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.     

On an estimate for Bloch and Doob norm and covering problem in Doob's class

For any fixed 0 < π ? 2π, let D(π) be the family of all holomorphic functions in the unit disk Δ which satisfy (i)f(0) = 0 and (ii) $\text{lim inf}{}_{\mathrm{z\; \to \; \pi ¦f(z)¦\; ?\; 1}}$, for all π lying on some arc A_f ? ?Δ with arclength $\text{¦A}{}_{\mathrm{f}}\text{¦ ? π}$. We show that for each 0 < ε < 1, there is a π₀ > 0 such that for any f?D(π) with π < π₀, the Bloch and Doob norm respectively satisfy

$\text{6f6}{}_{\mathrm{B}}\text{=}\text{sup}\text{z?Δ}\text{|f′(z)| (1?|z|}{}^2\text{) > 2(1 ? ε)}\text{log}\text{1+}\text{cos}\text{(}\text{p}\text{2}\text{1?}\text{cos}\text{(}\text{p}\text{2}{}^{\mathrm{?1}}$

$\text{6f6}{}_{\mathrm{D}}\text{=}\text{sup}\text{z?Δ}\text{|f′(z)| (1?|z|) > (1 ? ε)}\text{log}\text{1}\text{1?}\text{cos}\text{(}\text{p}\text{2}{}^{\mathrm{?1}}$

These two estimates do not hold with ε = 0.     

Positive definite distributions on K\GK

Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space $\text{C}$¹(G). We show that the corresponding property is no longer true for the space of double cosets $\text{K}\text{G}\text{K}$. If G is of real-rank 1, we construct liner functionals $\text{T}{}_{\mathrm{p}}\text{? (C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{))′}$ for each p, 0 < p ? 2, such that $\text{T}{}_{\mathrm{p}}\text{(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$ but T_p does not extend to a continuous functional on $\text{C}{}^{\mathrm{p}}\text{(K}\text{G}\text{K}\text{)}$. In particular, if p ? 1, T_v does not extend to a continuous functional on $\text{C}{}^1\text{(K}\text{G}\text{K}\text{)}$. We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that $\text{T(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$. The main tool used is a theorem of Trombi-Varadarajan.     

Concavity of certain maps on positive definite matrices and applications to Hadamard products

If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Löwner and if Φ₁ and Φ₂ are concave maps among positive definite matrices, then the following map involving tensor products:

$\text{(A,B)?f[Φ}{}_1\text{(A)}{}^{\mathrm{?1}}\text{?Φ}{}_2\text{(B)]·(Φ}{}_1\text{(A)?I)}$

is proved to be concave. If Φ₁ is affine, it is proved without use of positivity that the map

$\text{(A,B)?f[Φ}{}_1\text{(A)?Φ}{}_2\text{(B)}{}^{\mathrm{?1}}\text{]·(Φ}{}_1\text{(A)?I)}$

is convex. These yield the concavity of the map

$\text{(A,B)?A}{}^{\mathrm{1?p}}\text{?B}{}^{\mathrm{p}}$

(0<p?1) (Lieb's theorem) and the convexity of the map

$\text{(A,B)?A}{}^{\mathrm{1+p}}\text{?B}{}^{\mathrm{?p}}$

(0<p?1), as well as the convexity of the map

$\text{(A,B)?(A·log[A])?I?A?log[B]}$

.These concavity and convexity theorems are then applied to obtain unusual estimates, from above and below, for Hadamard products of positive definite matrices.     

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.     

Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

Enumeration of pairs of permutations

Let π = (a₁, a₂, …, a_n), ? = (b₁, b₂, …, b_n) be two permutations of $\text{Z}{}_{\mathrm{n}}\text{= {1, 2, …, n}}$. A rise of π is pair a_i, a_i+1 with a_i < a_i+1; a fall is a pair a_i, a_i+1 with a_i > a_i+1. Thus, for i = 1, 2, …, n ? 1, the two pairs a_i, a_i+1; b_i, b_i+1 are either both rises, both falls, the first a rise and the second a fall or the first a fall and the second a rise. These possibilities are denoted by RR, FF, RF, FR. The paper is concerned with the enumeration of pairs π, p with a given number of RR, FF, RF, FR. In particular if ω_n denotes the number of pairs with RR forbidden, it is proved that $\text{∑}{}^{\mathrm{\infty }}{}_0\text{ω}{}_{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{n!n!}\text{=}\text{1}\text{?(z)}$, $\text{?(z) = ∑}{}^{\mathrm{\infty }}{}_0\text{(-1)}{}^{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{n!n!}$. More precisely if ω(n, k) denotes the number of pairs π, p with exactly k occurences of RR(or FF, RF, FR) then $\text{1 + ∑}{}^{\mathrm{\infty }}{}_{\mathrm{n=1}}\text{z}{}^{\mathrm{n}}\text{n!n!}$$\text{∑}{}^{\mathrm{n?1}}{}_{\mathrm{k=0}}\text{ω(n, k)x}{}^{\mathrm{k}}\text{=}\text{(1 ? x)}\text{(?(z(1 ? x)) ? x)}$.     

Critical exponents for the Pucci's extremal operatorsLes exposants critiques pour l'opérateur extrémal de Pucci

In this Note we present some results on the existence of radially symmetric solutions for the nonlinear elliptic equation

(1)$\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}\text{(D}{}^2\text{u)+u}{}^{\mathrm{p}}\text{=0,u?0}\text{in}\text{R}{}^{\mathrm{N}}\text{.}$

Here N?3, p>1 and $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}$ denotes the Pucci's extremal operators with parameters 0<λ?Λ. The goal is to describe the solution set as function of the parameter p. We find critical exponents $\text{1<p}{}^{\mathrm{s}}{}_{\mathrm{+}}\text{<p}{}^1{}_{\mathrm{+}}\text{<p}{}^{\mathrm{p}}{}_{\mathrm{+}}$, that satisfy: (i) If $\text{1<p<p}{}^1{}_{\mathrm{+}}$ then there is no nontrivial solution of ($\text{1}$). (ii) If $\text{p=p}{}^1{}_{\mathrm{+}}$ then there is a unique fast decaying solution of ($\text{1}$). (iii) If $\text{p}{}^1\text{<p?p}{}^{\mathrm{p}}{}_{\mathrm{+}}$ then there is a unique pseudo-slow decaying solution to ($\text{1}$). (iv) If p^p₊<p then there is a unique slow decaying solution to ($\text{1}$). Similar results are obtained for the operator $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{?}}$. To cite this article: P.L. Felmer, A. Quaas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 909–914.     

Eigenvalues of p-summing and lp-type operators in Banach spaces

This paper is a study of the distribution of eigenvalues of various classes of operators. In Section 1 we prove that the eigenvalues (λ_n(T)) of a p-absolutely summing operator, p ? 2, satisfy

$\text{∑}\text{n∈N}\text{|λ}{}_{\mathrm{n}}\text{(T)|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{?π}{}_{\mathrm{p}}\text{(T).}$

This solves a problem of A. Pietsch. We give applications of this to integral operators in L_p-spaces, weakly singular operators, and matrix inequalities.In Section 2 we introduce the quasinormed ideal Π₂⁽ⁿ⁾, P = (p₁, …, p_n) and show that for T ∈ Π₂⁽ⁿ⁾, 2 = (2, …, 2) ∈ Nⁿ, the eigenvalues of T satisfy

$\text{∑}\text{i∈N}\text{|λ}{}_{\mathrm{i}}\text{(T)|}{}^{\mathrm{<mtext>2</mtext><mtext>n</mtext>}}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}\text{?π}{}^{\mathrm{n}}{}_2\text{(T).}$

More generally, we show that for T ∈ Π_p⁽ⁿ⁾, P = (p₁, …, p_n), pi ? 2, the eigenvalues are absolutely p-summable,

$\text{1}\text{p}\text{=}\text{∑}\text{i=1}\text{n}\text{1}\text{p}{}_{\mathrm{i}}\text{and}\text{∑}\text{n∈N}\text{|λ}{}_{\mathrm{n}}\text{(T)|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{?C}{}_{\mathrm{p}}\text{π}{}^{\mathrm{n}}{}_{\mathrm{P}}\text{(T).}$

