖s ? s₁‖ _θ . The result is new even for the classical case q ∈ L ₂, which corresponds to the case θ = 1.     

On the growth order of the rectangular partial sums of multiple non-orthogonal series

Пустьd-натуральное ч исло,Z ^d — множество на боров k=(k ₁, ...,k _d ), состоящих из неотрицательных цел ыхk _j ,Z ₊ ^d =k∈Z ^d :k≧1. Предположи м, что системаf _k (x):k∈Z ₊ ^d ? ?L₂(X,A, μ) и последовател ьностьa _k :k∈Z ₊ ^d . таковы, чт о для всех b∈Z^d и m∈Z ₊ ^d выполн ены неравенства (2) $$\left\| {\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k f_k (x)} } \right\|_2^2 \leqq w^2 (m)\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k^2 } $$ где последовательно сть {w(m): m∈Z ₊ ^d положительн а и не убывает. Например, есл иf _k (х) — квазистационарная система, то для соотве тствующей последовательности {ω(m) (2) имeeт Меcтo ДЛЯ ЛЮбОЙ ПОС ЛеДОВатеЛЬНОСТИ {a_k}. В работе получены оце нки порядка роста пря моугольных частных суммS _m (x)= =∑ a_kf_k(x) при maxmj→∞ как в случ ае {a_k}∈l₂, таки для {a_k}l₂. Эти оценки явля1≦k≦m 1≦j≦d ются новыми даже для о ртогональных кратны х рядов. Показано, что упомяну тые оценки в общем слу чае являются точными.     

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 .     

Approximation of functions of several variables by spherical Riesz means

For even N ≥ 2 and δ 2N-3 (for N-2 or 4 we assume that δ > (N-1)/2) we find asymptotic approximations for the quantity $$E_R^\delta (H_{\rm N}^\omega ) = \mathop {sup}\limits_{f \in H_{\rm N}^\omega } \parallel f(x) - S_R^\omega (x,f)\parallel _ \in (R \to \infty ),$$ , where S _R ^δ (x,f) is the spherical Riesz mean of order δ of the Fourier kernel of the functionf(x), and H _N ^ω is the class of periodic functions of N variables whose moduli of continuity do not exceed a given convex modulus of continuity ω(δ). For N 2 and δ > 1/2 the result is known.     

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 $m^{ - \tfrac{2} {7}}$ .     

Boundary noncrossings of additive Wiener fields∗

Let {W _i (t), t ∈ ?₊}, i = 1, 2, be two Wiener processes, and let W ₃ = {W ₃(t), t ∈ ? ₊ ² } be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary noncrossing probability P _f = P{W ₁(t ₁) + W ₂(t ₂) + W ₃(t) + f(t) ≤ u(t), t ∈ ? ₊ ² }, where f, u : ? ₊ ² → ? are two general measurable functions. We further show that, for large trend functions γf > 0, asymptotically, as γ → ∞, P _γf is equivalent to \( {P}_{\gamma}\underset{\bar{\mkern6mu}}{{}_f} \) , where \( \underset{\bar{\mkern6mu}}{f} \) is the projection of f onto some closed convex set of the reproducing kernel Hilbert space of the field W(t) = W ₁(t ₁) + W ₂(t ₂) + W ₃(t). It turns out that our approach is also applicable for the additive Brownian pillow.     

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.     

The maximal Fejér operator on the spacesH p (R×···×R)

It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH _p (R×···×R) toL _p (R ^d ) (1/2<p<∞) and is of weak type (H ₁ ^?i ,L ₁) (i=1,…,d), where the Hardy spaceH ₁ ^?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H ₁ ^?i ?L(logL) ^d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH _p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H _p (R×···×R) the Fejér means converge tof inH _p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too.     

On the degree of complex rational approximation to real functions

Iff∈C[?1, 1] is real-valued, letE _R ^mn (f) andE _C ^mn (f) be the errors in best approximation tof in the supremum norm by rational functions of type (m, n) with real and complex coefficients, respectively. We show that form≥n?1≥0 $$\gamma _{mn} = \inf \{ {{E_{mn}^C (f)} \mathord{\left/ {\vphantom {{E_{mn}^C (f)} {E_{mn}^R (f)}}} \right. \kern-\nulldelimiterspace} {E_{mn}^R (f)}}:f \in C[ - 1,1]\} = \tfrac{1}{2}.$$      

Weighted tight frames of exponentials on a finite interval

Given a finite intervalI?R, a characterization is given for those discrete sets of real numbers Λ and associated sequences {c _λ}_λ∈Λ, withc _λ>0, having the properties that every functionf∈L ²(I) can be expanded inL ²(I) as the unconditionally convergent series $$f = \sum\limits_{\lambda \in \Lambda } {\hat f} (\lambda )c_\lambda e^{2\pi i\lambda x} $$ and that the range of the mappingL ²(I)→L _μ ² :f→f has finite codimension inL _μ ² , iff denotes the Fourier transform off and μ is the measure μ = ∑_λ∈Λ c _λ δ_λ.     

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 $$E_n (f) = O(n^{ - \alpha } )\user2{}E_n^{(0)} (f,r) = O(n^{ - \alpha } )\user2{}0< \alpha< 3.$$      

Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials

In the spaceL _q, 1<q<∞ we establish estimates for the orders of the best approximations of the classes of functions of many variablesB _1,θ ^r andB _p,α ^r by orthogonal projections of functions from these classes onto the subspaces of trigonometric polynomials. It is shown that, in many cases, the estimates obtained in the present work are better in order than in the case of approximation by polynomials with harmonics from the hyperbolic cross.     

Some class of analytic functions related to conic domains

For q ∈ (0, 1) let the q-difference operator be defined as follows $$\partial _q f(z) = \frac{{f(qz) - f(z)}} {{z(q - 1)}} (z \in \mathbb{U}),$$ where \(\mathbb{U}\) denotes the open unit disk in a complex plane. Making use of the above operator the extended Ruscheweyh differential operator R _q ^λ f is defined. Applying R _q ^λ f a subfamily of analytic functions is defined. Several interesting properties of a defined family of functions are investigated.     

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.