On the zeros of derivatives of Bessel functions

In this paper we are interested in the behaviour respect tov of thekth positive zeroc′ _vk of the derivative of the general Bessel functionC _v(x)=J _v(x)cosα?Y _v(x)sinα, 0≤α<π, whereJ _v(x) andY _v(x) indicate the Bessel functions of first and second kind respectively. It is well known that forc′ _vk>∥v∥,c′ _vk increases asv increases. Here we prove several additional properties forc′ _vk. Our main result is thatc′ _vk is concave as a function ofv, whenc′ _vk>∥v∥>0. This implies the concavity ofc′ _vk for everyk=2,3, ?. In the case of the zerosJ′ _vk of _d ^dx J _v(x) we extend this property tok=1 for everyv≥0.     

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.     

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).     

The cancellation of Fourier coefficients of cusp forms over different sparse sequences

Let λ f(n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z)∈Sk(Γ).In this paper,we established nontrivial estimates for ∑n≤xλf(ni)λf(nj),where 1 ≤ i j ≤ 4.     

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.     

Elementary approximations for zeros of Bessel functions

Let j_vk, y_vk and c_vk denote the kth positive zeros of the Bessel functions J_v(x), Y_v(x) and of the general cylinder function C_v(x) = cos αJ_v(x)?sin αY_v(x), 0 ? α < π, respectively. In this paper we extend to c_vk, k = 2, 3,..., some linear inequalities presently known only for j_vk. In the case of the zeros y_vk we are able to extend these inequalities also to k = 1. Finally in the case of the first positive zero j_v1 we compare the linear enequalities given in [9] with some other known inequalities.     

An approximation for the zeros of Bessel functions

Summary LetC _vk be thekth positive zero of the cylinder functionC _v(x)=cosJ _v(x)–sinY _v(x), whereJ _v(x),Y _v(x) are the Bessel functions of first kind and second kind, resp., andv>0, 0<. Definej _vk byj _vk=C _vk with . Using the notation 1/K=, we derive the first two terms of the asymptotic expansion ofj _vk in terms of the powers of at the expense of solving a transcendental equation. Numerical examples are given to show the accuracy of this approximation.Dedicated to the memory of Professor Lothar CollatzThis work has been supported by the Hungarian Scientific Grant No. 6032/6319     

An upper bound for the zeros of the derivative of Bessel functions

Letj _vk ^′ denotes thekth positive zero of the derivativeJ _v ^′ (x)=dJ _v (x)/dx of Bessel functionJ _v (x) fork=1, 2,…. We establish the upper bound

$$j'_{\nu k}< \nu + a_k \left( {\nu + \frac{{{\rm A}_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} + \frac{3}{{10}}a_k^2 \left( {\nu + \frac{{A_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} , \nu \geqslant 0, k = 1,2, \ldots $$     

Riemann—Lebesgue theorem

Пусть?(x) — ограниченн ая функция на отрезке [0,1] и ее функция распределен ияΦ(t) удовлетворяет услов ию $$\Phi \left( t \right) + \Phi \left( { - t} \right) = 1.$$ Еслиf(x) — конечная поч ти всюду функция, то дл яF _n (t) — функции распределе ния произведенияf(x)?(nx) — вы полнены соотношения и В частности, еслиf(x) — и нтегрируемая функци я, то из (1) следует, что $$\mathop {\lim }\limits_{n \to \infty } \mathop \smallint \limits_0^1 f\left( x \right)\varphi \left( {nx} \right)dx = 0 $$      

Boundary-value problems for a nonlinear hyperbolic equation with variable coefficients and the Lévy Laplacian

For a nonlinear hyperbolic equation with variable coefficients and the infinite-dimensional Lévy Laplacian Δ _L , $$\beta \left( {\sqrt 2 \left\| x \right\|_H \frac{{\partial U(t,x)}} {{\partial t}}} \right)\frac{{\partial ^2 U(t,x)}} {{\partial t^2 }} + \alpha (U(t,x))\left[ {\frac{{\partial U(t,x)}} {{\partial t}}} \right]^2 = \Delta _L U(t,x),$$ we present algorithms for the solution of the boundary-value problem U(0, x) = u ₀, U(t, 0) = u ₁ and the exterior boundary-value problem U(0, x) = v ₀, \(\left. {U(t,x)} \right|_{\Gamma = v_1 }\) , \(\lim _{\left\| x \right\|_{H \to \infty } } \left. {U(t,x) = v_2 } \right|\) for the class of Shilov functions depending on the parameter t.     

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

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

Functional equations and Weierstrass sigma-functions

It is proved that if an entire function f: ? → ? satisfies an equation of the form α ₁(x)β ₁(y) + α ₂(x)β ₂(y) + α ₃(x)β ₃(y), x,y ∈ C, for some α _j , β _j : ? → ? and there exist no \({\widetilde \alpha _j}\) and ?\({\widetilde \beta _j}\) for which \(f\left( {x + y} \right)f\left( {x - y} \right) = {\overline \alpha _1}\left( x \right){\widetilde \beta _1}\left( y \right) + {\overline \alpha _2}\left( x \right){\widetilde \beta _2}\left( y \right)\), then f(z) = exp(Az ² + Bz + C) ? σ _Γ(z - z ₁) ? σ _Γ(z - z ₂), where Γ is a lattice in ?; σ _Γ is the Weierstrass sigma-function associated with Γ; A,B,C, z ₁, z ₂ ∈ ?; and \({z_1} - {z_2} \notin \left( {\frac{1}{2}\Gamma } \right)\backslash \Gamma \).     

