共查询到20条相似文献,搜索用时 62 毫秒
1.
Kai-Uwe Schmidt 《Journal of Fourier Analysis and Applications》2013,19(3):457-466
Littlewood raised the question of how slowly $\lVert f_{n}\rVert_{4}^{4}-\lVert f_{n}\rVert_{2}^{4}$ (where $\lVert.\rVert _{r}$ denotes the L r norm on the unit circle) can grow for a sequence of polynomials f n with unimodular coefficients and increasing degree. The results of this paper are the following. For $$g_n(z)=\sum_{k=0}^{n-1}e^{\pi ik^2/n} z^k $$ the limit of $(\lVert g_{n}\rVert_{4}^{4}-\lVert g_{n}\rVert_{2}^{4})/\lVert g_{n}\rVert_{2}^{3}$ is 2/π, which resolves a mystery due to Littlewood. This is however not the best answer to Littlewood’s question: for the polynomials $$h_n(z)=\sum_{j=0}^{n-1}\sum _{k=0}^{n-1} e^{2\pi ijk/n} z^{nj+k} $$ the limit of $(\lVert h_{n}\rVert_{4}^{4}-\lVert h_{n}\rVert_{2}^{4})/\lVert h_{n}\rVert_{2}^{3}$ is shown to be 4/π 2. No sequence of polynomials with unimodular coefficients is known that gives a better answer to Littlewood’s question. It is an open question as to whether such a sequence of polynomials exists. 相似文献
2.
Introduce the notation: $\mathbb{Z}$ is the set of integers, $\bar {\mathbb{Z}}={\mathbb{Z}} \cup \{-\infty, +\infty\},{\mathbb{R}}_+^2 =\{x=(x_1,x_2) \in {\mathbb{R}}^2; x_1>0,x_2>0\}$ , $g_{k,m} (x,\alpha,h)= \int\limits_0^1 {g_1 (\frac{(k+u)h_1 - x_1}{\alpha_1})g_2(\frac{(m+u)h_2 - x_2}{\alpha_2})}du$ , where $g_i :\mathbb{R} \to \mathbb{R},x \in \mathbb{R}^2 ,\alpha ,h \in \mathbb{R}_ + ^2 $ . Under certain conditions on the functions g 1, g 2, we prove that the system of functions $g_{k,m} (x,\alpha^(n), h^(n)) (k,m \in \bar {\mathbb{Z}})$ , where $\alpha ^{\left( n \right)} ,h^{\left( n \right)} \in \mathbb{R}_ + ^2 $ are arbitrary infinitesimal sequences, is complete in the space C $\mathbb{R}^2 $ of uniformly continuous bounded functions f equipped with the norm $||f|| = \mathop {\sup }\limits_{x \in \mathbb{R}^2 } |f(x)|$ . Starting with the functions g k,m , it is possible to construct a method for uniform approximating in $\mathbb{R}^2 $ any continuous function of bounded variation in the sense of Hardy. An error estimate is derived in terms of the second order moduli of continuity. Based on the obtained results, we discuss in detail the accuracy of uniform approximation of functions of several variables by linear functions. The error estimates are derived by using second order moduli of continuity. We pay a particular attention to sharpness of constants. Bibliography: 8 titles. 相似文献
3.
For p > 1, we establish existence and asymptotic behavior of a positive continuous solution to the following boundary value problem $$\left\{\begin{array}{ll}\frac{1}{A} \left( A\Phi _{p}(u^{\prime})\right) ^{\prime}+a_{1}(r)u^{\alpha _{1}}+a_{2}(r)u^{\alpha _{2}}=0, \, {\rm in}\, (0,\infty ),\\ {\rm lim}_{r\rightarrow 0} A\Phi _{p}(u^{\prime})(r)=0, {\rm lim}_{r\rightarrow \infty } u(r)=0,\end{array}\right.$$ where \({\alpha _{1}, \alpha _{2} < p -1, \Phi _{p}(t) = t|t| ^{p-2},A}\) is a positive differentiable function and a 1, a 2 are two positive functions in \({C_{\rm loc}^{\gamma}((0, \infty )), 0 < \gamma < 1,}\) satisfying some appropriate assumptions related to Karamata regular variation theory. Also, we obtain an uniqueness result when \({\alpha _{1}, \alpha _{2} \in (1-p,p-1)}\) . Our arguments combine a method of sub and supersolutions with Karamata regular variation theory. 相似文献
4.
