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This article is concerned with investigations on a phase transition which is related to the (finite) Ramsey theorem and the Paris-Harrington theorem. For a given number-theoretic function g, let R-c(d)(g)(k) be the least natural number R such that for all colourings P of the d-element subsets of {0, . . . ,R - 1} with at most c colours there exists a subset H of {0, . . . , R - 1} such that P has constant value on all d-element subsets of H and such that the cardinality of H is not smaller than max{k, g(min(H))}. If g is a constant function with value e, then R-c(d)(g)(k) is equal to the usual Ramsey number R-c(d)(max{e, k}); and if g is the identity function, then R-c(d)(g)(k) is the corresponding Paris-Harrington number, which typically is much larger than R-c(d)(k). In this article we give for all d >= 2 a sharp classification of the functions g for which the function m bar right arrow R-m(d)(g)(m) grows so quickly that it is no longer provably total in the subsystem of Peano arithmetic, where the induction scheme is restricted to formulas with at most (d - 1)-quantifiers. Such a quick growth will in particular happen for any function g growing at least as fast as i bar right arrow epsilon . log(. . . (log(i). . .)(sic)(d-1)-times (where epsilon > 0 is fixed) but not for the function g(i) = 1/log*(i) . log(. . . (log( i). . .).(sic)(d-1)-times (Here log* denotes the functional inverse of the tower function.) To obtain such results arid even sharper bounds we employ certain suitable transfinite iterations of nonconstructive lower bound functions for Ramsey numbers. Thereby we improve certain results from the article A classification of rapidly growing Ramsey numbers (PAMS 132 (2004), 553-561) of the first author, which were obtained by employing constructive ordinal partitions.

In the present paper, we generalize the method suggested in an earlier paper by the author and overcome its main deficiency. First, we modify the well-known Prony method, which subsequently will be utilized for recovering exactly the locations of jump discontinuities and the associated jumps of a piecewise constant function by means of its Fourier coefficients with respect to any system of the classical orthogonal polynomials. Next, we will show that the method is applicable to a wider class of functions, namely, to the class of piecewise smooth functions—for functions which piecewise belong to C 2 [ − 1 , 1 ] C^2[-1,1] , the locations of discontinuities are approximated to within O ( 1 / n ) O(1/n) by means of their Fourier-Jacobi coefficients. Unlike the previous one, the generalized method is robust, since its success is independent of whether or not a location of the discontinuity coincides with a root of a classical orthogonal polynomial. In addition, the error estimate is uniform for any [ c , d ] ⊂ ( − 1 , 1 ) [c,d]\subset (-1,1) . To the end, we discuss the accuracy, stability, and complexity of the method and present numerical examples.

The discrete Cesàro operator C C associates to a given complex sequence s = { s n } s = \{s_n\} the sequence C s ≡ { b n } Cs \equiv \{b_n \} , where b n = s 0 + ⋯ + s n n + 1 , n = 0 , 1 , … b_n = \frac {s_0 + \dots + s_n}{n +1}, n = 0, 1, \ldots . When s s is a convergent sequence we show that { C n s } \{C^n s \} converges under the sup-norm if, and only if, s 0 = lim n → ∞ s n s_0 = \lim _{n\rightarrow \infty } s_n . For its adjoint operator C ∗ C^* , we establish that { ( C ∗ ) n s } \{(C^*)^n s\} converges for any s ∈ ℓ 1 s \in \ell ^1 . The continuous Cesàro operator, C f ( x ) ≡ 1 x ∫ 0 x f ( s ) d s C\!f (x) \ \equiv \ \frac {1}{x} \int _{0}^ {x}\, f(s) ds , has two versions: the finite range case is defined for f ∈ L ∞ ( 0 , 1 ) f \in L^\infty (0,1) and the infinite range case for f ∈ L ∞ ( 0 , ∞ ) f \in L^\infty (0, \infty ) . In the first situation, when f : [ 0 , 1 ] → C f: [0, 1] \rightarrow \mathbb {C} is continuous we prove that { C n f } \{C^n f \} converges under the sup-norm to the constant function f ( 0 ) f(0) . In the second situation, when f : [ 0 , ∞ ) → C f: [0, \infty )\rightarrow \mathbb {C} is a continuous function having a limit at infinity, we prove that { C n f } \{C^n f \} converges under the sup-norm if, and only if, f ( 0 ) = lim x → ∞ f ( x ) f(0) = \lim _{x\rightarrow \infty }f(x) .

