Extension de quelques theoremes sur les densites de series d'elements de n a des series de sous-ensembles finis de n

For a sequence A = {A_k} of finite subsets of N we introduce: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{n}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$, where A(m) is the number of subsets A_k ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation $\text{a ∪ b, (a ∩ b), (a 1 b = a ∪ b ? a ∩ b)}$ constitutes a finite semi-group N^∪ (semi-group N^∩) (group $\text{N}{}^1$). For N^∪, N^∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)^?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {A_kr}, where A_kr | A_kr+1 for r = 1, 2, …. In Section 6 we consider for $\text{N∪, N∩, N1}$ analogues of Rohrbach inequality: $\text{2n}\text{? g(n) ? 2}\text{n}$, where g(n) = min k over the subsets {a₁ < … < a_k} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = a_i + a_j.Pour une série A = {A_k} de sous-ensembles finis de N on introduit les densités: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{m}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$ où A(m) est le nombre d'ensembles A_k ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations $\text{a ∪ b, a ∩ b, a 1 b = a ∪ b ? a ∩ b}$, un semi-groupe fini N^∪, N^∩ ou un groupe N¹ respectivement. Pour N^∪, N^∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)^?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {A_kr}, où A_kr|A_kr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N^∪, $\text{N}{}^{\mathrm{\cap }}\text{, N}{}^1$ les analogues de l'inégalité de Rohrbach: $\text{2n}\text{? g(n) ? 2}\text{n}$, où g(n) = min k pour les ensembles {a₁ < … < a_k} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = a_i + a_j.     

Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁…,z_n), z_i ? $\text{C}$, i=1, …, n, is uniquely determined from the magnitude of f(x₁…,x_n): | f(x₁…,x_n)|, x_i ? $\text{R}$, i=1,…, n, except for (1) linear shifts: ^{i(α1z1+…+α_n2n+β}), β, α_i?$\text{R}$, i=1,…, n; and (2) conjugation: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

Spectral representation for Schrödinger operators with long-range potentials

A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R³, is obtained, where V(x) is a long-range potential: $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(1</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, grad $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(3</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, $\text{ΛV(x) = O(¦ x}\text{s}\text{?}{}^{\mathrm{?\delta }}\text{) (δ > 0), Λ}$ being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator $\text{F}$ from PL₂(R³) onto $\text{L}{}_2\text{((0, ∞); L}{}_2\text{(Ω)), P}$ being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ),

$\text{(α(H)(Pf,g)=}\text{∫}\text{0}\text{∞}\text{(α(λ)(Ff)(λ),(Fg)(λ))L}{}_2\text{(ω) dλ}$

.     

Estimating functions of canonical correlation coefficients

Let ρ²₁,…,ρ²_p be the squares of the population canonical correlation coefficients from a normal distribution. This paper is concerned with the estimation of the parameters δ₁,…,δ_p, where $\text{δ}{}_{\mathrm{i}}\text{=}\text{ρ}{}^2{}_{\mathrm{i}}\text{(1 ? ρ}{}^2{}_{\mathrm{i}}\text{)}$, i = 1,…,p, in a decision theoretic way. The approach taken is to estimate a parameter matrix Δ whose eigenvalues are δ₁,…,δ_p, given a random matrix F whose eigenvalues have the same distribution as $\text{r}{}^2{}_{\mathrm{i}}\text{(1 ? r}{}^2{}_{\mathrm{i}}\text{)}$, i = 1,…,p, where r₁,…,r_p are the sample canonical correlation coefficients.     

Kolmogorov-Landau inequalities for monotone functions

Presented in this report are two further applications of very elementary formulae of approximate differentiation. The first is a new derivation in a somewhat sharper form of the following theorem of V. M. Olovyani?nikov: LetN_n (n ? 2) be the class of functionsg(x) such thatg(x), g′(x),…, g⁽ⁿ⁾(x) are ? 0, bounded, and nondecreasing on the half-line ?∞ < x ? 0. A special element ofN_nis

$\text{g}{}_1\text{(x) = 0}\text{if}\text{?∞ < x < ?1, g}{}_1\text{(x) = (1 + x)}{}^{\mathrm{n}}\text{if}\text{?1 ? x ? 0}$

. Ifg(x) ∈ N_nis such that

$\text{g(0) ? g}{}_1\text{(0) = 1, g}{}^{\mathrm{(n)}}\text{(0) ? g}{}_1{}^{\mathrm{(n)}}\text{(0) = n!}$

, then

$\text{g}{}^{\mathrm{(v)}}\text{(0) ? g}{}_1{}^{\mathrm{(v)}}\text{(0)}$

for

1$\text{v = 1,…, n ? 1}$

. Moreover, if we have equality in (1) for some value of v, then we have there equality for all v, and this happens only if $\text{g(x) = g}{}_1\text{(x)}$ in (?∞, 0].The second application gives sufficient conditions for the differentiability of asymptotic expansions (Theorem 4).     