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.     

Generalized <Emphasis Type="Italic">n</Emphasis>-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

Let n ≥ 2 and let Ω ? ? ⁿ be an open set. We prove the boundedness of weak solutions to the problem

$$u \in W_0^1 L^\Phi \left( \Omega \right) and - div\left( {\Phi '\left( {\left| {\nabla u} \right|} \right)\frac{{\nabla u}}{{\left| {\nabla u} \right|}}} \right) + V\left( x \right)\Phi '\left( {\left| u \right|} \right)\frac{u}{{\left| u \right|}} = f\left( {x,u} \right) + \mu h\left( x \right) in \Omega ,$$

where ? is a Young function such that the space W ₀ ¹ L ^Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L ^Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω = ? ⁿ .

     

On the power of the zeros of Bessel functions

For ν≥0 let c_νk be the k-th positive zero of the cylinder functionC _v(t)=J _v(t)cosα-Y _v(t)sinα, 0≤α>π whereJ _ν(t) andY _ν(t) denote the Bessel functions of the first and the second kind, respectively. We prove thatC _v,k ^1+H(x) is convex as a function of ν, ifc _νk≥x>0 and ν≥0, whereH(x) is specified in Theorem 1.1.     

Pointwise estimates for Lagrange interpolation polynomials

Let f ∈ C[?1, 1]. Let the approximation rate of Lagrange interpolation polynomial of f based on the nodes $ \left\{ {\cos \frac{{2k - 1}} {{2n}}\pi } \right\} \cup \{ - 1,1\} $ be Δ _{n + 2}(f, x). In this paper we study the estimate of Δ _{n + 2}(f,x), that keeps the interpolation property. As a result we prove that $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left| {T_n (x)} \right|\ln (n + 1) + \omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}\left| {T_n (x)} \right|} \right)} \right\}, $$ where T _n (x) = cos (n arccos x) is the Chebeyshev polynomial of first kind. Also, if f ∈ C ^r [?1, 1] with r ≧ 1, then $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$      

Copositive pointwise approximation

We prove that if a functionf ∈C ⁽¹⁾ (I),I: = [?1, 1], changes its signs times (s ∈ ?) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ?, there exists an algebraic polynomialP _n =P _n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω _k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k ? 1, there exists a polynomialP _n =P _n (x) of degree ≤n such thatP _n (x) ≥ 0,x ∈I, and |f(x) ?P _n (x)| ≤c(k)ω _k (f;n ^?2 +n ^?1 √1 ?x ²),x ∈I.     

On exact boundary zero-controlability of two-dimensional Navier-Stokes equations

For two-dimensional Navier-Stokes equations defined in a bounded domain Ω and for an arbitrary initial vector field, we construct the boundary Dirichlet condition that is tangent to the boundary ?Ω of Ω and satisfies the property: the solutionυ(t, x) of the mentioned boundary-value problem equals zero at a certain finite time momentT. Moreover, $$\parallel x(t, \cdot )\parallel _{L_2 (\Omega )} \leqslant c\exp \left( {\tfrac{{ - k}}{{(T - t)^2 }}} \right)ast \to T,$$ wherec > 0,k > 0 constants.     

Kriterien in der Theorie der Gleichverteilung

It is the aim of this paper to introduce two new notions of discrepancy. They are defined by the formulas $$\begin{gathered} \Delta _N^r \left( {\omega ;f} \right) = \mathop {\sup }\limits_{\left| z \right| = r} \left| {\left( {{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}} \right)\sum\limits_{n = 1}^N {f\left( {z e^2 \pi i\omega \left( n \right)} \right)} - f\left( 0 \right)} \right|, and \hfill \\ \delta _N^r \left( {\omega ;f} \right) = \mathop {\sup }\limits_{\left| z \right| = r} \left| {\left( {{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}} \right)\sum\limits_{n = 1}^N {f\left( {z \omega \left( n \right)} \right)} \cdot z - \int\limits_0^z {f\left( \zeta \right)d\zeta } } \right|, \hfill \\ \end{gathered} $$ wheref is a holomorphic function defined in the unit disc withf ^(k) (0)≠0 for allk∈?,r<1 is a positive number, and ω is a sequence in [0, 1]. The first of these discrepancies can be generalized for multidimensional sequences. ω is uniform distributed if and only if lim _N→∞ Δ _N ^r (ω;f)=0 resp. lim _N→∞δ _N ^r (ω;f)=0. These results are proved in a quantitative way by estimating the classical discrepancyD _N (ω) by means ofΔ _N ^r (ω;f) and δ _N ^r (ω;f): $$\begin{gathered} \Delta _N^r \left( {\omega ;f} \right) \ll D_N \left( \omega \right) \ll \Phi \left( {\Delta _N^r \left( {\omega ;f} \right)} \right), \hfill \\ \delta _N^r \left( {\omega ;f} \right) \ll D_N \left( \omega \right) \ll \Psi \left( {\delta _N^r \left( {\omega ;f} \right)} \right). \hfill \\ \end{gathered} $$ The functions Φ and Ψ only depend onf andr. These estimations are based on the inequalities ofKoksma-Hlawka andErdös-Turán.