For T > 0, by applying Arthur's analytic truncation AT to the constant function 1 on G(F)\G(A)1, we obtain a characteristic function of a compact subset i(T) of G(F)\G(A)1. The aim of this paper is to give a general formula for the volume of a(T), which is an exponential polynomial function of T. Consequently, by letting T -* oo, we obtain a new way to evaluate the volume of fundamental domain for G(F)\G(A)1. While a(T) seems to be quite arbitrary, its volume admits beautiful structures: (i) Surprisingly enough, the coefficients of T are special values of zeta functions. (ii) The volume is an alternating sum associated to standard parabolic subgroups. ((ii) is suggested by geometry after all, a(T) is obtained from G(F)\G(A)1 by truncating off cuspidal regions which are known to be parametrized by standard parabolic subgroups.) Besides all basic properties of Arthur's analytic truncation, our method is based on a fundamental formula of Jacquet-Lapid-Rogawski on an integration of truncated Eisenstein series associated to cusp forms. Their formula, an advanced version of the Rankin-Selberg method, is obtained using the so-called Bernstein principle via regularized integration over cones.

We extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz'ya-Shaposhnikova (2002), on limits of Sobolev spaces, to the setting of interpolation scales. This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points. A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained. We also give a new approach to the problem of how to recognize constant functions via Sobolev conditions.

In the present paper we generalize Eckhoff's method, i.e., the method for approximating the locations of discontinuities and the associated jumps of a piecewise smooth function by means of its Fourier-Chebyshev coefficients. A new method enables us to approximate the locations of discontinuities and the associated jumps of a discontinuous function, which belongs to a restricted class of the piecewise smooth functions, by means of its Fourier-Jacobi coefficients for arbitrary indices. Approximations to the locations of discontinuities and the associated jumps are found as solutions of algebraic equations. It is shown as well that the locations of discontinuities and the associated jumps are recovered exactly for piecewise constant functions with a finite number of discontinuities. In addition, we study the accuracy of the approximations and present some numerical examples.

A topological space Y is said to be g-contractible provided that there exists a continuous onto function f: Y → Y such that f is homotopic to a constant function. Answering a question by Sam B. Nadler, Jr., in this paper we construct a metric continuum Z such that its hyperspace of subcontinua C(Z) is not g-contractible.

We show that the property (P) for every Darboux function g: R → R there exists a continuous nowhere constant function f: R → R such that f + g is Darboux follows from the following two propositions: (A) for every subset S of R of cardinality c there exists a uniformly continuous function f: R → [0, 1] such that f[S] = [0, 1], (B) for an arbitrary function h: R → R whose image h[R] contains a nontrivial interval there exists an A C R of cardinality c such that the restriction h | A of h to A is uniformly continuous, which hold in the iterated perfect set model.

Diamond, Pomerance and Rubel (1981) proved that there are subsets M M of the complex plane such that for any two entire functions f f and g g if f [ M ] = g [ M ] f[M]=g[M] , then f = g f=g . Baraducci and Dikranjan showed in 1993 that the continuum hypothesis (CH) implies the existence of a similar set M ⊂ R M\subset \mathrm {R} for the class C n ( R ) C_n(\mathbb {R}) of continuous nowhere constant functions from R \mathrm {R} to R \mathrm {R} , while it follows from the results of Burke and Ciesielski (1997) and Ciesielski and Shelah that the existence of such a set is not provable in ZFC. In this paper we will show that for several well-behaved subclasses of C ( R ) C(\mathrm {R}) , including the class D 1 D^1 of differentiable functions and the class A C AC of absolutely continuous functions, a set M M with the above property can be constructed in ZFC. We will also prove the existence of a set M ⊂ R M\subset \mathbb {R} with the dual property that for any f , g ∈ C n ( R ) f,g\in C_n(\mathrm {R}) if f − 1 [ M ] = g − 1 [ M ] f^{-1}[M]=g^{-1}[M] , then f = g f=g .