S. A. Iskhokov 《Differential Equations》2008,44(2):241-255
Let Ω be an arbitrary open set in R n , and let σ(x) and g i (x), i = 1, 2, ..., n, be positive functions in Ω. We prove a embedding theorem of different metrics for the spaces W p r (Ω, σ, $ \vec g $ ), where r ∈ N, p ≥ 1, and $ \vec g $ (x) = (g 1(x), g 2(x), ..., g n (x)), with the norm $$ \left\| {u;W_p^r (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\left\| {u;L_{p,r}^r (\Omega ;\sigma ,\vec g)} \right\|^p + \left\| {u;L_{p,r}^0 (\Omega ;\sigma ,\vec g)} \right\|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where $$ \left\| {u;L_{p,r}^m (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\sum\limits_{\left| k \right| = m} {\int\limits_\Omega {(\sigma (x)g_1^{k_1 - r} (x)g_2^{k_2 - r} (x) \cdots g_n^{k_n - r} (x)\left| {u^{(k)} (x)} \right|)^p dx} } } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ We use this theorem to prove the existence and uniqueness of a minimizing element U(x) ∈ W p r (Ω, σ, $ \vec g $ ) for the functional $$ \Phi (u) = \sum\limits_{\left| k \right| \leqslant r} {\frac{1} {{p_k }}\int\limits_\Omega {a_k (x)} \left| {u^{(k)} (x)} \right|^{p_k } } dx - \left\langle {F,u} \right\rangle , $$ where F is a given functional. We show that the function U(x) is a generalized solution of the corresponding nonlinear differential equation. For the case in which Ω is bounded, we study the differential properties of the generalized solution depending on the smoothness of the coefficients and the right-hand side of the equation. 相似文献
5.
In our paper we find all functions ${f : \mathbb {R} \times \mathbb {R}^{3} \rightarrow \mathbb {H}}$ and ${g : \mathbb {R}^{3} \rightarrow \mathbb {H}}$ satisfying ${f (r, {\bf v}) f (s, {\bf w}) = -\langle{\bf v},{\bf w}\rangle + f (rs, s{\bf v} + r{\bf w} + {\bf v} \times {\bf w})}$ ${(r, s \in \mathbb {R}, {\bf v},{\bf w} \in \mathbb {R}^{3})}$ , and ${g({\bf v})g({\bf w}) = -\langle{\bf v}, {\bf w}\rangle + g({\bf v} \times {\bf w})}$ $({{\bf v},{\bf w} \in \mathbb {R}^{3}})$ . These functional equations were motivated by the well-known identities for vector products and quaternions, which can be obtained from the solutions f (r, (v 1, v 2, v 3)) = r + v 1 i + v 2 j + v 3 k and g((v 1 ,v 2, v 3)) = v 1 i + v 2 j + v 3 k. 相似文献
6.
LetH r be anr-uniform hypergraph. Letg=g(n;H r ) be the minimal integer so that anyr-uniform hypergraph onn vertices and more thang edges contains a subgraph isomorphic toH r . Lete =f(n;H r ,εn) denote the minimal integer such that everyr-uniform hypergraph onn vertices with more thane edges and with no independent set ofεn vertices contains a subgraph isomorphic toH r . We show that ifr>2 andH r is e.g. a complete graph then $$\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} f(n;H^r ,\varepsilon n) = \mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} g(n;H^r )$$ while for someH r with \(\mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} g(n;H^r ) \ne 0\) $$\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} f(n;H^r ,\varepsilon n) = 0$$ . This is in strong contrast with the situation in caser=2. Some other theorems and many unsolved problems are stated. 相似文献
7.
Jiawang Nie 《Mathematical Programming》2013,137(1-2):225-255
Given polynomials f (x), g i (x), h j (x), we study how to minimize f (x) on the set $$S = \left\{ x \in \mathbb{R}^n:\, h_1(x) = \cdots = h_{m_1}(x) = 0,\\ g_1(x)\geq 0, \ldots, g_{m_2}(x) \geq 0 \right\}.$$ Let f min be the minimum of f on S. Suppose S is nonsingular and f min is achievable on S, which are true generically. This paper proposes a new type semidefinite programming (SDP) relaxation which is the first one for solving this problem exactly. First, we construct new polynomials ${\varphi_1, \ldots, \varphi_r}$ , by using the Jacobian of f, h i , g j , such that the above problem is equivalent to $$\begin{gathered}\underset{x\in\mathbb{R}^n}{\min} f(x) \hfill \\ \, \, {\rm s.t.}\; h_i(x) = 0, \, \varphi_j(x) = 0, \, 1\leq i \leq m_1, 1 \leq j \leq r, \hfill \\ \quad \, \, \, g_1(x)^{\nu_1}\cdots g_{m_2}(x)^{\nu_{m_2}}\geq 0, \, \quad\forall\, \nu \,\in \{0,1\}^{m_2} .\hfill \end{gathered}$$ Second, we prove that for all N big enough, the standard N-th order Lasserre’s SDP relaxation is exact for solving this equivalent problem, that is, its optimal value is equal to f min. Some variations and examples are also shown. 相似文献
8.
