The asymptotic distribution of lattice points in hyperbolic space

Suppose x and y are two points in the upper half-plane H⁺, and suppose Γ is a discontinuous group of conformal automorphisms of H⁺ having compact fundamental domain S. Denote by N_T(x, y) the number of points of the form γy (γ?Γ) in the closed disc of hyperbolic radius T centered about x, and set $\text{Q}{}_{\mathrm{T}}\text{(x, y) = N}{}_{\mathrm{T}}\text{(x, y) ?}\text{V(T)}\text{A}\text{,}\text{where}\text{V(T)}$ is the hyperbolic area of the disc, and A is the hyperbolic area of S. The asymptotic behavior of the quantity ?_LxL(Q_T(x,y))² is estimated in terms of small eigenvalues of the Laplacian on functions automorphic under Γ.     

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.     

Arnold's canonical matrices and the asymptotic simplification of ordinary differential equations

Let A(x,ε) be an n×n matrix function holomorphic for |x|?x₀, 0<ε?ε₀, and possessing, uniformly in x, an asymptotic expansion $\text{A(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{A}{}_{\mathrm{r}}\text{(x) ε}{}^{\mathrm{r}}$, as ε→0⁺. An invertible, holomorphic matrix function P(x,ε) with an asymptotic expansion $\text{P(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{P}{}_{\mathrm{r}}\text{(x)ε}{}^{\mathrm{r}}$, as ε→0⁺, is constructed, such that the transformation y = P(x,ε)z takes the differential equation $\text{ε}{}^{\mathrm{h}}\text{dy}\text{dx}\text{= A(x,ε)y,h}$ a positive integer, into $\text{ε}{}^{\mathrm{h}}\text{dz}\text{dx}\text{= B(x,ε)z}$, where B(x,ε) is asymptotically equal, to all orders, to a matrix in a canonical form for holomorphic matrices due to V.I. Arnold.     

On essential self-adjointness for Schrödinger operators with wildly oscillating potentials

A sufficient condition is given for the operator T₀: C^∞₀(R^m) → L²(R^m) given by

$\text{T}{}_0\text{∑}\text{K?1}\text{M}\text{(}\text{i?}\text{?x}{}_1\text{+b}{}_1\text{)a}{}_{\mathrm{1K}}\text{(}\text{i?}\text{?N}{}_{\mathrm{k}}\text{+b}{}_{\mathrm{k}}\text{)+q}$

to be essentially self-adjoint. This condition is sufficiently general to admit certain potentials q having unbounded oscillations in a neighborhood of ∞.     

Barriers for uniformly elliptic equations and the exterior cone condition

We consider the mixed boundary value problem $\text{Au = f}\text{in}\text{Ω, B}{}_0\text{u = g}{}_0\text{in}\text{Γ}{}^{\mathrm{?}}\text{, B}{}_1\text{u = g}{}_1\text{in}\text{Γ}{}^{\mathrm{+}}$, where Ω is a bounded open subset of $\text{R}$ⁿ whose boundary Γ is divided into disjoint open subsets Γ⁺ and Γ^? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on $\text{Ω}$ and B_j, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on $\text{Γ}{}^{\mathrm{+}}$. The coefficients of the operators and Γ⁺, Γ^? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset $\text{T}$ of the reals such that, if $\text{D}{}^{\mathrm{s}}\text{= {u ? H}{}^{\mathrm{s}}\text{(Ω): Au = 0}}$ then for $\text{s ? =}\text{1}\text{2}\text{(}\text{mod}\text{1), (B}{}_0\text{,B}{}_1\text{): D}{}^{\mathrm{s}}\text{→ H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{?}}\text{) × H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>3</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{+}}\text{)}$ is a Fredholm operator if and only if s ∈$\text{T}$ . Moreover, $\text{T}$ = ?_xew$\text{T}$_x, where the sets $\text{T}$_x are determined algebraically by the coefficients of the operators at x. If n = 2, $\text{T}$_x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, $\text{T}$_x is either an open interval of length 1 or is empty; and finally, if n ? 4, $\text{T}$_x is an open interval of length 1.     

