Long time small solutions to nonlinear parabolic equations

A sharp result on global small solutions to the Cauchy problem $$u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $$ In Rⁿ is obtained under the the assumption thatf is C^1+r forr>2/n and ‖u ₀‖C²(R ⁿ ) +‖u ₀‖W ₁ ² (R ⁿ ) is small. This implies that the assumption thatf is smooth and ‖u ₀ ‖W ₁ ^k (R ⁿ )+‖u ₀‖W ₂ ^k (R ⁿ ) is small fork large enough, made in earlier work, is unnecessary.     

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.     

n-Widths of Sobolev spaces inL p

LetW _p ^(r) ={f:f∈C ^r?1[0, 1],f ^(r?1) abs.cont., ∥f ^(r)∥ _p <∞}, and setB _p ^(r) ={f:f∈W _p ^(r) ,∥f ^(r)∥ _p ≤1}. We find the exact Kolmogorov, Gel'fand, linear, and Bernsteinn-widths ofB _p ^(r) inL ^p for allp∈(1, ∞). For the Kolmogorovn-width we show that forn≥r there exists an optimal subspace of splines of degreer?1 withn?r fixed simple knots depending onp.     

L p-approximation by Kantorovič operators

Оператор Канторович а дляf∈L _p(I), I=[0,1], определяе тся соотношением $$P_n (f,x) = (n + 1)\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} x^k (1 - x)^{n - 1} \int\limits_{I_k } {f(t)dt,} $$ гдеI _k=[k/(n}+1),(k+1)/(n+ 1)],n∈N. Доказывается, что есл ир>1 иf∈W _p ² (I), т.е.f абсол ютно непрерывна наI иf″∈L _p(I), то $$\left\| {P_n f - f} \right\|_p = O(n^{ - 1} ).$$ Далее, установлено, чт о еслиf∈L _p(I),p>1 и ∥P _n f-f∥_р=О(n ^?1), тоf∈S, гдеS={f∶f аб-солютно непрерывна наI, x(1?x)f′(x)=∝ ₀ ^x h(t)dt, гдеh∈L _p(I) и ∝ ₀ ¹ h(t)dt=0}. Если жеf∈L_p(I),p>1, то из условия ∥P _n(f)?f∥_pL=o(n^?1) вытекает, чтоf постоянна почти всюду.     

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 .

     

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 $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

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.     

A weight space invariant with respect to a singular linear operator

For the singular operator $$Su = \int_a^b {\frac{{K(x, s) u (s)}}{{s - x}}} ds$$ invariant weight spacesλ _α ^β , _p (u(x)∈λ _α ^β , _p if 1⁰,u (x) ρ (x)∈ H _β ⁰ , 2⁰.‖u‖L _p(ρ₀)<∞, ρ (x) = (x?a) (b ?x)^1+β, ρ₀(x)=(b?x)^α(p?1), 0<α, β<1,p>1H ₀ ^β is a Hölder space. Multiplicative inequalities of the type of Kh. Sh. Mukhtarov are also obtained.     

Rows and diagonals of the Walsh array for entire functions with smooth Maclaurin series coefficients

Let \(f(z): = \sum\nolimits_{j = 0}^\infty {a_j z^J } \) be entire, witha _j≠0,j large enough, \(\lim _{J \to \infty } a_{j + 1} /a_J = 0\) , and, for someq∈C, \(q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q\) asj→∞. LetE _mn(f; r) denote the error in best rational approximation off in the uniform norm on |z‖≤r, by rational functions of type (m, n). We study the behavior ofE _mn(f; r) asm and/orn→∞. For example, whenq above is not a root of unity, or whenq is a root of unity, butq _m has a certain asymptotic expansion asm→∞, then we show that, for each fixed positive integern, ,m→∞. In particular, this applies to the Mittag-Leffler functions \(f(z): = \sum\nolimits_{j = 0}^\infty {z^j /\Gamma (1 + j/\lambda )} \) and to \(f(z): = \sum\nolimits_{j = 0}^\infty {z^j /(j!)^{I/\lambda } } \) , λ>0. When |q‖<1, we also handle the diagonal case, showing, for example, that ,n→∞. Under mild additional conditions, we show that we can replace 1+0(1) ⁿ by 1+0(1). In all cases we show that the poles of the best approximants approach ∞ asm→∞.     

