In 1954, A. Novikoff studied the asymptotic behavior of the Pollaczek polynomials Pn(x; a, b) when , where t > 0 is fixed. He divided the positive t-axis into two regions, 0 < t < (a + b)1/2 and t > (a + b)1/2, and derived an asymptotic formula in each of the two regions. Furthermore, he found an asymptotic formula for the zeros of these polynomials. Recently M. E. H. Ismail (1994) reconsidered this problem in an attempt to prove a conjecture of R. A. Askey and obtained a two-term expansion for these zeros. Here we derive an infinite asymptotic expansion for , which holds uniformly for 0 < ε ≤ t ≤ M < ∞, and show that Ismail's result is incorrect. 相似文献
We look at the decomposition of arbitrary f in L2( R ) in terms of the family of functions φmn(x) = π?1/4exp{ ? 1/2imnab + i max ? 1/2(x ? nb)2}, with a, b > 0. We derive bounds and explicit formulas for the minimal expansion coefficients in the case where ab = 2π/N, N an integer ≧ 2. Transported to the Hilbert space F of entire functions introduced by V. Bargmann, these results are expressed as inequalities of the form We conjecture that these inequalities remain true for all a, b such that ab < 2π. 相似文献
Let Q(D) be a class of functions q, q(0) = 0, |q(z)| < 1 holomorphic in the Reinhardt domain D ? C n, a and b — arbitrary fixed numbers satisfying the condition — 1 ≤ b < a ≤ 1. ??(a, b; D) — the class of functions p such that p ? ??(a, b; D) iff for some q ? Q(D) and every z ? D. S*(a, b; D) — the class of functions f such that f ? S*(a, g; D) iff Sc(a, b; D) — the class of functions q such that q ? Sc(a, b; D) iff , where p ε ??(a, b; D) and K is an operator of the form for z=z1,z2,…zn. The author obtains sharp bounds on |p(z)|, f(z)| g(z)| as well as sharp coefficient inequalities for functions in ??(a, b; D), S*(a, b; D) and Sc(a, b; D). 相似文献
Positive entire solutions of the equation \(\Delta _p u = u^{ - q} in \mathbb{R}^N (N \geqslant 2)\) where 1 < p ≤ N, q > 0, are classified via their Morse indices. It is seen that there is a critical power q = qc such that this equation has no positive radial entire solution that has finite Morse index when q > qc but it admits a family of stable positive radial entire solutions when 0 < q ≤ qc. Proof of the stability of positive radial entire solutions of the equation when 1 < p < 2 and 0 < q ≤ qc relies on Caffarelli–Kohn–Nirenberg’s inequality. Similar Liouville type result still holds for general positive entire solutions when 2 < p ≤ N and q > qc. The case of 1 < p < 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p = 2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator. 相似文献
In this paper we condiser non-negative solutions of the initial value problem in ?N for the system where 0 ? δ ? 1 and pq > 0. We prove the following conditions. Suppose min(p,q)≥1 but pq1.
(a) If δ = 0 then u=v=0 is the only non-negative global solution of the system.
(b) If δ>0, non-negative non-globle solutions always exist for suitable initial values.
(c) If 0<?1 and max(α, β) ≥ N/2, where qα = β + 1, pβ = α + 1, then the conclusion of (a) holds.
(d) If N > 2, 0 < δ ? 1 and max (α β) < (N - 2)/2, then global, non-trivial non-negative solutions exist which belong to L∞(?N×[0, ∞]) and satisfy 0 < u(X, t) ? c∣x∣?2α and 0 < v(X, t) ? c ∣x∣?2bT for large ∣x∣ for all t > 0, where c depends only upon the initial data.
(e) Suppose 0 > δ 1 and max (α, β) < N/2. If N> = 1,2 or N > 2 and max (p, q)? N/(N-2), then global, non-trivial solutions exist which, after makinng the standard ‘hot spot’ change of variables, belong to the weighted Hilbert space H1 (K) where K(x) ? exp(¼∣x∣2). They decay like e[max(α,β)-(N/2)+ε]t for every ε > 0. These solutions are classical solutions for t > 0.
(f) If max (α, β) < N/2, then threre are global non-tivial solutions which satisfy, in the hot spot variables where where 0 < ε = ε(u0, v0) < (N/2)?;max(α, β). Suppose min(p, q) ? 1.
(g) If pq ≥ 1, all non-negative solutions are global. Suppose min(p, q) < 1.
(h) If pg > 1 and δ = 0, than all non-trivial non-negative maximal solutions are non-global.
(i) If 0 < δ ? 1, pq > 1 and max(α,β)≥ N/2 all non-trivial non-negative maximal solutions are non-global.
(j) If 0 < δ ≥ 1, pq > 1 and max(α,β) < N/2, there are both global and non-negative solutions.
We also indicate some extensions of these results to moe general systems and to othere geometries. 相似文献
Summary Leta, b > 0 be positive real numbers. The identric meanI(a, b) of a andb is defined byI = I(a, b) = (1/e)(bb/aa)1/(b–a), fora b, I(a, a) = a; while the logarithmic meanL(a, b) ofa andb isL = L(a, b) = (b – a)/(logb – loga), fora b, L(a, a) = a. Let us denote the arithmetic mean ofa andb byA = A(a, b) = (a + b)/2 and the geometric mean byG =G(a, b) =
. In this paper we obtain some improvements of known results and new inequalities containing the identric and logarithmic means. The material is divided into six parts. Section 1 contains a review of the most important results which are known for the above means. In Section 2 we prove an inequality which leads to some improvements of known inequalities. Section 3 gives an application of monotonic functions having a logarithmically convex (or concave) inverse function. Section 4 works with the logarithm ofI(a, b), while Section 5 is based on the integral representation of means and related integral inequalities. Finally, Section 6 suggests a new mean and certain generalizations of the identric and logarithmic means. 相似文献
We consider the eigenvalue problem for t ? [0, b], where an = |a|n sgna, a ? ?, λ ? ?, the constants μ, v are real such that 0 ≤ μ < n and derive asymptotic estimates for solutions of the differential equation in the definite case q(t)> 0 which corresponds to the well-known WKB-approximation in the linear case n = 1, μ = 0. In the second part we investigate the asymptotic distribution of the eigenvalues in the general case of two -point boundary conditions and refine these results for the so called separated boundary conditions. 相似文献
For fixed c > 1 and for arbitrary and independent a,b ≧ 1 let Z2|b( cosh(x/a)−c) ≦ y < 0}. We investigate the asymptotic behaviour of R(a,b) for a,b → ∞. In the special case b = o(a5/6) the lattice rest has true order of magnitude
.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
Abstract
Let q ≥ 3 be an odd number, a be any fixed positive integer with (a, q) = 1. For each integer b with 1 ≤ b < q and (b, q) = 1, it is clear that there exists one and only one c with 0 < c < q such that bc ≡ a (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bc ≡ a (mod q) for 1 ≤ b, c < q in which b and c are of opposite parity, and let
. The main purpose of this paper is to study the distribution properties of E(a, q), and give a sharper hybrid mean-value formula involving E(a, q) and general Kloosterman sums.
This work is supported by the NSF and the PSF of P. R. China 相似文献