Quantitative aspect of correction theorems. II

Assume that 0<ε≤1, F ∈ C( ), E={≠0}, δ>0. Then there exists a function G with uniformly convergent Fourier series such that |G|+|F−G|≤(1+δ)|F|, m{F≠G}≤εmE, and . Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 83–91.     

Non-commutative Clarkson Inequalities for n-Tuples of Operators

Let A ₀, ... , A _n−1 be operators on a separable complex Hilbert space , and let α₀,..., α _n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,

Carleson measures and the heat equation

Let { }, where { } is the open unit disk on the complex plane { }. In G, we consider analytic solutions u(t, z) ({ }, { }) of the heat equation 2u_t=u_zz with initial data f(z)=u(0, z) belonging to the Fock space F, i.e., to the space of entire functions square summable with the weight e−|z|2.Conditions on a nonnegative measure μ on G are described under which for all f ∈ F we have { } Bibliography: 17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 146–155. Translated by S. V. Kislyakov.     

On the decay rate of (p, A)-lacunary series

A power series with radius of convergence equal 1 is called a (p,A)-lacunary one if n_k ≥ Ak^p, A > 0, 1 < p < ∞. It is proved that if 1 < p < 2 and f(x) is a (p,A)-lacunary series that satisfies the condition

A remark on the coefficients of bounded polynomials

Let be such that |p(e^iq)|≤1 for ϕ∈R and |p(1)|=a∈[0,1]. An inequality of Dewan and Govil for the sum |a_v|+|a_n|, 0≤u<v≤n is sharpened.     

Compactification of supports of energy weak solutions to quasilinear parabolic equations of nonstationary filtration type with strong convection

The paper deals with localization properties of solutions to the Cauchy problem with the initial data u₀(x) ∈ L₂(ℝ_n) for a wide class of equations in the divergence form. This class contains, e.g., the following equation:

Some sharp estimates for the second continuity modulus for periodic functions and for functions extended from a segment

Let C be the space of 2π-periodic continuous real functions with the uniform norm, let H_n be the set of trigonometric polynomials of order not more than n, let ω₂(f) be the second continuity modulus for a function f∈C, and let T_n(f) be the best approximation polynomial of order n for f∈C. Set ; U:C→C; . In this paper, for h sufficiently large we find the values C(U,h) for some positive operators U. For example, C(A₀,h) and C(T₀,h) are found. For n=1,2,3 we find the values for some linear positive operators U:C→H_n. We establish relations between C(T₀,h) and exact constants in the inequality ω₂(f,h₁)≤C(h₁;h)ω₂(f,h) for some h and h₁ such that 0<h<h₁≤π. For a seminorm P invariant with respect to the shift and majorized by the uniform norm, analogs of C(U,h) are estimated from above. We investigate the problem of extension of a function defined on a segment with preservation of the second continuity modulus. The relation

On Sikkema-Kantorovič polynomials of order K

Let a_n≥0 and F(u)∈C [0,1], Sikkema constructed polynomials: , ifα _n ≡0, then B_n (0, F, x) are Bernstein polynomials. Let , we constructe new polynomials in this paper: Q _n ^(k) (α _n ,f(t))=d ^k /dx ^k B _n+k (α _n ,F _k (u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα _n ≡0, k=1, then Q_n ⁽¹⁾ (0, f(t), x) are Kantorovič polynomials P_n(f). Ifα _n =0, k=2, then Q_n ⁽²⁾, (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is: Theorem 2. Let 1≤p≤∞, in order that for every f∈L^P [0, 1], , it is sufficient and necessary that , § 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define^{[1–2],[8–10]}: . As usual, for the space L^p [a,b](1≤p<∞), we have and L[a, b]=l¹[a, b]. Letα _n ⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials ^{[3] [4]}. The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports.     

Strong monotonicity of operator functions

LetA, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspace such that is invariant forA andB, and . We will use this to show strong monotonicity or strong convexity of operator functions. We will see that when 0≤A≤B, andB−A is of finite rank,A ^t ≤B ^t for somet>1 if and only if the null space ofB−A is invariant forA.     