We also consider the distribution of eigenvalues of p-nuclear operators on L_r-spaces.In Section 3 we prove the Banach space analog of the classical Weyl inequality, namely

$\text{∑}\text{n∈N}\text{|λ}{}_{\mathrm{n}}\text{(T)|}{}^{\mathrm{p}}\text{? C}{}_{\mathrm{p}}\text{∑}\text{n∈N}\text{α}{}_{\mathrm{n}}\text{(T)}{}^{\mathrm{p}}$

, 0 < p < ∞, where α_n denotes the Kolmogoroff, Gelfand of approximation numbers of the operator T. This solves a problem of Markus-Macaev.Finally we prove that Hilbert space is (isomorphically) the only Banach space X with the property that nuclear operators on X have absolutely summable eigenvalues. Using this result we show that if the nuclear operators on X are of type l₁ then X must be a Hilbert space.     

On a conjecture by D. Ž. D̄okovic

It is shown that if $\text{A?Ω}{}_{\mathrm{n}}\text{?{J}{}_{\mathrm{n}}\text{}}$ satisfies

$\text{nkσ}{}_{\mathrm{k}}\text{(A)?(n?k+1)}{}^2\text{σ}{}_{\mathrm{k?1}}\text{(A)}$

$\text{(k=1,2,…,n)}$

, where σ_k(A) denotes the sum of all kth order subpermanent of A, then Per[λJ_n+(1?λ)A] is strictly decreasing in the interval 0<λ<1.     

Two-parameter Turan inequalities for ultraspherical and Laguerre polynomials

It is known that the classical orthogonal polynomials satisfy inequalities of the form U_n²(x) ? U_{n + 1}(x) U_{n ? 1}(x) > 0 when x lies in the spectral interval. These are called Turan inequalities. In this paper we will prove a generalized Turan inequality for ultraspherical and Laguerre polynomials. Specifically if P_n^λ(x) and L_n^α(x) are the ultraspherical and Laguerre polynomials and $\text{F}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(x) =}\text{P}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(x)}\text{P}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(1)}\text{, G}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) =}\text{L}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x)}\text{L}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(0)}\text{,}\text{then}\text{F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? F}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > 0, ? 1 < x < 1, ?}\text{1}\text{2}\text{< α ? β ? α + 1}\text{and}\text{G}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) G}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? G}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) G}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > 0, x > 0, 0 < α ? β ? α + 1}$. We also prove the inequality $\text{(n + 1) F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? nF}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > A}{}_{\mathrm{n}}\text{[F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x)]}{}^2\text{, ?1 < x < 1, ?}\text{1}\text{2}\text{< α ? β < α + 1,}\text{where}\text{A}{}_{\mathrm{n}}$ is a positive constant depending on α and β.     

L-spaces and S-spaces in P(ω)

We show that CH implies that $\text{P}\text{(ω)}$, when equipped with the Vietoris topology, has a subspace which is an L-space and a subspace which is an S-space. This is an immediate consequence of the following purely combinatorial result: CH implies the existence of an ω₁-sequence 〈x_α: α < ω₁〉 in $\text{P}\text{(ω)}$ such that (1) if α<β<ω₁, then $\text{X}{}_{\mathrm{\beta }}\text{?}{}^1\text{X}{}_{\mathrm{\alpha }}$; (2) if I ?ω₁ is unaccountable, then there are distinct α, β ∈ I with X_β ?X_α.     

On a class of random Schrödinger operators

Considered are random Schrödinger operators on L²(R^d) that are stationary and metrically transitive with respect to a lattice e.g., independent identically distributed. A method of carrying over results from the case of potentials metrically transitive with respect to R^d is presented. Among these results are the Thouless formula and Kotani's theory.     