On the absolute continuity of Schrödinger operators with spherically symmetric,long-range potentials,II

Let H = ?Δ + V, where the potential V is spherically symmetric and can be decomposed as a sum of a short-range and a long-range term, V(r) = V_S(r) + V_L. Let λ = lim sup_r→∞V_L(r) < ∞ (we allow λ = ? ∞) and set λ⁺ = max(λ, 0). Assume that for some r₀, V_L(r) ?C^2k(r₀, ∞) and that there exists δ > 0 such that $\text{(}\text{d}\text{dr}\text{)}{}^{\mathrm{j}}\text{V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r) · (λ}{}^{\mathrm{+}}\text{? V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r) + 1)}{}^{\mathrm{?1}}\text{= O(r}{}^{\mathrm{?j\delta }}\text{), j = 1,…, 2k,}\text{as}\text{r → ∞}$. Assume further that $\text{∝}{}_1{}^{\mathrm{\infty }}\text{(}\text{dr}\text{¦ V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r)¦}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{) = ∞}$ and that 2kδ > 1. It is shown that: (a) The restriction of H to C^∞(Rⁿ) is essentially self-adjoint, (b) The essential spectrum of H contains the closure of (λ, ∞). (c) The part of H over (λ, ∞) is absolutely continuous.     

A Feynman-Kac gauge for solvability of the Schrödinger equation

Let {X_t, t ≥ 0} be Brownian motion in $\text{R}$^d (d ≥ 1). Let D be a bounded domain in $\text{R}$^d with C² boundary, ?D, and let q be a continuous (if d = 1), Hölder continuous (if d ≥ 2) function in D?. If the Feynman-Kac “gauge” E_x{exp(∝₀^τ_Dq(X_t)dt)1_A(X_τD)}, where τ_D is the first exit time from D, is finite for some non-empty open set A on ?D and some x?D, then for any $\text{? ? C}{}^0\text{(?D}$), is the unique solution in $\text{C}{}^2\text{(D) ∩ C}{}^0\text{(}\text{D}\text{?}\text{)}$ of the Schrödinger boundary value problem $\text{(}\text{1}\text{2}\text{Δ}\text{+ q)φ = 0}\text{in}\text{D, φ = ?}\text{on}\text{?D}$.     

A divided difference inequality for n-convex functions

For a₍₁₎ ? a₍₂₎ ? ··· ? a_(n) ? 0, b₍₁₎ ? b₍₂₎ ? ··· ? b_(n) ? 0, the ordered values of a_i, b_i, i = 1, 2,…, n, m fixed, m ? n, and p ? 1 it is shown that

$\text{∑}\text{1}\text{n}\text{a}{}_{\mathrm{i}}\text{b}{}_{\mathrm{i}}\text{?}\text{∑}\text{1}\text{m}\text{a}{}^{\mathrm{p}}{}_{\mathrm{(i)}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{∑}\text{1}\text{m?k?1}\text{b}{}^{\mathrm{q}}{}_{\mathrm{(i)}}\text{+}\text{b}{}^{\mathrm{q}}{}_{\mathrm{[m?k]}}\text{(k+1)}{}^{\mathrm{<mtext>q</mtext><mtext>p</mtext>}}{}^{\mathrm{<mtext>1</mtext><mtext>q</mtext>}}$

where $\text{1}\text{p}\text{+}\text{1}\text{q}\text{= 1, b}{}_{\mathrm{[j]}}\text{= b}{}_{\mathrm{(j)}}\text{+ b}{}_{\mathrm{(j\; +\; 1)}}\text{+ ··· + b}{}_{\mathrm{(n)}}\text{,}\text{and}\text{k}$ is the integer such that $\text{b}{}_{\mathrm{(m\; ?\; k\; ?\; 1)}}\text{?}\text{b[m ? k]}\text{(k + 1)}$ and $\text{b}{}_{\mathrm{(m\; ?\; k)}}\text{<}\text{b}{}_{\mathrm{[m\; ?\; k\; +\; 1]}}\text{k}$. The inequality is shown to be sharp. When p < 1 and a_(i)'s are in increasing order then the inequality is reversed.     