Let ${K=\mathbb{Q}(\theta)}$ be an algebraic number field with θ in the ring A K of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ${\mathbb{Q}}$ of rational numbers. For a rational prime p, let ${\bar{f}(x)\,=\,\bar{g}_{1}(x)^{e_{1}}....\bar{g}_{r}(x)^{e_{r}}}$ be the factorization of the polynomial ${\bar{f}(x)}$ obtained by reducing coefficients of f(x) modulo p into a product of powers of distinct irreducible polynomials over ${\mathbb{Z}/p\mathbb{Z}}$ with g i (x) monic. Dedekind proved that if p does not divide [ ${A_{K}:\mathbb{Z}}$ [θ]], then ${pA_{K}=\wp_{1}^{e_{1}}\ldots\wp_{r}^{e_{r}}}$ , where ${\wp_{1},\ldots,\wp_{r}}$ are distinct prime ideals of A K , ${\wp_{i}=pA_{K}+g_{i}(\theta)A_{K}}$ having residual degree equal to the degree of ${\bar{g}_{i}(x)}$ . He also proved that p does not divide [ ${A_{K}:\mathbb{Z}}$ [θ]] if and only if for each i, either e i = 1 or ${\bar{g}_{i}(x)}$ does not divide ${\bar{M}(x)}$ where ${M(x)=\frac{1}{p}(f(x)-g_{1}(x)^{e_{1}}....g_{r}(x)^{e_{r}})}$ . Our aim is to give a weaker condition than the one given by Dedekind which ensures that if the polynomial ${\bar{f}(x)}$ factors as above over ${\mathbb{Z}/p\mathbb{Z}}$ , then there are exactly r prime ideals of A K lying over p, with respective residual degrees ${\deg \bar {g}_{1}(x),...,\deg \bar {g}_{r}(x)}$ and ramification indices e 1, ..., e r . In this paper, the above problem has been dealt with in a more general situation when the base field is a valued field (K, v) of arbitrary rank and K(θ) is any finite extension of K. 相似文献
9.
Zhu Yaochen 《数学学报(英文版)》1991,7(2):127-134
In the present note the algebraic independence of certain continued fractions is proved. Especially, we prove that the Böhmer-Mahler's series \(\sum\limits_{K = 1}^\infty {\left[ {\omega _v k} \right]} {\text{ }}g_\mu ^{ - k} \left( {1 \leqslant \mu \leqslant s,1 \leqslant v \leqslant t} \right)\) are algebraically independent, where \(\mathop \omega \nolimits_1 {\text{ , }}...{\text{ , }}\mathop \omega \nolimits_{\text{t}} \) , ..., \(\mathop g\nolimits_1 {\text{ , }}...{\text{ , }}\mathop g\nolimits_s \) are some irrational numbers andg 1, ...,g s are distinct positive integers. 相似文献
10.
B. FISHER K. TAS 《数学学报(英文版)》2006,22(6):1639-1644
Let f and g be distributions and let gn = (g * δn)(x), where δn (x) is a certain converging to the Dirac delta function. The non-commutative neutrix product fog of f and g to be the limit of the sequence {fgn }, provided its limit h exists in the sense that sequence is defined N-lim n-∞(f(x)g,, (x), φ(x)〉 = (h(x), φ(x)},for all functions p in 2. It is proved that (x^λ+1n^px+)0(x^μ+1n^qx+)=x+^λμ1n^p+qx+,(x^λ-1n^qx-)=x-^λ+μ1n^p+qx-,for λ+μ〈-1; λ,μ, λ+μ≠-1,-2…and p,q=0,1,2…… 相似文献
11.
Michael Bildhauer Martin Fuchs 《Calculus of Variations and Partial Differential Equations》2010,37(1-2):167-186
We prove higher integrability and differentiability results for local minimizers u: ${\mathbb {R}^2\supset\Omega\to\mathbb {R}^M}$ , M ≥ 1, of the splitting-type energy ${\int_{\Omega}[h_1(|\partial_1 u|)+h_2(|\partial_2 u|)]\,{\rm d}x}$ . Here h 1, h 2 are rather general N-functions and no relation between h 1 and h 2 is required. The methods also apply to local minimizers u: ${\mathbb {R}^2\supset\Omega \to \mathbb {R}^2}$ of the functional ${\int_{\Omega}[h_1(|{\rm div}\,{\rm u}|)+h_2(|\varepsilon^D(u)|)]\,{\rm d}x}$ so that we can include some variants of so-called nonlinear Hencky-materials. Further extensions concern non-autonomous problems. 相似文献
12.
