Zeros of analytic functions and norms of inverse matrices

Letk _n be the smallest constant such that for anyn-dimensional normed spaceX and any invertible linear operatorT ∈L(X) we have $|\det (T)| \cdot ||T^{ - 1} || \le k_n |||T|^{n - 1} $ . LetA ₊ be the Banach space of all analytic functionsf(z)=Σ _k≥0 a _kz^k on the unit diskD with absolutely convergent Taylor series, and let ‖f‖A ₊=σ _k≥0 |ακ|; define ? _n on $\overline D ^n $ by $ \begin{array}{l} \varphi _n \left( {\lambda _1 ,...,\lambda _n } \right) \\ = inf\left\{ {\left\| f \right\|_{A + } - \left| {f\left( 0 \right)} \right|; f\left( z \right) = g\left( z \right)\prod\limits_{i = 1}^n {\left( {\lambda _1 - z} \right), } g \in A_ + , g\left( 0 \right) = 1 } \right\} \\ \end{array} $ . We show thatk _n=sup {? _n(λ₁,…, λ _n ); (λ₁,…, λ _n )∈ $\overline D ^n $ }. Moreover, ifS is the left shift operator on the space ?∞:S(x ₀,x ₁, …,x _p, …)=(x ₁,…,x _p,…) and if J_n(S) denotes the set of allS-invariantn-dimensional subspaces of ?∞ on whichS is invertible, we have $k_n = \sup \{ |\det (S|_E )|||(S|_E )^{ - 1} ||E \in J_n (S)\} $ . J. J. Schäffer (1970) proved thatk _n≤√en and conjectured thatk _n=2, forn≥2. In factk ₃>2 and using the preceding results, we show that, up to a logarithmic factor,k _n is of the order of √n whenn→+∞.     

Chebyshev's inequalities for functions monotone in the mean

LetR be the reals ≥ 0. LetF be the set of mapsf: {1, 2, ?,n} →R. Choosew ∈ F withw _i = w(i) > 0. PutW _i = w₁ + ? + w_i. Givenf ∈ F, define \(\bar f\) ∈F by $$\bar f\left( i \right) = \frac{{\left\{ {w_i f\left( 1 \right) + \ldots + w_i f\left( i \right)} \right\}}}{{W_i }}.$$ Callf mean increasing if \(\bar f\) is increasing. Letf ₁, ?, f_t be mean decreasing andf _t+1,?: f_t+u be mean increasing. Put $$k = W_n^u \min \left\{ {w_i^{u - 1} W_i^{t - u} } \right\}.$$ Then $$k\mathop \sum \limits_{i = 1}^n w_i f_1 \left( i \right) \ldots f_{t + u} \left( i \right) \leqslant \mathop \prod \limits_{j = 1}^{t + u} (\mathop \sum \limits_{i = 1}^n w_i f_1 (i)).$$      

On extensions of polynomial functions

Let X, Y be two linear spaces over the field ? of rationals and let D ≠ ? be a (?—convex subset of X. We show that every function ?: D → Y satisfying the functional equation $${\mathop\sum^{n+1}\limits_{j=0}}(-1)^{n+1-j}\Bigg(^{n+1}_{j}\Bigg)f\Bigg((1-{j\over {n+1}})x+{j\over{n+1}}y\Bigg)=0,\ \ \ x,y\in\ D,$$ admits an extension to a function F: X → Y of the form $$F(x)=A^o+A^1(x)+\cdot\cdot\cdot+A^n(x),\ \ \ x\in\ X,$$ where A ^o ∈ Y, A^k(x) ? A_k(x,…,x), x ∈ X, and the maps A _k: X ^k → Y are k—additive and symmetric, k ∈ {1,…, n}. Uniqueness of the extension is also discussed.     

Gysin maps and cycle classes for Hodge cohomology

Iff:X→Y is a projective morphism between regular varieties over a field, we construct Gysin maps $$f_ * :H^i \left( {X,\Omega _{X/Z}^j } \right) \to H_{f(x)}^{i + d} \left( {X,\Omega _{Y/Z}^j } \right)$$ for the Hodge cohomology groups, whered-dimY-dimX. These Gysin maps have the expected properties, and in particular may be used to construct a cycle class map $$Cl_X :CH^i \left( {X,S} \right) \to H^i \left( {X,\Omega _{X/Z}^i } \right)$$ whereX is quasi-projective over a field,S is the singular locus, andCH ⁱ(X, S) is the relative Chow group of codimension-i cycles modulo rational equivalence. Simple properties of this cycle map easily imply the infinite dimensionality theorem for the Chow group of zero cycles of a normal projective varietyX overC with \(H^n \left( {X,\mathcal{O}_X } \right) \ne 0\) , wheren=dimX. One also recovers examples of Nori of affinen-dimensional varieties which support indecomposable vector bundles of rankn.     

