A note on uniqueness problem for meromorphic mappings with 2N + 3 hyperplanes

In this paper,a uniqueness theorem for meromorphic mappings partially sharing 2N+3 hyperplanes is proved.For a meromorphic mapping f and a hyperplane H,set E(H,f) = {z|ν(f,H)(z) 0}.Let f and g be two linearly non-degenerate meromorphic mappings and {Hj}j2=N1+ 3be 2N + 3 hyperplanes in general position such that dim f-1(Hi) ∩ f-1(Hj) n-2 for i = j.Assume that E(Hj,f) E(Hj,g) for each j with 1 j 2N +3 and f = g on j2=N1+ 3f-1(Hj).If liminfr→+∞ 2j=N1+ 3N(1f,Hj)(r) j2=N1+ 3N(1g,Hj)(r) NN+1,then f ≡ g.     

An extremal problem for algebraic polynomials in the symmetric discrete Gegenbauer-Sobolev space

We study discrete Sobolev spaces with symmetric inner product $$\left\langle {f,g} \right\rangle _\alpha = \int_{ - 1}^1 {f g d\mu _\alpha } + M[f(1)g(1) + f( - 1)g( - 1)] + K[f'(1)g'(1) + f'( - 1)g'( - 1)]$$ , where M ≥ 0, k ≥ 0, and $$d\mu _\alpha (x) = \frac{{\Gamma (2\alpha + 2)}}{{2^{2\alpha + 1} \Gamma ^2 (\alpha + 1)}}(1 - x^2 )^\alpha dx, \alpha > - 1$$ , is the Gegenbauer probability measure. We obtain the solution of the following extremal problem: Calculate $$\mathop {\inf }\limits_{a_0 ,a_1 ,...,a_{N - r} } \left\{ {\langle P_N^{(r)} ,P_N^{(r)} \rangle _\alpha ,1 \leqslant r \leqslant N - 1, P_N^{(r)} (x) = \sum\limits_{j = N - r + 1}^N {a_j^0 x^j } + \sum\limits_{j = 0}^{N - r} {a_j x^j } } \right\}$$ , where the a _j ⁰ , j = N ? r + 1, N ? r + 2, ..., N ? 1, N, a _N ⁰ > 0, are fixed numbers, and find the extremal polynomial.     

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.     

Bounds for the spectral abscissa of an element in a Banach algebra

For an arbitrary element x with spectrum sp(x) in a Banach algebra with identity e ≠ 0 we define the upper (lower) spectral abscissa \(\mathop {\sigma + (x)}\limits_{( - )} = \mathop {\max }\limits_{(\min )} \operatorname{Re} \lambda ,\lambda \in sp(x)\) . With the aid of the spectral radius \(\rho (x) = \mathop {\max }\limits_{\lambda \in sp(x)} \left| \lambda \right| = \mathop {\lim }\limits_{n \to + \infty } \parallel x^n {{1 - } \mathord{\left/ {\vphantom {{1 - } n}} \right. \kern-0em} n}\) we prove the following bounds: γ_?(x)?σ_?(x)?Γ_?(x)?₊(x)?σ₊(x)?γ₊(x), Γ_(±)(x)=(2δ_(±))^?1 (ρ _δ ² )_(±)?δ _(±) ² ?ρ ₀ ² )(δ_(±)≠0), γ_(±)(x)= (±)ρ_δ(±)?δ_(±), δ₊?0, δ_??0 ρ _(±) ^δ = ρ(x+eδ_(±)). We mention a case where equality is achieved, some corollaries,and discuss the sharpness of the bounds: for every ? > 0 there is a δ: ¦δ¦ ≥ρ ₀ ² /2?, such that Δ: = ¦γ_(±) x?Γ_(±) x¦?ε and conversely, if the bounds are computed for some δ ≠ 0, then △ ≤ρ ₀ ² /2 ¦δ¦. An example is considered.     