Invariants of classCk for finite coxeter groups and their representations in terms of anisotropic spaces

This article is devoted to the study of representations of C^k { } functions f invariant with respect to finite Coxeter groups W in the form f=F op, where p is a base in the algebra of W-invariant polynomials. We examine the lowering of smoothness of F as compared with f and conclude that this lowering has an anisotropic nature and that, more precisely, at each point Po it is described by a vector { }. We examine the cases W=A_n, B_n, D_n, { }; in each case the greatest component { } of { } is equal to the Coxeter number of the stabilizer W_yo of the point yo, where po=p(yo). Bibliography: 22 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 46–70. Translated by S. Yu. Pilyugin.     

Remainder estimates for the approximation numbers of weighted Hardy operators acting onL 2

We consider the weighted Hardy integral operatorT:L ²(a, b) →L ²(a, b), −∞≤a<b≤∞, defined by . In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa _n(T) ofT. In this paper, we show that under suitable conditions onu andv, where ∥w∥_p=(∫ _a ^b |w(t)|^p dt)^1/p. Research supported by NSERC, grant A4021. Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic.     

Some inequalities for polynomials satisfying p(z)≡znp(1/z)

Let be a polynomial degreen and let . Then according to Bernstein’s inequality ‖p’‖≤n‖p‖. It is a well known open problem to obtain inequality analogous to Bernstein’s inequality for the class II_n of polynomials satisfying p(z)≡zⁿp(1/z). Here we obtain an inequality analogous to Bernstein’s inequality for a subclass of II_n. Our results include several of the known results as special cases.     

Stick breaking process generated by virtual permutations with Ewens distribution

Given a sequence x of points in the unit interval, we associate with it a virtual permutation w=w(x) (that is, a sequence w of permutationsw _n such that for all n=1,2,..., w_n−1=w′_n is obtained from w_n by removing the last element n from its cycle). We introduce a detailed version of the well-known stick breaking process generating a random sequence x. It is proved that the associated random virtual permutation w(x) has a Ewens distribution. Up to subsets of zero measure, the space of virtual permutations is identified with the cube [0, 1]^∞. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 162–180.     

Lower semicontinuity of some functionals under PDE constraints: An \mathcal{A}-quasiconvex pair

The problem of establishing necessary and sufficient conditions for l.s.c. under PDE constraints is studied for a special class of functionals:

Regolarità hölderiana delle derivate dei minimi di funzionali con parte principale quadratica

Riassunto In questo lavoro si prova la regolarità h?lderiana delle derivate, fino all'ordinek, dei minimi locali dei funzionali sotto opportune ipotesi suA _ij ^αβ e sug.

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Summary In this paper we prove h?lder-continuity of the derivates, up to orderk, of local minima of functionals under suitable hypotheses forA _ij ^αβ andg.

     

On exact constant in a one-dimensional embedding theorem

In the case 1≤p<q≤∞, the question on the exact constant in the embedding of the space W _p ¹ (0,1) into the space L_q(0,1) is studied, i.e.,

Asymptotic normality of a quadratic measure of orthogonal series type density estimate

LetX ₁,...,X _n be i.i.d. random variable with a common densityf. Let be an estimate off(x) based on a complete orthonormal basis {φ _k :k≧0} ofL ₂[a, b]. A Martingale central limit theorem is used to show that , where and .     

On the zeros of a class of generalised Dirichlet series-XIV

We prove a general theorem on the zeros of a class of generalised Dirichlet series. We quote the following results as samples. Theorem A.Let 0<θ<1/2and let {a _n }be a sequence of complex numbers satisfying the inequality for N = 1,2,3,…,also for n = 1,2,3,…let α _n be real and |α_n| ≤ C(θ)where C(θ) > 0is a certain (small)constant depending only on θ. Then the number of zeros of the function in the rectangle (1/2-δ⩽σ⩽1/2+δ,T⩽t⩽2T) (where 0<δ<1/2)is ≥C(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided T ≥T ₀(θ,δ)a large positive constant. Theorem B.In the above theorem we can relax the condition on a _n to and |a_N| ≤ (1/2-θ)^-1.Then the lower bound for the number of zeros in (σ⩾1/3−δ,T⩽t⩽2T)is > C(θ,δ) Tlog T(log logT)^-1.The upper bound for the number of zeros in σ⩾1/3+δ,T⩽t⩽2T) isO(T)provided for every ε > 0. Dedicated to the memory of Professor K G Ramanathan