Zeros ofA p functions and related classes of analytic functions

The functionf(z), analytic in the unit disc, is inA ^p if \(\int {\int {_{\left| z \right|< 1} \left| {f(z)} \right|^p dxdy< \infty } } \) . A necessary condition on the moduli of the zeros ofA ^p functions is shown to be best possible. The functionf(z) belongs toB ^p if \(\int {\int {_{\left| z \right|< 1} \log ^ + \left| {f(z)} \right|)^p } } \) . Let {z _n } be the zero set of aB ^p function. A necessary condition on |z _n | is obtained, which, in particular, implies that Σ(1?|z _n |)^1+(1/p)+g <∞ for all ε>0 (p≧1). A condition on the Taylor coefficients off is obtained, which is sufficient for inclusion off inB ^p. This in turn shows that the necessary condition on |z _n | is essentially the best possible. Another consequence is that, forq≧1,p<q, there exists aB ^p zero set which is not aB ^q zero set.     

Convergence of polyharmonic splines on semi-regular grids \mathbb{Z}{{\boldsymbol{\times}} \boldsymbol{a}} {\mathbb{Z}^{\it\boldsymbol{n}}} for {\boldsymbol{a}\rightarrow\mathbf {0}}

Let p, n ∈ ? with 2p ≥ n + 2, and let I _a be a polyharmonic spline of order p on the grid ? × a? ⁿ which satisfies the interpolating conditions $I_{a}\left( j,am\right) =d_{j}\left( am\right) $ for j ∈ ?, m ∈ ? ⁿ where the functions d _j : ? ⁿ → ? and the parameter a > 0 are given. Let $B_{s}\left( \mathbb{R}^{n}\right) $ be the set of all integrable functions f : ? ⁿ → ? such that the integral $$ \left\| f\right\| _{s}:=\int_{\mathbb{R}^{n}}\left| \widehat{f}\left( \xi\right) \right| \left( 1+\left| \xi\right| ^{s}\right) d\xi $$ is finite. The main result states that for given $\mathbb{\sigma}\geq0$ there exists a constant c>0 such that whenever $d_{j}\in B_{2p}\left( \mathbb{R}^{n}\right) \cap C\left( \mathbb{R}^{n}\right) ,$ j ∈ ?, satisfy $\left\| d_{j}\right\| _{2p}\leq D\cdot\left( 1+\left| j\right| ^{\mathbb{\sigma}}\right) $ for all j ∈ ? there exists a polyspline S : ? ⁿ⁺¹ → ? of order p on strips such that $$ \left| S\left( t,y\right) -I_{a}\left( t,y\right) \right| \leq a^{2p-1}c\cdot D\cdot\left( 1+\left| t\right| ^{\mathbb{\sigma}}\right) $$ for all y ∈ ? ⁿ , t ∈ ? and all 0 < a ≤ 1.     

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 $$\gamma _{mn} = \inf \{ {{E_{mn}^C (f)} \mathord{\left/ {\vphantom {{E_{mn}^C (f)} {E_{mn}^R (f)}}} \right. \kern-\nulldelimiterspace} {E_{mn}^R (f)}}:f \in C[ - 1,1]\} = \tfrac{1}{2}.$$      

A sharp weightedL 2-estimate for the solution to the time-dependent Schrödinger equation

For Ξ∈R ⁿ ,t∈R andf∈S(R ⁿ ) define $\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$ . We determine the optimal regularitys ₀ such that $\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$ holds whereC is independent off∈S(R ⁿ ) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11]. Our theorems can be generalized to the case where the exp(it|ξ|²) is replaced by exp(it|ξ|^a),a≠2. The proof uses Parseval's formula onR, orthogonality arguments arising from decomposingL ²(R ⁿ ) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity.     

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.     

A short note on the Frobenius norm of the commutator

This note mainly aims to improve the inequality, proposed by Böttcher and Wenzel, giving the upper bound of the Frobenius norm of the commutator of two particular matrices in ? ^n×n . We first propose a new upper bound on basis of the Böttcher and Wenzel’s inequality. Motivated by the method used, the inequality ‖XY ? XY‖ _F ² ≤ 2‖X‖ _F ² ‖Y‖ _F ² is finally improved into $$ \left\| {XY - YX} \right\|_F^2 \leqslant 2\left\| X \right\|_F^2 \left\| Y \right\|_F^2 - 2[tr(X^T Y)]^2 . $$ . In addition, a further improvement is made.     