Elementary proof of the transformation formula for Lambert series involving generalized Dedekind sums

An elementary proof is given of the author's transformation formula for the Lambert series $\text{G}{}_{\mathrm{p}}\text{(x) =}\text{Σ}{}_{\mathrm{n?1}}{}^{\mathrm{<mtext>\infty </mtext>}}\text{n}{}^{\mathrm{?p}}\text{x}{}^{\mathrm{n}}\text{(1?x}{}^{\mathrm{n}}\text{)}$ relating G_p(e^2πiτ) to G_p(e^2πiAτ), where p > 1 is an odd integer and $\text{Aτ =}\text{(aτ + b)}\text{(cτ + d)}$ is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function $\text{log}\text{η(τ) =}\text{πiτ}\text{12}\text{? G}{}_1\text{(e}{}^{\mathrm{2\pi i\tau }}\text{)}$, and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions.     

Series inversion of some convolution transforms

We study degeneration for ? → + 0 of the two-point boundary value problems

$\text{τ?}{}^{\mathrm{\pm }}\text{u := ?((au′)′ + bu′ + cu) ± xu′ ? κu = h, u(±1) = A ± B}$

, and convergence of the operators T_?⁺ and T_?^? on $\text{L}$²(?1, 1) connected with them, T_?^±u := τ_?^±u for all

$\text{u?D(T}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{, D(T}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{) := {u ? L}{}^2\text{(?1, 1) ∣ u″ ? L}{}^2\text{(?1, 1) \&; u(?1) = u(1) = O}, T}{}_0{}^{\mathrm{+}}\text{u: = xu′}$

for all

$\text{u?D(T}{}_{\mathrm{O}}{}^{\mathrm{+}}\text{), D(T}{}_{\mathrm{O}}{}^{\mathrm{+}}\text{) := {u ? L}{}^2\text{(?1, 1) ∣ xu′ ? L}{}^2\text{(?1, 1) \&; u(?1) = u(1) = O}}$

. Here ? is a small positive parameter, λ a complex “spectral” parameter; a, b and c are real $\text{b}$^∞-functions, a(x) ? γ > 0 for all x? [?1, 1] and h is a sufficiently smooth complex function. We prove that the limits of the eigenvalues of T_?⁺ and of T_?^? are the negative and nonpositive integers respectively by comparison of the general case to the special case in which a  1 and b  c  0 and in which we can compute the limits exactly. We show that (T_?⁺ ? λ)^?1 converges for ? → +0 strongly to (T₀⁺ ? λ)^?1 if $\text{R e λ > ?}\text{1}\text{2}$. In an analogous way, we define the operator T_?⁺, n (n ? $\text{N}$ in the Sobolev space H₀^?n(? 1, 1) as a restriction of τ_?⁺ and prove strong convergence of (T⁺_?,n ? λ)^?1 for ? → +0 in this space of distributions if $\text{R e λ > ?n ?}\text{1}\text{2}$. With aid of the maximum principle we infer from this that, if h?$\text{C}$¹, the solution of τ_?⁺u ? λu = h, u(±1) = A ± B converges for ? → +0 uniformly on [?1, ? ?] ∪ [?, 1] to the solution of xu′ ? λu = h, u(±1) = A ± B for each p > 0 and for each λ ? $\text{C}$ if ? ?$\text{N}$.Finally we prove by duality that the solution of τ_?^?u ? λu = h converges to a definite solution of the reduced equation uniformly on each compact subset of (?1, 0) ∪ (0, 1) if h is sufficiently smooth and if 1 ? ?$\text{N}$.     