The minimal generating space of a polynomial

Given a polynomial $\text{P(X}{}_1\text{,…,X}{}_{\mathrm{N}}\text{)∈}\text{R}\text{[}\text{X}\text{]}$, we calculate a subspace G_p of the linear space 〈X〉 generated by the indeterminates which is minimal with respect to the property $\text{P∈}\text{R}\text{[}\text{G}{}_{\mathrm{p}}\text{]}$ (the algebra generated by G_p, and prove its uniqueness. Furthermore, we use this result to characterize the pairs (P,Q) of polynomials P(X₁,…,X_n) and Q(X₁,…,X_n) for which there exists an isomorphism T:〈X〉 →〈X〉 that “separates P from Q,” i.e., such that for some k(1<k<n) we can write P and Q as $\text{P}{}^1\text{(Y}{}_1\text{,…,Y}{}_{\mathrm{k}}\text{)}$ and $\text{Q}{}^1\text{(Y}{}_{\mathrm{<mtext>k+</mtext><mtext>1</mtext>}}\text{,…,Y}{}_{\mathrm{n}}\text{)}$ respectively, where $\text{Y}\text{=T}\text{X}$.     

Qualitative behavior of special functions on compact symmetric spaces

For a symmetric space $\text{G}\text{K}$ of compact type, the highest-weight vectors for representations of G occurring in $\text{L}{}^2\text{(G}\text{K)}$ become heavily concentrated near certain submanifolds of $\text{G}\text{K}$ as the highest weight goes to infinity. This fact is applied to obtain estimates for the spectral measures of the operators qλ = P_λqP_λ, where $\text{P}{}_{\mathrm{\lambda }}\text{: L}{}^2\text{(}\text{G}\text{K}\text{) → V}{}_{\mathrm{\lambda }}$ is an orthogonal projection onto a G-irreducible summand, and q: G/K → $\text{R}$ is a continuous function acting on $\text{L}{}^2\text{(G}\text{K)}$ by multiplication.     

The sum of aΩ(n) over integers n ≤ x with all prime factors between a and y

For a > 0 let $\text{ψ}{}_{\mathrm{a}}\text{(x, y) =}\text{Σ}\text{a}{}^{\mathrm{\Omega (n)}}$, the sum taken over all n, 1 ≤ n ≤ x such that if p is prime and p|n then a < p ≤ y. It is shown for u < about ($\text{log log}\text{x}\text{log log log}\text{x}$) that $\text{ψ}{}_{\mathrm{a}}\text{(x, x}{}^{\mathrm{<mtext>1</mtext><mtext>u</mtext>}}\text{) ? x(}\text{log}\text{x)}{}^{\mathrm{a?1}}\text{p}{}_{\mathrm{a}}\text{(u)}$, where p_a(u) solves a delay differential equation much like that for the Dickman function p(u), and the asymptotic behavior of p_a(u) is worked out.     

Universal bounds for global solutions of a forced porous medium equation

We show that in a smooth bounded domain $\text{Ω⊂}\text{R}{}^{\mathrm{n}}$, n⩾2, all global nonnegative solutions of u_t−Δu^m=u^p with zero boundary data are uniformly bounded in $\text{Ω×(τ,∞)}$ by a constant depending on $\text{Ω,p}$ and τ but not on u₀, provided that 1<m<p<[(n+1)/(n−1)]m. Furthermore, we prove an a priori bound in $\text{L}{}^{\mathrm{\infty }}\text{(Ω×(0,∞))}$ depending on $\text{||u}{}_0\text{||}{}_{\mathrm{L\infty (\Omega )}}$ under the optimal condition 1<m<p<[(n+2)/(n−2)]m.     

Partition theorems for systems of finite subsets of integers

We study generalizations of Ramsey theorem to systems of finite subsets of ω. A system $\text{L}$ of finite subsets of ω is called to be Ramsey if for every partition $\text{L}$=$\text{L}$₁∪$\text{L}$₂ there exists an infinite set Y?ω such that $\text{L}{}_1\text{∩[Y]}{}^{\mathrm{<\omega }}\text{=0 or}\text{L}{}^2\text{∩[Y]}{}^{\mathrm{<\omega }}\text{=0}$. We give some sufficient conditions for a system to be Ramsey. We also prove a theorem which concerns partitions into infinitely many classes. This may be regarded as a common generalization of Erdös-Rado and Nash-Williams theorems.