Ranges of certain functionals in classes of regular functions

LetP _κ,n (λ,β) be the class of functions \(g(z) = 1 + \sum\nolimits_{v = n}^\infty {c_\gamma z^v }\) , regular in ¦z¦<1 and satisfying the condition $$\int_0^{2\pi } {\left| {\operatorname{Re} \left[ {e^{i\lambda } g(z) - \beta \cos \lambda } \right]} \right|} /\left( {1 - \beta } \right)\cos \lambda \left| {d\theta \leqslant \kappa \pi ,} \right.z = re^{i\theta } ,$$ , 0 < r < 1 (κ?2,n?1, 0?Β<1, -π<λ<π/2;M _κ,n (λ,β,α),n?2, is the class of functions \(f(z) = z + \sum\nolimits_{v = n}^\infty {a_v z^v }\) , regular in¦z¦<1 and such thatF _α(z)∈P _κ,n?1(λ,β), where \(F_\alpha (z) = (1 - \alpha )\frac{{zf'(z)}}{{f(z)}} + \alpha (1 + \frac{{zf'(z)}}{{f'(z)}})\) (0?α?1). Onr considers the problem regarding the range of the system {g ^(v?1)(z_?)/(v?1)!}, ?=1,2,...,m,v=1,2,...,N _?, on the classP _κ,1(λ,β). On the classesP _κ,n (λ,β),M _κ,n (λ,β,α) one finds the ranges of C_v, v?n, a_m, n?m?2n-2, and ofg(?),F _?(?), 0<¦ξ¦<1, ξ is fixed.     

Unitary SK1 of semiramified graded and valued division algebras

We prove formulas for SK₁(E, τ), which is the unitary SK₁ for a graded division algebra E finite-dimensional and semiramified over its center T with respect to a unitary involution τ on E. Every such formula yields a corresponding formula for SK₁(D, ρ) where D is a division algebra tame and semiramified over a Henselian valued field and ρ is a unitary involution on D. For example, it is shown that if ${\sf{E} \sim \sf{I}_0 \otimes_{\sf{T}_0}\sf{N}}$ where I ₀ is a central simple T ₀-algebra split by N ₀ and N is decomposably semiramified with ${\sf{N}_0 \cong L_1\otimes_{\sf{T}_0} L_2}$ with L ₁, L ₂ fields each cyclic Galois over T ₀, then $${\rm SK}_1(\sf{E}, \tau) \,\cong\ {\rm Br}(({L_1}\otimes_{\sf{T}_0} {L_2})/\sf{T}_0;\sf{T}_0^\tau)\big/ \left[{\rm Br}({L_1}/\sf{T}_0;\sf{T}_0^\tau)\cdot {\rm Br}({L_2}/\sf{T}_0;\sf{T}_0^\tau) \cdot \langle[\sf{I}_0]\rangle\right].$$      

Bulk universality for generalized Wigner matrices

Consider N × N Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure ν _ij with a subexponential decay. Let ${\sigma_{ij}^2}$ be the variance for the probability measure ν _ij with the normalization property that ${\sum_{i} \sigma^2_{ij} = 1}$ for all j. Under essentially the only condition that ${c\le N \sigma_{ij}^2 \le c^{-1}}$ for some constant c?>?0, we prove that, in the limit N → ∞, the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale M ^?1.     

Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

Suppose that X={X _t :t≥0} is a supercritical super Ornstein-Uhlenbeck process, that is, a superprocess with an Ornstein-Uhlenbeck process on $\mathbb{R}^{d}$ corresponding to $L=\frac{1}{2}\sigma^{2}\Delta-b x\cdot\nabla$ as its underlying spatial motion and with branching mechanism ψ(λ)=?αλ+βλ ²+∫_(0,+∞)(e ^?λx ?1+λx)n(dx), where α=?ψ′(0+)>0, β≥0, and n is a measure on (0,∞) such that ∫_(0,+∞) x ² n(dx)<+∞. Let $\mathbb{P} _{\mu}$ be the law of X with initial measure μ. Then the process W _t =e ^?αt ∥X _t ∥ is a positive $\mathbb{P} _{\mu}$ -martingale. Therefore there is W _∞ such that W _t →W _∞, $\mathbb{P} _{\mu}$ -a.s. as t→∞. In this paper we establish some spatial central limit theorems for X. Let $\mathcal{P}$ denote the function class $$ \mathcal{P}:=\bigl\{f\in C\bigl(\mathbb{R}^d\bigr): \mbox{there exists } k\in\mathbb{N} \mbox{ such that }|f(x)|/\|x\|^k\to 0 \mbox{ as }\|x\|\to\infty \bigr\}. $$ For each $f\in\mathcal{P}$ we define an integer γ(f) in term of the spectral decomposition of f. In the small branching rate case α<2γ(f)b, we prove that there is constant $\sigma_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, $$\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_1(f)\bigr), \quad t\to\infty, \end{aligned}$$ where W ^? has the same distribution as W _∞ conditioned on no-extinction and $G_{1}(f)\sim \mathcal{N}(0,\sigma_{f}^{2})$ . Moreover, W ^? and G ₁(f) are independent. In the critical rate case α=2γ(f)b, we prove that there is constant $\rho_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, $$\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{t^{1/2}\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_2(f)\bigr), \quad t\to\infty, \end{aligned}$$ where W ^? has the same distribution as W _∞ conditioned on no-extinction and $G_{2}(f)\sim \mathcal{N}(0, \rho_{f}^{2})$ . Moreover W ^? and G ₂(f) are independent. We also establish two central limit theorems in the large branching rate case α>2γ(f)b. Our central limit theorems in the small and critical branching rate cases sharpen the corresponding results in the recent preprint of Mi?o? in that our limit normal random variables are non-degenerate. Our central limit theorems in the large branching rate case have no counterparts in the recent preprint of Mi?o?. The main ideas for proving the central limit theorems are inspired by the arguments in K. Athreya’s 3 papers on central limit theorems for continuous time multi-type branching processes published in the late 1960’s and early 1970’s.     