Asymptotic evaluation of two families of integrals

In this paper the integrals $\text{f}{}_{\mathrm{mv}}\text{(τ) = ∝}{}_0{}^{\mathrm{\infty }}\text{exp}\text{[?(t + τ)]t}{}^{\mathrm{v}}\text{(}\text{ln}\text{t)}{}^{\mathrm{m}}\text{(t + τ)}{}^{\mathrm{?1}}\text{dt}\text{and}\text{g}{}_{\mathrm{mv}}\text{(τ) = ∝}{}_0{}^{\mathrm{\infty }}\text{exp}\text{[? ¦ ? τ ¦]t}{}^{\mathrm{v}}\text{(}\text{ln}\text{t)}{}^{\mathrm{m}}\text{(t ? τ)}{}^{\mathrm{?1}}\text{dt}$ are investigated for positive real values of the variable τ. Here, m is a nonnegative integer, v is a complex variable with Re(v) > ?1. Both integrals are related to the complex integral Φ_mγ(z) = ∝₀^∞exp[?(t ? z)]t^?γ(ln t)^m(t ? z)^?1dt with 0 ? Re(γ) < 1, the behavior of which is analyzed in detail. The results are applied to obtain asymptotic representations for f_mn(τ) and g_mn(τ), m and n both nonnegative integers, near τ = 0. The latter integrals play a role in the study of the equations of neutron transport and radiative transfer.     

On asymptotic solutions of the renewal equation

Consider the renewal equation in the form (¹) $\text{u(t) = g(t) + ∝}{}_{\mathrm{o}}{}^{\mathrm{t}}\text{u(t ? τ) ?(τ) dτ}$, where $\text{?(t)}$ is a probability density on [0, ∞) and lim_{t → ∞}g(t) = g₀. Asymptotic solutions of (¹) are given in the case when f(t) has no expectation, i.e., $\text{∝}{}_0{}^{\mathrm{\infty }}\text{t?(t)dt = ∞}$. These results complement the classical theorem of Feller under the assumption that f(t) possesses finite expectation.     

Weak-1 generators for a class of subalgebras of l1

Let α ? 0 and let $\text{D(}\text{α}\text{) = {f(z) = ∑}{}_0{}^{\mathrm{\infty }}\text{α}{}_{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{¦ ∑}{}_0{}^{\mathrm{\infty }}\text{(n + 1)}{}^{\mathrm{<mtext>\alpha </mtext>}}\text{¦ a}{}_{\mathrm{n}}\text{¦ < ∞}}$. Then D(α) is a subalgebra of l¹. We discuss the weak-1 generators of D(α). We use some of our techniques to prove that if ? is a weak-1 generator of H^∞ and ∥ ? ∥_∞ ? 1, then the composition operator C_? on the Dirichlet space has dense range.     

On the probability that k positive integers are relatively prime II

Given a set S of positive integers let $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t)}$ denote the number of k-tuples 〈m₁, …, m_k〉 for which $\text{m}{}_{\mathrm{i}}\text{∈ S ? [1, t]}$ and (m₁, …, m_k) = 1. Also let $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n)}$ denote the probability that k integers, chosen at random from $\text{S ? [1, n]}$, are relatively prime. It is shown that if P = {p₁, …, p_r} is a finite set of primes and S = {m : (m, p₁ … p_r) = 1}, then $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t) = (td(S))}{}^{\mathrm{k}}\text{Π}{}_{\mathrm{\nu ?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) + O(t}{}^{\mathrm{k?1}}\text{)}$ if k ≥ 3 and $\text{Z}{}_2{}^{\mathrm{S}}\text{(t) = (td(S))}{}^2\text{Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^2\text{) + O(t}\text{log}\text{t)}$ where d(S) denotes the natural density of S. From this result it follows immediately that $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) = (ζ(k))}{}^{\mathrm{?1}}\text{Π}{}_{\mathrm{p\in P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$ as n → ∞. This result generalizes an earlier result of the author's where $\text{P = ?}$ and S is then the whole set of positive integers. It is also shown that if S = {p₁^x₁ … p_r^x_r : x_i = 0, 1, 2,…}, then $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → 0}$ as n → ∞.     