On the class of limits of lacunary trigonometric series

Let (n _k ) _k≧1 be a lacunary sequence of positive integers, i.e. a sequence satisfying n _k+1/n _k > q > 1, k ≧ 1, and let f be a “nice” 1-periodic function with ∝ ₀ ¹ f(x) dx = 0. Then the probabilistic behavior of the system (f(n _k x)) _k≧1 is very similar to the behavior of sequences of i.i.d. random variables. For example, Erd?s and Gál proved in 1955 the following law of the iterated logarithm (LIL) for f(x) = cos 2πx and lacunary $ (n_k )_{k \geqq 1} $ : (1) $$ \mathop {\lim \sup }\limits_{N \to \infty } (2N\log \log N)^{1/2} \sum\limits_{k = 1}^N {f(n_k x)} = \left\| f \right\|_2 $$ for almost all x ∈ (0, 1), where ‖f‖₂ = (∝ ₀ ¹ f(x)² dx)^1/2 is the standard deviation of the random variables f(n _k x). If (n _k ) _k≧1 has certain number-theoretic properties (e.g. n _k+1/n _k → ∞), a similar LIL holds for a large class of functions f, and the constant on the right-hand side is always ‖f‖₂. For general lacunary (n _k ) _k≧1 this is not necessarily true: Erd?s and Fortet constructed an example of a trigonometric polynomial f and a lacunary sequence (n _k ) _k≧1, such that the lim sup in the LIL (1) is not equal to ‖f‖₂ and not even a constant a.e. In this paper we show that the class of possible functions on the right-hand side of (1) can be very large: we give an example of a trigonometric polynomial f such that for any function g(x) with sufficiently small Fourier coefficients there exists a lacunary sequence (n _k ) _k≧1 such that (1) holds with √‖f‖ ₂ ² + g(x) instead of ‖f‖₂ on the right-hand side.     

Convolutions of random measures on compact groups

LetG be a compact group andM ₁(G) be the convolution semigroup of all Borel probability measures onG with the weak topology. We consider a stationary sequence {μ _n } _n=?∞ ^+∞ of random measures μ _n =μ _n (ω) inM ₁(G) and the convolutions $$v_{m,n} (\omega ) = \mu _m (\omega )* \cdots *\mu _{n - 1} (\omega ), m< n$$ and $$\alpha _n^{( + k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n,n + i} (\omega ),} \alpha _n^{( - k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n - i,n} (\omega )} $$ We describe the setsA _m ⁺ (ω) andA _n ⁺ (ω) of all limit points ofv _m,n(ω) asm→?∞ orn→+∞ and the setA _∞(ω) of its two-sided limit points for typical realizations of {μ _n (ω)} _n=?∞ ^+∞ . Using an appropriate random ergodic theorem we study the limit random measures ρ _n ^(±) (ω)=lim _k→∞ α _n ^(±k) (ω).     

Atomic decomposition characterizations of weighted multiparameter Hardy spaces

Let w ?? A _??. In this paper, we introduce weighted-(p, q) atomic Hardy spaces H _w ^p,q (? ⁿ ×? ^m ) for 0 < p ? 1, q >q _w and show that the weighted Hardy space H _w ^p (? ⁿ × ? ^m ) defined via Littlewood-Paley square functions coincides with H _w ^p,q (? ⁿ × ? ^m ) for 0 < p ? 1, q > q _w . As applications, we get a general principle on the H _w ^p (? ⁿ × ? ^m ) to L _w ^p (? ⁿ ×? ^m ) boundedness and a boundedness criterion for two parameter singular integrals on the weighted Hardy spaces.     

Convergence of the Vallée-Poussin means for Fourier-Jacobi sums

Letf εC[?1, 1], ?1<α,β≤0, let $f \in C[ - 1, 1], - 1< \alpha , \beta \leqslant 0$ , letS _n ^{α, β} (f, x) be a partial Fourier-Jacobi sum of ordern, and let $$\nu _{m, n}^{\alpha , \beta } = \nu _{m, n}^{\alpha , \beta } (f) = \nu _{m, n}^{\alpha , \beta } (f,x) = \frac{1}{{n + 1}}[S_m^{\alpha ,\beta } (f,x) + ... + S_{m + n}^{\alpha ,\beta } (f,x)]$$ be the Vallée-Poussin means for Fourier-Jacobi sums. It was proved that if 0<a≤m/n≤b, then there exists a constantc=c(α, β, a, b) such that ‖ν _{m, n} ^{α, β} ‖ ≤c, where ‖ν _{m, n} ^{α, β} ‖ is the norm of the operator ν _{m, n} ^{α, β} inC[?1,1].     