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 ? ?$\text{N}$.     

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).     

A theory for optimal regularization in the finite dimensional case

Consider the matrix problem $\text{Ax = y + ε =}\text{y}\text{?}$ in the case where A is known precisely, the problem is ill conditioned, and ε is a random noise vector. Compute regularized “ridge” estimates,$\text{x}\text{?}{}_{\mathrm{\lambda }}\text{= (A}{}^1\text{A + λI)}{}^{-1}\text{A}{}^1\text{y}\text{?}$,where ¹ denotes matrix transpose. Of great concern is the determination of the value of λ for which x?_λ “best” approximates $\text{x}{}_{\mathrm{<mtext>0</mtext>}}\text{= A}{}^{\mathrm{+}}\text{y}$. Let $\text{Q = 6}\text{x}\text{?}{}_{\mathrm{\lambda }}\text{? x}{}_{\mathrm{<mtext>0</mtext>}}\text{6}{}^2$,and define λ₀ to be the value of λ for which Q is a minimum. We look for λ₀ among solutions of dQ/dλ = 0. Though Q is not computable (since ε is unknown), we can use this approach to study the behavior of λ₀ as a function of y and ε. Theorems involving “noise to signal ratios” determine when λ₀ exists and define the cases λ₀ > 0 and λ₀ = ∞. Estimates for λ₀ and the minimum square error Q₀ = Q(λ₀) are derived.     

Global existence for a delay differential equation

An elastic-plastic bar with simply connected cross section Q is clamped at the bottom and given a twist at the top. The stress function u, at a prescribed cross section, is then the solution of the variational inequality (0.1) $\text{min}{}_{\mathrm{<mtext>v</mtext><mtext>?</mtext><mtext>K</mtext>}}\text{{∝}{}_{\mathrm{Q}}\text{¦}\text{▽}\text{v¦}{}^2\text{? 2θ}{}_1\text{∝}{}_{\mathrm{Q}}\text{v} = ∝}{}_{\mathrm{Q}}\text{¦}\text{▽}\text{u¦}{}^2\text{? 2θ}{}_1\text{∝}{}_{\mathrm{Q}}\text{u, u}\text{?}\text{K,}\text{where}\text{(0.2) K = {v}\text{?}\text{H}{}_0{}^1\text{(Q), ¦}\text{▽}\text{v¦ ? 1}\text{a.e.}\text{}}\text{and}\text{θ}{}_1$ is equal to the angle of the twist (after normalizing the units). Introducing the Lagrange multiplier λ_θ1, the unloading problem consists in solving the variational inequality (0.3) is the twisting angle for the unloaded bar; θ₂ < θ₁. Let (0.4) $\text{K}{}^1\text{= {v}\text{?}\text{H}{}_0{}^1\text{(Q), ?d(x) ? v(x) ? d(x)},}\text{where}\text{d(x) =}\text{dist}\text{.(x, ?Q)}$, and denote by $\text{u}{}^1\text{, w}{}^1$ the solutions of (0.1), (0.3), respectively, when K is replaced by $\text{K}{}^1$. The following results are well known for the loading problem (0.1):(0.5) $\text{u = u}{}^1$; (0.6) the plastic set $\text{P = (X}\text{?}\text{Q}\text{?}\text{; ¦}\text{▽}\text{u(x)¦ = 1}}$ is connected to the boundary. In this paper we show that, in general, (0.7) $\text{w ≠ w}{}^1$; (0.8) the plastic set $\text{P}\text{?}\text{= {x}\text{?}\text{Q}\text{?}\text{; ¦}\text{▽}\text{w(x)¦ = 1}}$ is not connected to the boundary. That is, we construct domains Q for which (0.7) and (0.8) hold for a suitable choice of θ₁, θ₂.     

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),

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

.     