Asymptotics for the minimization of a Ginzburg-Landau functional

LetΩ ? ?² be a smooth bounded simply connected domain. Consider the functional $$E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| {\nabla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 } $$ on the classH _g ¹ ={u εH ¹(Ω; ?);u=g on ?Ω} whereg:?Ω? → ? is a prescribed smooth map with ¦g¦=1 on ?Ω? and deg(g, ?Ω)=0. Let uu _ε be a minimizer for E_ε onH _g ¹ . We prove that u_ε → u₀ in \(C^{1,\alpha } (\bar \Omega )\) as ε → 0, where u₀ is identified. Moreover \(\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2 \) .     

The completeness of systems of functions of the Mittag-Leffler type for weighted uniform approximation in a complex domain

For a givenρ(1/2 <ρ < + ∞) let us set L _ρ = {z: |arg z| = π/(2ρ)} and assume that a real valued measurable function ?(t) such that ?(t) ≥ 1(t ∈ L _ρ ) and \(\mathop {\lim }\limits_{|t| \to + \infty } \varphi (t) = + \infty (t \in L_\rho )\) is defined on L _ρ . Let C _? (L _ρ ) denote the space of continuous functionsf(t) on L _ρ such that \(\lim \tfrac{{f(t)}}{{\varphi (t)}} = 0\) , where the norm of an elementf is defined as: \(\parallel f\parallel = \mathop {\sup }\limits_{t \in L_\rho } \tfrac{{|f(t)|}}{{\varphi (t)}}\) . In this note we pose the question about the completeness of the system of functions of the Mittag-Leffler type {Eρ(ut; μ)} (μ ≥ 1, 0 ≤ u ≤a) or, what is the same thing, of the system of functions \(p(t) = \int_0^a {E_\rho (ut;\mu )d\sigma (u)} \) in C _? (L _ρ ). The following theorem is proved: The system of functions of the Mittag-Leffler type is complete in C _? (L _ρ ) if and only if sup |p(z)| ≡ +∞, z ∈ L _ρ , where the supremum is taken over the set of functions p(t) such that ∥p(t) (t + 1)^?1 ∥ ≤ 1.     

Padé tables of entire functions of very slow and smooth growth,II

Letf(z):=Σ _j=0 ^∞ a _j z ^j , where a_j ≠ 0,j large enough, and for someq ε C such that ¦q¦ $$q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q,j \to \infty .$$ Define for m,n = 0,1,2,..., the Toeplitz determinant $$D(m/n): = \det (a_{m - j + k} )_{j,k = 1}^n .$$ Given ? > 0, we show that form large enough, and for everyn = 1,2,3,..., $$(1 - \varepsilon )^n \leqslant \left| {{{D(m/n)} \mathord{\left/ {\vphantom {{D(m/n)} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right| \leqslant (1 + \varepsilon )^n .$$ We apply this to show that any sequence of Padé approximants {[m _k /n _k ]} ₁ ^∞ tof, withm _k →∞ ask→ ∞, converges locally uniformly in C. In particular, the diagonal sequence {[n/n]} ₁ ^∞ converges throughout C. Further, under additional assumptions, we give sharper asymptotics forD(m/n).     