A method for summing Fourier integrals of functions from H p (E 2n + ), 0 < p < ∞

Suppose that H ^p (E _2n ⁺ ) is the Hardy space for the first octant $$E_{2n}^ + = \{ z \in \mathbb{C}^n :\operatorname{Im} z_j > 0, j = 1, \ldots ,n\} $$ and P _? ^l (f, x), l > 0, is the generalized Abel-Poisson means of a function f ? H ^p (E _2n ⁺ ). In this paper, we prove the inequalities $$C_1 (l,p)\widetilde\omega _l (\varepsilon ,f)_p \leqslant \left\| {f(x) - P_\varepsilon ^l (f,x)} \right\|_p \leqslant C_2 (l,p)\omega _l (\varepsilon ,f)_p ,$$ where $\widetilde\omega _l (\varepsilon ,f)_p $ and ω _l (?, f) _p are the integral moduli of continuity of lth order. For n = 1 and an integer l, this result was obtained by Soljanik.     

Comparison Inequalities for One Sided Normal Probabilities

Several sharp upper and lower bounds for the ratio of two normal probabilities $\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(1)}_i\leq \mu_i\bigr\}\Biggr)\Big/\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(0)}_i\leq \mu_i\bigr\}\Biggr)$ are given in this paper for various cases, where (ξ ₁ ⁽⁰⁾ ,ξ ₂ ⁽⁰⁾ ,…,ξ _n ⁽⁰⁾ ) and (ξ ₁ ⁽¹⁾ ,ξ ₂ ⁽¹⁾ , …,ξ _n ⁽¹⁾ ) are standard normal random variables with covariance matrices R ⁰=(r _ij ⁰ ) and R ¹=(r _ij ¹ ), respectively.     

ON possible Turán densities

The Turán density π(F) of a family F of k-graphs is the limit as n → ∞ of the maximum edge density of an F-free k-graph on n vertices. Let Π _∞ ^(k) consist of all possible Turán densities and let Π _fin ^(k) ? Π _∞ ^(k) be the set of Turán densities of finite k-graph families. Here we prove that Π _fin ^(k) contains every density obtained from an arbitrary finite construction by optimally blowing it up and using recursion inside the specified set of parts. As an application, we show that Π _fin ^(k) contains an irrational number for each k ≥ 3. Also, we show that Π _∞ ^(k) has cardinality of the continuum. In particular, Π _∞ ^(k) ≠ Π _fin ^(k) .     

The push-out space of immersed spheres

Let f: M ^m → ? ^m+1 be an immersion of an orientable m-dimensional connected smooth manifold M without boundary and assume that ξ is a unit normal field for f. For a real number t the map f _tξ: M ^m → ? ^m+1 is defined as f _tξ(p) = f(p) + tξ(p). It is known that if f _tξ is an immersion, then for each p ∈ M the number of the focal points on the line segment joining f(p) to f _tξ(p) is a constant integer. This constant integer is called the index of the parallel immersion f _tξ and clearly the index lies between 0 and m. In case f: $\mathbb{S}^m \to \mathbb{R}^{m + 1} $ is an immersion, we study the presence of a component of index μ in the push-out space Ω(f). If there exists a component with index μ = m in Ω(f) then f is known to be a strictly convex embedding of $\mathbb{S}^m $ . We reveal the structure of Ω(f) when $f(\mathbb{S}^m )$ is convex and nonconvex. We also show that the presence of a component of index μ in Ω(f) enables us to construct a continuous field of tangent planes of dimension μ on $\mathbb{S}^m $ and so we see that for certain values of μ there does not exist a component of index μ in Ω(f).     

Polynomial approximation inL p(S) forp>0

For a simple polytopeS inR ^d andp>0 we show that the best polynomial approximationE _n(f)_p≡E_n(f)L_p(S) satisfies $$E_n \left( f \right)_p \leqslant C\omega _S^r \left( {f,\frac{1}{n}} \right)p,$$ where ω _S ^r is a measure of smoothness off. This result is the best possible in the sense that a weak-type converse inequality is shown and a realization of ω _S ^r (f,t)_p via polynomial approximation is proved.     

Expansions in Series of Suitable Systems of Functions

The paper is devoted to the study of expansions of functions ?, which are holomorphic in a circle Ω around 0, in a series of the form where the given system of functions (f_k) _k=0 ^? has the property that each function f_k has a representation as a power series of the form with a ₀ ^(k) = 1. It is the question if and where an expansion of the form (*) holds. We present sufficient conditions for the sequence (a _n ^(k) ) _k,n=0 ^∞ , under which there exists a subset Ω₀ of Ω such that for arbitrary ? the expansion (*) holds on Ω₀ and the series converges absolutely uniformly on compact subsets of Ω₀. The obtained results can be applied to prove expansions of analytic functions in a series of a suitable system of m-fold products of Bessel functions. Expansions of this type have been treated by many authors in special cases, but we are able to state a general expansion theorem.     

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 $m^{ - \tfrac{2} {7}}$ .