Minimum variance estimators for misclassification probabilities in discriminant analysis

Let α(n₁, n₂) be the probability of classifying an observation from population Π₁ into population Π₂ using Fisher's linear discriminant function based on samples of size n₁ and n₂. A standard estimator of α, denoted by T₁, is the proportion of observations in the first sample misclassified by the discriminant function. A modification of T₁, denoted by T₂, is obtained by eliminating the observation being classified from the calculation of the discriminant function. The UMVU estimators, $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$, of ET₁ = τ₁(n₁, n₂) and ET₂ = τ₂(n₁, n₂) = α(n₁ ? 1, n₂) are derived for the case when the populations have multivariate normal distributions with common dispersion matrix. It is shown that $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$ are nonincreasing functions of D², the Mahalanobis sample distance. This result is used to derive the sampling distributions and moments of $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$. It is also shown that α is a decreasing function of Δ² = (μ₁ ? μ₂)′Σ^?1(μ₁ ? μ₂). Hence, by truncating $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$ (or any estimator) at the value of α for Σ = 0, new estimators are obtained which, for all samples, are as close or closer to α.     

Decomposability of symmetric multilinear functions

Let V denote a finite dimensional vector space over a field K of characteristic 0, let T_n(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let S_n(V) denote the subspace of T_n(V) whose members are the fully symmetric members of T_n(V). If $\text{L}$_n denotes the symmetric group on {1,2,…,n} then we define the projection $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{: T}{}_{\mathrm{n}}\text{(V) → S}{}_{\mathrm{n}}\text{(V)}$ by the formula , where P_σ : T_n(V) → T_n(V) is defined so that P_σ(A)(y₁,y₂,…,y_n = A(y_σ(1),y_σ(2),…,y_σ(n)) for each A?T_n(V) and y_i?V, 1 ? i ? n. If $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, then x₁?x₂? … ?x_n denotes the member of T_n(V) such that $\text{(x}{}_1\text{?x}{}_2\text{· ? ? ?x}{}_{\mathrm{n}}\text{)(y}{}_1\text{,y}{}_2\text{,…,y}{}_{\mathrm{n}}\text{) = П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{x}{}_{\mathrm{i}}\text{(y}{}_{\mathrm{i}}\text{)}$ for each y₁ ,₂,…,y_n in V, and x₁·x₂… x_n denotes $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{(x}{}_1\text{?x}{}_2\text{? … ?x}{}_{\mathrm{n}}\text{)}$. If B? S_n(V) and there exists $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, such that B = x₁·x₂…x_n, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of S_n(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C.     

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)}$.     

Stationary queues with one autonomous server

Given a stationary stochastic continuous demand of service σ(θ_tω) dt with ∫ σ(ω)P(dω) < 1, we construct real stationary point processes $\text{(T}{}_{\mathrm{n}}\text{, n ∈}\text{Z}\text{)[T}{}_{\mathrm{n}}\text{< T}{}_{\mathrm{n}}\text{+1, lim}{}_{\mathrm{\pm \infty }}\text{T}{}_{\mathrm{n}}\text{= ±∞]}$ such that

for a given constant D \2>0. These point processes correspond to a service discipline for which a single server services during the time intervals [T_n, T_n+1[ the demand of service accumulated during the proceding intervals [T_n?1, T_n[ and take a rest of fixed duration D.     

Periodic solutions of a quasilinear parabolic differential equation

This paper treats the quasilinear, parabolic boundary value problem $\text{u}{}_{\mathrm{xx}}\text{? u}{}_{\mathrm{t}}\text{= ??(x, t, u)}$u(0, t) = ?₁(t); u(l, t) = ?₂(t) on an infinite strip $\text{{(x, t) ¦ 0 < x < l, ?∞ < t < ∞}}$ with the functions $\text{?(x, t, u), ?}{}_1\text{(t), ?}{}_2\text{(t)}$ being periodic in t. The major theorem of the paper gives sufficient conditions on $\text{?(x, t, u)}$ for this problem to have a periodic solution u(x, t) which may be constructed by successive approximations with an integral operator. Some corollaries to this theorem offer more explicit conditions on $\text{?(x, t, u)}$ and indicate a method for determining the initial estimate at which the iteration may begin.     