Remark on an infinite semipositone problem with indefinite weight and falling zeros

In this work, we consider the positive solutions to the singular problem $$ \left\{\begin{array}{ll} -\Delta u = am(x)u-f(u) - \dfrac{c}{u^{\alpha}} & {\rm in}\;\Omega,\\ u=0 & {\rm on}\; \partial\Omega, \end{array} \right. $$ where 0?<?α?<?1,a?>?0 and c?>?0 are constants, Ω is a bounded domain with smooth boundary $\partial\Omega$ , Δ is a Laplacian operator, and $f:[0,\infty] \longrightarrow{\mathbb R}$ is a continuous function. The weight functions m(x) satisfies m(x)?∈?C(Ω) and m(x)?>?m ₀?>?0 for x?∈?Ω and also ||m||_?∞??=?l?<?∞. We assume that there exist A?>?0, M?>?0, p?>?1 such that alu???M?≤?f(u)?≤?Au ^p for all u?∈?[0,?∞?). We prove the existence of a positive solution via the method of sub-supersolutions when $m_{0}a>\frac{2\lambda_{1} }{1+\alpha}$ and c is small. Here λ ₁ is the first eigenvalue of operator ??Δ with Dirichlet boundary conditions.     

On the distribution of integer random variables related by a certain linear inequality: III

We consider tuples {N _jk }, j = 1, 2, ..., k = 1, ..., q _j , of nonnegative integers such that $$ \sum\limits_{j = 1}^\infty {\sum\limits_{k = 1}^{q_j } {jN_{jk} } } \leqslant M. $$ Assuming that q _j ～ j ^d?1, 1 < d < 2, we study how the probabilities of deviations of the sums $ \sum\nolimits_{j = j_1 }^{j_2 } {\sum\nolimits_{k = 1}^{q_j } {N_{jk} } } $ N _jk from the corresponding integrals of the Bose-Einstein distribution depend on the choice of the interval [j ₁,j ₂].     

Two Parameter Asymptotic Spectra in the Uniformly Elliptic Case

In this article we study the abstract two parameter eigenvalue problem $$\begin{gathered} T_1 u_1 = \left( {\lambda _1 V_{11} + \lambda _2 V_{12} } \right)u_1 , \left\| {u_1 } \right\| = 1 \hfill \\ T_2 u_2 = \left( {\lambda _1 V_{21} + \lambda _2 V_{22} } \right)u_2 , \left\| {u_2 } \right\| = 1 \hfill \\ \end{gathered}$$ where, in the Hilbert spaces H_j, T_j is self-adjoint, bounded below and has compact resolvent, and V_jk are self-adjoint bounded operators, (?1)^j+kV_jk >> 0, j, k = 1, 2. An eigenvalue λ for this problem is a point in R² satisfying both equations. Under appropriate conditions, the eigenvalues λⁿ = (λ₁ ⁿ, λ₂ ⁿ) are countable and in R². We aim to describe the set of limit points of λⁿ/∥λⁿ∥, as ∥λⁿ∥ → ∞, in terms of the V_jk.     

The “Full Müntz Theorem” inL p[0, 1] for 0<p<∞

Denote by span {f ₁,f ₂, …} the collection of all finite linear combinations of the functionsf ₁,f ₂, … over ?. The principal result of the paper is the following. Theorem (Full Müntz Theorem in L_p(A) for p ∈ (0, ∞) and for compact sets A ? [0, 1] with positive lower density at 0). Let A ? [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ _j ) _j=1 ^∞ is a sequence of distinct real numbers greater than ?(1/p). Then span {x ^λ1,x ^λ2,…} is dense in L_p(A) if and only if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ . Moreover, if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ , then every function from the L_p(A) closure of {x ^λ1,x ^λ2,…} can be represented as an analytic function on {z ∈ ? \ (?∞,0] : |z| < r_A} restricted to A ∩ (0, r_A) where $r_A : = \sup \left\{ {y \in \mathbb{R}:\backslash ( - \infty ,0]:\left| z \right|< r_A } \right\}$ (m(·) denotes the one-dimensional Lebesgue measure). This improves and extends earlier results of Müntz, Szász, Clarkson, Erdös, P. Borwein, Erdélyi, and Operstein. Related issues about the denseness of {x ^λ1,x ^λ2,…} are also considered.     