Multiobjective programming and penalty functions

This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form $$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$ wheree ?R ^m, with each component equal to unity;f:R ⁿ →R ^m, represents them objective functions {f ⁱ} defined onX \( \subseteq \) R ⁿ; λ ∈R ¹, λ>0;P:R ⁿ →R ¹ X \( \subseteq \) Z \( \subseteq \) R ⁿ,P(x)≦0, ∨x ∈R ⁿ,P(x) = 0 ?x ∈X. The paper studies properties of {E (Z, λ _r )} for a sequence of positive {λ _r } converging to 0 in relationship toE(X), whereE(Z, λ _r ) is the efficient set ofZ with respect tog(·, λ_r) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.     

Estimates of maximal functions measuring local smoothness

Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. Letf ∈L(I ⁿ ) (I≡[0,1]. Set $${\mathcal{N}}_\eta f(x) = \sup \frac{1}{{\left| Q \right|\eta (\left| Q \right|^{1/n} )}} \smallint _Q \left| {f(t) - f(x)} \right|dt,$$ , where the supremum is taken over all cubes containing the pointx. Forη=t ^α (0<α≤1) this definition was given by A.Calderón. In the paper we prove estimates of the maximal functions ${\mathcal{N}}_\eta f$ , along with some embedding theorems. In particular, we prove the following Sobolev type inequality: if $$1 \leqslant p< q< \infty , \theta \equiv n(1/p - 1/q)< 1, and \eta (t) \leqslant t^\theta \sigma (t),$$ , then $$\parallel {\mathcal{N}}_\sigma {f} {\parallel_{q,p}} \leqslant c \parallel {\mathcal{N}}_\eta {f} {\parallel_p} .$$ . Furthermore, we obtain estimates of ${\mathcal{N}}_\eta f$ in terms of theL ^p -modulus of continuity off. We find sharp conditions for ${\mathcal{N}}_\eta f$ to belong toL ^p (I ⁿ ) and the Orlicz class?(L), too.     

Exponential laws for ultrametric partially differentiable functions and applications

We establish exponential laws for certain spaces of differentiable functions over a valued field $\mathbb{K}$ . For example, we show that $$C^{(\alpha ,\beta )} \left( {U \times V,E} \right) \cong C^\alpha \left( {U,C^\beta \left( {V,E} \right)} \right)$$ if α ∈ (?₀ ∪ {∞}) ⁿ , β ∈ (?₀ ∪ {∞}) ^m , $U \subseteq \mathbb{K}^n$ and $V \subseteq \mathbb{K}^m$ are open (or suitable more general) subsets, and E is a topological vector space. As a first application, we study the density of locally polynomial functions in spaces of partially differentiable functions over an ultrametric field (thus solving an open problem by Enno Nagel), and also global approximations by polynomial functions. As a second application, we obtain a new proof for the characterization of C ^r -functions on (? _p ) ⁿ in terms of the decay of their Mahler expansions. In both applications, the exponential laws enable simple inductive proofs via a reduction to the one-dimensional, vector-valued case.     

Simultaneous approximation by algebraic polynomials

Some estimates for simultaneous polynomial approximation of a function and its derivatives are obtained. These estimates are exact in a certain sense. In particular, the following result is derived as a corollary: Forf∈C ^r[?1,1],m∈N, and anyn≥max{m+r?1, 2r+1}, an algebraic polynomialP _n of degree ≤n exists that satisfies $$\left| {f^{\left( k \right)} \left( x \right) - P_n^{\left( k \right)} \left( {f,x} \right)} \right| \leqslant C\left( {r,m} \right)\Gamma _{nrmk} \left( x \right)^{r - k} \omega ^m \left( {f^{\left( r \right)} ,\Gamma _{nrmk} \left( x \right)} \right),$$ for 0≤k≤r andx ∈ [?1,1], where ω^υ(f^(k),δ) denotes the usual vth modulus of smoothness off ^(k), and Moreover, for no 0≤k≤r can (1?x ²)( ^{r?k+1)/(r?k+m)}(1/n²)^{(m?1)/(r?k+m)} be replaced by (1-x²)^αkn^2αk-2, with α_k>(r-k+a)/(r-k+m).     

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.     

Approximation to continuous functions by Walsh-Fourier sums

Пусть ω(δ) - модуль непре рывности,H _ω — класс 1-периодических непре рывных функцийf (t), модуль непрерывнос ти которыхω(f, δ)≦ω(δ), и п устьS _n (f,x)=S _n (f),D _n (t), L_n – соответственно ча стичные суммы ряда Фу рье-Уолшаf(t), ядра Дирихле и конст анты Лебега порядкап (п- 1, 2, ...). Доказаны Теорема 1.Для любой не прерывной функции f $$\mathop {\lim \inf }\limits_{k \to \infty } \frac{{\parallel f - S_k (f)\parallel }}{{\omega (f,1/k)L_k }} = 0$$ Теорема 2.Пусть 0n=[nβ]+1 (n=1,2, ...). Пусть, далее, М_n — множество тех k (2ⁿk<2ⁿ⁺¹), для которых Тогда в каждом классе $$\int\limits_0^{2^{ - m_n } } {\left| {D_k (t)} \right|dt \geqq AL_k } $$ найдется функция f, т акая, что $$\mathop {\lim \inf }\limits_{k \in \mathop \cup \limits_n M_n } \frac{{\left\| {f - S_k (f)} \right\|}}{{\omega (1/k)L_k }} \geqq B(A) > 0$$      