13.
Let ${\mathfrak{a}}$ be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. It is shown that ${{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))= {\rm Ann}_R(M/T_R(\mathfrak{a}, M))}$ , where ${T_R(\mathfrak{a}, M)}$ is the largest submodule of M such that ${{\rm cd}(\mathfrak{a}, T_R(\mathfrak{a}, M)) < {\rm cd}(\mathfrak{a}, M)}$ . Several applications of this result are given. Among other things, it is shown that there exists an ideal ${\mathfrak{b}}$ of R such that ${{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))={\rm Ann}_R(M/H_{\mathfrak{b}}^{0}(M))}$ . Using this, we show that if ${ H_{\mathfrak{a}}^{{\rm dim} R}(R)=0}$ , then ${{{\rm Att}_R} H^{{\rm dim} R-1}_{\mathfrak a}(R)= \{\mathfrak{p} \in {\rm Spec} R | \,{\rm cd}(\mathfrak{a}, R/\mathfrak{p}) = {\rm dim} R-1\}.}$ These generalize the main results of Bahmanpour et al. (see [2, Theorem 2.6]), Hellus (see [7, Theorem 2.3]), and Lynch (see [10, Theorem 2.4]). 相似文献
14.
We elaborate Weiermann-style phase transitions for well-partial-orderings (wpo) determined by iterated finite sequences under Higman-Friedman style embedding with Gordeev’s symmetric gap condition. For every d-times iterated wpo ${\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}\right)}$ in question, d >? 1, we fix a natural extension of Peano Arithmetic, ${T \supseteq \sf{PA}}$ , that proves the corresponding second-order sentence ${\sf{WPO}\left({\rm S}{\textsc{eq}}^{d}, \trianglelefteq _{d}\right) }$ . Having this we consider the following parametrized first-order slow well-partial-ordering sentence ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}, r\right):}$ $$\left( \forall K > 0 \right) \left( \exists M > 0\right) \left( \forall x_{0},\ldots ,x_{M}\in {\rm S}\text{\textsc{eq}}^{d}\right)$$ $$\left( \left( \forall i\leq M\right) \left( \left| x_{i}\right| < K + r \left\lceil \log _{d} \left( i+1\right) \right\rceil \right)\rightarrow \left( \exists i < j \leq M \right) \left(x_{i} \trianglelefteq _{d} x_{j}\right) \right)$$ for a natural additive Seq d -norm |·| and r ranging over EFA-provably computable positive reals, where EFA is an abbreviation for IΔ 0?+?exp. We show that the following basic phase transition clauses hold with respect to ${T = \Pi_{1}^{0}\sf{CA}_{ < \varphi ^{_{\left( d-1\right) }} \left(0\right) }}$ and the threshold point1.
- If r <? 1 then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }$ is provable in T.
- If ${r > 1}$ then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }$ is not provable in T.
- ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1\right)}$ is still provable in T = PA (actually in EFA).
- If ${\alpha < \varepsilon _{0}}$ , then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1_{\alpha}\right)}$ is provable in T = PA.
- ${\sf{SWP}\left( {\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2},1_{\varepsilon _{0}}\right)}$ is not provable in T = PA.
15.
J. Nahas 《Calculus of Variations and Partial Differential Equations》2013,46(1-2):427-437
We show that smooth, radially symmetric wave maps U from ${\mathbb {R}^{2+1}}$ to a compact target manifold (N, where ? r U and ? t U have compact support for any fixed time, scatter. The result will follow from the work of Christodoulou and Tahvildar-Zadeh, and Struwe, upon proving that for ${(\lambda^{\prime} \in (0,1),}$ energy does not concentrate in the set $$K_{\frac{5}{8}T,\frac{7}{8}T}^{\lambda^{\prime}} = \{(x,t) \in \mathbb{R}^{2+1} \vert \quad|x| \leq \lambda^{\prime} t, t \in [(5/8)T,(7/8)T] \}.$$ 相似文献
16.