An integral operator connected with the Helmholtz equation

Let Lu be the integral operator defined by $\text{(L}{}_{\mathrm{k?}}\text{)(x, y) = ∝ s ∝ ?(x′, y′)(}\text{e}{}^{\mathrm{ik?}}\text{?}\text{) dx′ dy′, (x, y) ? S}$ where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ? 0, Im k ? 0, and ?² = (x ?x′)² + (y ? y′)². We define $\text{q(x, y) = [}\text{dist}\text{((x, y), ?S)]}\text{1}\text{2}\text{, (x, y) ? S; L}{}_2\text{(q, S) = {? : ∝ s ∝ ¦ ?(x, y)¦}{}^2\text{q(x, y) dx dy < ∞}; W}{}_2{}^1\text{(q, S) = {? : ? ? L}{}_2\text{(q, S),}\text{??}\text{?x}\text{,}\text{?f}\text{?y}\text{? L}{}_2\text{(q, S)}}$, where in the definition of W₂¹(q, S) the derivatives are taken in the sense of distributions. We prove that L_k is a continuous 1-l mapping of L₂(q, S) onto W₂¹(q, S).     

Comparaison de deux indicateurs de la régularité en variation des queues de distributionsComparison between two indicators for the variation regularity of tails of distributions

We compare assumptions used in [4] in order to study the rate of convergence to 0, as u→s₊(F), of , where $\text{F}{}_{\mathrm{u}}$ is the survival function of the excesses over u, s₊(F)=sup{x,F(x)<1} is the upper end point of the distribution function (d.f.) F and $\text{G}{}_{\mathrm{\gamma }}$ is the survival function of the Generalized Pareto Distribution, with assumptions used in [2] in order to study the rate of convergence to 0, as n→+∞, of $\text{d}\text{?}{}_{\mathrm{n}}\text{=}\text{sup}{}_{\mathrm{<mtext>x\in </mtext><mtext>R</mtext>}}\text{|F}{}^{\mathrm{n}}\text{(x)?H}{}_{\mathrm{\gamma }}\text{(}\text{x?α}{}_{\mathrm{n}}\text{σ}{}_{\mathrm{n}}\text{)|}$, where H_γ is the d.f. of an extreme value distribution. In each case, an indicator linked to regular variation assumptions had been introduced. We characterize situations where these two indicators coincide, and others where they are different. To cite this article: R. Worms, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 709–712.     

Construction of a recurrence relation for modified moments

In this paper we are constructing a recurrence relation of the form

$\text{∑}\text{i=0}\text{r}\text{ω}{}_{\mathrm{i}}\text{(k)m}{}_{\mathrm{k+i}}{}^{\mathrm{\{\lambda \}}}\text{[f] = ω(k)}$

for integrals (called modified moments)

$\text{m}{}_{\mathrm{k}}{}^{\mathrm{\{\lambda \}}}\text{[f]df=}\text{∫}\text{?1}\text{1}\text{f(x)C}{}_{\mathrm{k}}{}^{\mathrm{(\lambda )}}\text{(x)dx (k = 0,1,…)}$

in which C_k^(λ) is the k-th Gegenbauer polynomial of order $\text{λ(λ > ?}\text{1}\text{2}\text{)}$, and f is a function satisfying the differential equation

$\text{∑}\text{i=0}\text{n}\text{P}{}_{\mathrm{i}}\text{(x)f}{}^{\mathrm{(i)}}\text{(x) = p(x) (?1?x?1)}$

of order n, where p₀, p₁, …, p_n ? 0 are polynomials, and m_k^〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense.     

Concrete model theory for a class of operators

The operator $\text{L?(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∝}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∝}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ acting on H=∝₀^2π⊕L²(v_t), where m and v_t, 0 ? t ? 2π are measures on [0, 2π] with m smooth and e(s, t) = exp[?∝_t^s∝_Tdv_λ(θ) dm(λ)], satisfies $\text{rank}\text{(I ? LL}{}^1\text{) =}\text{rank}\text{(I ? L}{}^1\text{L) = 1}$. It is, therefore, unitarily equivalent to a scalar Sz.-Nagy-Foia? canonical model. The purpose of this paper is to determine the model explicitly and to give a formula for the unitary equivalence.