On rational approximations of functions of complex variable integrable over plane domains

пУстьE — ИжМЕРИМОЕ пО лЕБЕгУ ОгРАНИЧЕННОЕ МНОжЕстВО пОлОжИтЕльНОИ плОЩА ДИ mes₂ E кОМплЕксНОИ плОск ОстИ с. кАк ОБыЧНО, пРИp≧1 ОБОжНАЧИМ ЧЕРЕжL ^p (E) БА НАхОВО пРОстРАНстВО ИжМЕРИ Мых пО лЕБЕгУ НАE кОМплЕксНОжНАЧНых Ф УНкцИИf с сУММИРУЕМО Иp—стЕпЕНьУ Их МОДУль И ОБыЧНОИ НОРМОИ \(\left\| \cdot \right\|_p = \left\| \cdot \right\|_{L_p (E)}\) . ЧЕР ЕжL ^p R _n (f,E) ОБОжНАЧИМ НАИМЕН ьшЕЕ УклОНЕНИЕf?L ^p (E) От РАц ИОНАльНых ФУНкцИИ ст ЕпЕНИ ≦n кОМплЕксНОгО пЕРЕМЕ ННОгОz пО НОРМЕ ∥ · ∥. пОлОжИМf(z)=0 Дльz?¯CE,E _δ —δ-ОкРЕстНОсть МНО жЕстВАE (δ>0), И $$\omega _p (\delta ,f) = \mathop {\sup {\mathbf{ }}}\limits_{\left| h \right|< \delta } \{ \int\limits_{E_\sigma } {\int {{\mathbf{ }}|f(z + h) - f(z)|^p } d\sigma } \} ^{1/p} .$$ тЕОРЕМА.пУсть 1≦p<2,f?L _p (E),n≧4.тОгДА $$\begin{array}{*{20}c} {L^p R_n (f,E) \leqq 12\omega _p \left( {\frac{{\delta + \ln n}}{{\sqrt n }},f} \right){\mathbf{ }}npu{\mathbf{ }}p = 1,} \\ {L^p R_n (f,E) \leqq \frac{{24}}{{(p - 1)(2 - p)}}\omega _p (n^{(p - 2)/2p} ,f){\mathbf{ }}npu{\mathbf{ }}1< p< 2,} \\ {L^1 R_n (\bar z,[0,1] \times [0,1]) \geqq \frac{1}{{32\sqrt n }}.} \\ \end{array} $$ .     

Estimation of the remainder of a cubature formula on a Chebyshev grid for two-variable functions

For a cubature formula of the form $$\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {f(x,y)dxdy = \frac{{4\pi ^2 }} {{mn}}\sum\limits_{i = 0}^{n - 1} {\sum\limits_{j = 0}^{m - 1} {f\left( {\frac{{2\pi i}} {n},\frac{{2\pi j}} {m}} \right) + R_{n,m} (f)} } } }$$ on a Chebyshev grid, the remainder R _n,m (f) is proved to satisfy the sharp estimate $$\mathop {\sup }\limits_{f \in H\left( {r_1 ,r_2 } \right)} \left| {R_{n,m} (f)} \right| = O\left( {n^{ - r_1 + 1} + m^{ - r_1 + 1} } \right)$$ in some class of functions H(r ₁, r ₂) defined by a generalized shift operator. Here, r ₁, r ₂ > 1; ??^?1 ?? n/m ?? ?? with ?? > 0; and the constant in the O-term depends only on ??.     

Another approach to characterizations of generalized triangle inequalities in normed spaces

In this paper, we consider a generalized triangle inequality of the following type: $$\left\| {x_1 + \cdots + x_n } \right\|^p \leqslant \frac{{\left\| {x_1 } \right\|^p }} {{\mu _1 }} + \cdots + \frac{{\left\| {x_2 } \right\|^p }} {{\mu _n }}\left( {for all x_1 , \ldots ,x_n \in X} \right),$$ where (X, ‖·‖) is a normed space, (µ₁, ..., µ _n ) ∈ ? ⁿ and p > 0. By using ψ-direct sums of Banach spaces, we present another approach to characterizations of the above inequality which is given by [Dadipour F., Moslehian M.S., Rassias J.M., Takahasi S.-E., Nonlinear Anal., 2012, 75(2), 735–741].