Оценки отклонений оп тимальных сеток вL p -норме и теория квад ратурных формул

Let E^s=[0, 1]^s be then-dimensional unit cube, 1<p<∞, anda=(a ₁, ...,a _s ) some set of natural numbers. Denote byL _p ^(a) , (E ^s ) the class of functionsf: E ^s → C for which $$\left\| {\frac{{\partial ^{b_1 + \cdots + b_s } f}}{{\partial x_1^{b_1 } \cdots \partial x_s^{b_s } }}} \right\|_p \leqslant 1,$$ where $$0< b_1< a_1 , ..., 0< b_s< a_s .$$ Set $$R_p^{\left( a \right)} \left( N \right) = \mathop {\inf }\limits_{card \mathfrak{S} = N} R_\mathfrak{S} \left( {L_p^{\left( a \right)} \left( {E^s } \right)} \right),$$ where $R_\mathfrak{S} \left( {L_p^{\left( a \right)} \left( {E^s } \right)} \right)$ is the error of the quadrature formulas on the mesh $\mathfrak{S}$ (for the classL _p ^(a) (E ^s )), consisting of N nodes and weights, and the infimum is taken with respect to all possibleN nodes and weights. In this paper, the two-sided estimate $$\frac{{\left( {\log N} \right)^{{{\left( {l - 1} \right)} \mathord{\left/ {\vphantom {{\left( {l - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{N^d }} \ll _{p, a} R^{\left( a \right)} \left( N \right) \ll _{p, a} \frac{{\left( {\log N} \right)^{{{\left( {l - 1} \right)} \mathord{\left/ {\vphantom {{\left( {l - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{N^d }}$$ is proved for every natural numberN > 1, whered=min{a ₁, ...,a _s }, whilel is the number of those components of a which coincide withd. An analogous result is proved for theL _p -norm of the deviation of meshes.     

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→∞.     

A limit theorem for expectations conditional on a sum

Let ξ₁, ξ₂, ξ₃,... be a sequence of independent random variables, such that μ _j ?E[ξ _j ], 0<α?Var[ξ _j ] andE[|ξ _j ?μ _j |^2+δ] for some δ, 0<δ?1, and everyj?1. IfU and ξ₀ are two random variables such thatE[ξ ₀ ² ]<∞ andE[|U|ξ ₀ ² ]<∞, and the vector 〈U,ξ〉 is independent of the sequence {ξ _j :j?1}, then under appropriate regularity conditions $$E\left[ {U\left| {\xi _0 + S_n } \right. = \sum\limits_{j = 1}^n {\mu _j + c_n } } \right] = E[U] + O\left( {\frac{1}{{s_n^{1 + \delta } }}} \right) + O\left( {\frac{{|c_n |}}{{s_n^2 }}} \right)$$ whereS _n ?ξ₁+ξ₂+?+ξ _n ,μ _j ?E[ξ _j ],s _n ² ?Var[S _n ], andc _n =O(s _n ).     

Scanning control of a vibrating string

We construct scanning feedback controls {γ _i (t)} for the vibrating string equation $$\begin{gathered} y_{tt} (x,t) = y_{xx} (x,t) + Ry(x,t) + \sum\limits_{i = 1}^N {\phi (x - \gamma _i } (t))y(x,t), \hfill \\ 0< x< 1,y = 0 at x = 0,1. \hfill \\ \end{gathered} $$ so that (y, y _t ) → (0,0) ast → ∞ in the weak topology ofH ₀ ¹ (0,1) ×L ₂ (0,1). In particular we show that ifφ is an even polynomial of degreeN with nonpositive coefficients that forR <π ² we can find such stabilizingγ _i (t), i=1,?,N.     

Equiconvergence of spectral decompositions of Hill operators

We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator L = ?d ²/dx ² + v(x), x ∈ L ¹([0, π], with H _per ^?1 -potential and the free operator L ⁰ = ?d ²/dx ², subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that $\left\| {S_N - S_N^0 :L^a \to L^b } \right\| \to 0if1 < a \leqslant b < \infty ,1/a - 1/b < 1/2,$ , where S _N and S _N ⁰ are the N-th partial sums of the spectral decompositions of L and L ⁰. Moreover, if v ∈ H ^?α with 1/2 < α < 1 and $\frac{1} {a} = \frac{3} {2} - \alpha $ , then we obtain the uniform equiconvergence ‖S _N ?S _N ⁰ : L ^a → L ^∞‖ → 0 as N → ∞.