V. I. Arnautov 《Journal of Mathematical Sciences》2012,185(2):176-183
Let R(+, ·) be a nilpotent ring and $ \left( {\mathfrak{M}, < } \right) $ be the lattice of all ring topologies on R(+, ·) or the lattice of all such ring topologies on R(+, ·) in each of which the ring R possesses a basis of neighborhoods of zero consisting of subgroups. Let ?? and ??? be ring topologies from $ \mathfrak{M} $ such that $ \tau = {\tau_0}{ \prec_\mathfrak{M}}{\tau_1}{ \prec_\mathfrak{M}} \cdots { \prec_\mathfrak{M}}{\tau_n} = \tau ^{\prime} $ . Then k????n for every chain $ \tau = {\tau ^{\prime}_0} < {\tau ^{\prime}_1} < \cdots < {\tau ^{\prime}_k} = \tau ^{\prime} $ of topologies from $ \mathfrak{M} $ , and also n?=?k if and only if $ {\tau ^{\prime}_i}{ \prec_\mathfrak{M}}{\tau ^{\prime}_{i + 1}} $ for all 0????i?<?k. 相似文献
17.
В. И. БУслАЕВ 《Analysis Mathematica》1977,3(1):11-44
LetB σ be the class of entire functions of exponential type σ, real valued and bounded in modulus by 1 in the real line. A setG of functions defined on the segment [-T-r, T+r], wherer is a fixed positive number, is called an (ε, δ)-net of the classB σ on the segment [-т, т] if for any f?B σ there existsg?G such that for anyx?[-T,T] $$\left| {f(x) - g(x)} \right| \leqq \frac{\varepsilon }{{2r}}\int\limits_{x - r}^{x + r} {\left| {f(t)} \right|dt + \delta .} $$ The main result consists in the following: For any positive σ, r, ε≦1, δ≦1 and sufficiently largeT we have $$H_{\varepsilon ,\delta } (B_\sigma ,T) \leqq \frac{{2\sigma T}}{\pi }\log \frac{{c(\sigma r)}}{{\max (\varepsilon ,\delta )}},$$ where c(σr) depends only on the product σr. The main tool of the proof of this inequality is the following estimate of the derivative of a polynomialP(x) with real coefficients: $$\left\| {P'(x)} \right\|_{L_p ( - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}) \leqq } c\left( {q + 1 + \sum\limits_{i = 1}^{n - q} {\frac{1}{{\left| {a_i } \right|^2 }}} } \right)\left\| {P(x} \right\|_{L_p ( - 1,1)} ,$$ whereq is the number of roots of the polynomialP(x) lying in the disk ¦z¦<1; a1, ..., an?g are the other roots, с is an absolute constant, and 1≦p≦∞. 相似文献
18.
19.
Wolfgang Müller 《Monatshefte für Mathematik》1989,108(4):301-323
LetK be a quadratic number field with discriminantD and denote byF(n) the number of integral ideals with norm equal ton. Forr≥1 the following formula is proved $$\sum\limits_{n \leqslant x} {F(n)F(n + r) = M_K (r)x + E_K (x,r).} $$ HereM k (r) is an explicitly determined function ofr which depends onK, and for every ε>0 the error term is bounded by \(|E_K (x,r)|<< |D|^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2} + \varepsilon } x^{{5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-0em} 6} + \varepsilon } \) uniformly for \(r<< |D|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} x^{{5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-0em} 6}} \) Moreover,E k (x,r) is small on average, i.e \(\int_X^{2X} {|E_K (x,r)|^2 dx}<< |D|^{4 + \varepsilon } X^{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2} + \varepsilon } \) uniformly for \(r<< |D|X^{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-0em} 4}} \) . 相似文献
20.
Consider the real Clifford algebra ${\mathbb{R}_{0,n}}$ generated by e 1, e 2, . . . , e n satisfying ${e_{i}e_{j} + e_{j}e_{i} = -2\delta_{ij} , i, j = 1, 2, . . . , n, e_{0}}$ is the unit element. Let ${\Omega}$ be an open set in ${\mathbb{R}^{n+1}}$ . u(x) is called an h-regular function in ${\Omega}$ if $$D_{x}u(x) + \widehat{u}(x)h = 0, \quad\quad (0.1)$$ where ${D_x = \sum\limits_{i=0}^{n} e_{i}\partial_{xi}}$ is the Dirac operator in ${\mathbb{R}^{n+1}}$ , and ${\widehat{u}(x) = \sum \limits_{A} (-1)^{\#A}u_{A}(x)e_{A}, \#A}$ denotes the cardinality of A and ${h = \sum\limits_{k=0}^{n} h_{k}e_{k}}$ is a constant paravector. In this paper, we mainly consider the Hilbert boundary value problem (BVP) for h-regular functions in ${\mathbb{R}_{+}^{n+1}}$ . 相似文献