On the L2-stability of a class of nonlinear systems

Sufficient conditions are given for the L₂-stability of a class of feedback systems consisting of a linear operator G and a nonlinear gain function, either odd monotone or restricted by a power-law, in cascade, in a negative feedback loop. The criterion takes the form of a frequency-domain inequality, Re[1 + Z(jω)] G(jω) ? δ > 0 ?ω? (?∞, +∞), where Z(jω) is given by, Z(jω) = β[Y₁(jω) + Y₂(jω)] + (1 ? β)[Y₃(jω) ? Y₃(?jω)], with 0 ? β ? 1 and the functions y₁(·), y₂(·) and y₃(·) satisfying the time-domain inequalities, $\text{∝}{}_{\mathrm{?\infty }}{}^{\mathrm{+\infty }}\text{¦y}{}_1\text{(t) + y}{}_2\text{(t)¦ dt ? 1 ? ?, y}{}_1\text{(·) = 0, t < 0}$, y₂(·) = 0, t > 0 and ? > 0, and $\text{∝}{}_0{}^{\mathrm{\infty }}\text{¦y}{}_3\text{(t)¦ dt <}\text{1}\text{2c}{}_2$, c₂ being a constant depending on the order of the power-law restricting the nonlinear function. The criterion is derived using Zames' passive operator theory and is shown to be more general than the existing criteria.     

Temporal exponential decay for the Stark effect in atoms

Let $\text{H}{}_1\text{= ?∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{(Δ}{}_{\mathrm{i}}\text{+ V(}\text{x}{}_{\mathrm{i}}\text{)) + ∑}{}_{\mathrm{1\; ?\; i\; <j\; ?\; N}}\text{¦}\text{x}{}_{\mathrm{i}}\text{?}\text{x}{}_{\mathrm{j}}\text{¦}{}^{\mathrm{?1}}\text{, V(}\text{x}{}_{\mathrm{i}}\text{) = N ∝ ¦}\text{x}\text{?}\text{y}\text{¦}{}^{\mathrm{?1}}\text{?(}\text{y}\text{)d}\text{y}$, with ? a normalized Gaussian. Suppose $\text{E}$ ≠ 0 and that $\text{H = H}{}_1\text{+}\text{E}\text{· (∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}\text{)}$ has no eigenfunctions in L²($\text{R}$^3N. If H₁ψ = μψ with μ < infσ_ess(H₁), then (ψ, e^?itHψ) decays exponentially at a rate governed by the positions of the resonances of H.     

Operator properties of Sobolev imbeddings over unbounded domains

For an open set Ω ? $\text{R}$^N, 1 ? p ? ∞ and λ ∈ $\text{R}$⁺, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators $\text{U}$, 1 ? p, q ? ∞ and a quasibounded domain Ω ? $\text{R}$^N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the given Banach ideal $\text{U}$: Assume the quasibounded domain fulfills condition C_k^l for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any $\text{x ? Ω}$ to the boundary ?Ω tends to zero as $\text{O(¦ x ¦}{}^{\mathrm{?l}}\text{)}$ for $\text{¦ x ¦ → ∞}$, and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ $\text{N}$, μ > λ _S($\text{U}$; p,q:N) and v > N/l · λ_D($\text{U}$;p,q), one has that $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ belongs to the Banach ideal $\text{U}$. Here λ_D($\text{U}$;p,q;N)∈$\text{R}$⁺ and λ_S($\text{U}$;p,q;N)∈$\text{R}$⁺ are the D-limit order and S-limit order of the ideal $\text{U}$, introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings l_pⁿ → l_qⁿ for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω fulfills condition C₁^l.For an open set Ω in $\text{R}$^N, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in $\text{R}$^N and give sufficient conditions on λ such that the Sobolev imbedding operator $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded open set in $\text{R}$^N.     

Hermite series estimates of a probability density and its derivatives

The following estimate of the pth derivative of a probability density function is examined: $\text{Σ}{}_{\mathrm{k\; =\; 0}}{}^{\mathrm{N}}\text{a}\text{?}{}_{\mathrm{k}}\text{h}{}_{\mathrm{k}}\text{(x)}$, where h_k is the kth Hermite function and $\text{a}\text{?}{}_{\mathrm{k}}\text{= (}\text{(?1)}{}^{\mathrm{p}}\text{n}\text{)}$Σ_{i = 1}ⁿh_k^(p)(X_i) is calculated from a sequence X₁,…, X_n of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n^?α) and O(n^?α log n), respectively, where $\text{α =}\text{2(r ? p)}\text{(2r + 1)}$. Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n^?β) and $\text{O(n}{}^{\mathrm{<mtext>?\beta </mtext><mtext>2</mtext>}}\text{log}\text{n)}$, respectively, where $\text{β =}\text{(2(r ? p) ? 1)}\text{(2r + 1)}$.