Let X be a compact Hausdorff space and let A be a linear subspace of C ( X ; R ) C(X;\mathbb {R}) containing the constant functions, and separating points from probability measures. Then the inf-lattice generated by A is uniformly dense in C ( X ; R ) C(X;\mathbb {R}) . We show that this is a corollary of the Choquet-Deny Theorem, thus simplifying the proof and extending to the nonmetric case a result of McAfee and Reny.

For a compact metric space X X , consider a linear subspace A A of C ( X ) C\left ( X \right ) containing the constant functions. One version of the Stone-Weierstrass Theorem states that, if A A separates points, then the closure of A A under both minima and maxima is dense in C ( X ) C\left ( X \right ) . By the Hahn-Banach Theorem, if A A separates probability measures, A A is dense in C ( X ) C\left ( X \right ) . It is shown that if A A separates points from probability measures, then the closure of A A under minima is dense in C ( X ) C\left ( X \right ) . This theorem has applications in economic theory.

We present some consequences of the inequality u > g \mathfrak {u} > \mathfrak {g} among cardinal invariants of the continuum, which has previously been shown to be consistent relative to ZFC. We are interested in its effect on two orderings of families of functions on the natural numbers; in particular we show that, under u > g \mathfrak {u} > \mathfrak {g} , there are exactly five equivalence classes for both orderings (excluding the families bounded by a fixed constant function). This implies, under the same hypothesis, the existence of exactly four classes of rarefaction of measure zero sets.

We show how a local version of a result due to A. Bernard on ultraseparating Banach function spaces can be used to give a short proof of the theorem stating that only affine functions operate on the real part of a function algebra. The paper [5] by K. de Leeuw and Y. Katznelson contains the following generalization of the Stone-Weierstrass Theorem: Let X be a compact Hausdorff space and B a subspace of CR(X), which separates the points of X and contains the constant functions. If there is a continuous non-affine function h defined on a subinterval I of R which operates on B, i.e., hob B if b E B and hob is defined, then B is dense in CR(X). At the same time J. Wermer [8] proved the following theorem: Let A be a function algebra on a compact Hausdorff space X. Then if Re A is an algebra, A = C(X). Here, ReA denotes the space of the real parts of functions in A. Another formulation of Wermer's result is that if the function t -4t2 operates on Re A, then A = C(X). These two results lead to the following conjecture: I Let A be a function algebra on the compact Hausdorff space X. If there is a non-affine function defined on a subinterval of R which operates on Re A, then A = C(X). (Here, it is not necessary to assume that the operating function is continuous because a function which operates on the real part of a function algebra is automatically continuous [6].) If one tries to apply directly the theorem of de Leeuw and Katznelson one only finds that Re A is a dense subspace of CR(X), i.e., that A is a Dirichlet algebra. Thus, more is needed to prove the conjecture. In [1] A. Bernard introduced a powerful method to attack the conjecture and related problems: Let ,6(N x X) denote the Stone-Cech compactification of N x X, and let l??(N, C(X)) denote the space of all bounded sequences Received by the editors August 31, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B 15, 46J 10. ? 1991 American Mathematical Society 0002-9939/91 $1.00 + $.25 per page

Let A A be a maximum modulus algebra on X X , and V V a maximal open subset of X X such that V V has the structure of one-dimensional variety on which functions from A A are analytic. Then, the restriction algebra A X ∖ V {A_{X\backslash V}} is again a maximum modulus algebra.

We give an example of a natural Banach function algebra on the unit disc such that a smaller disc is a peak set for the algebra, but it does not contain any peak point. A Banach function algebra on a compact Hausdorff space X is a Banach algebra A consisting of continuous functions on X, such that A separates points of X and contains the constant functions. If the norm of the algebra A coincides with the sup norm on X , it is called a uniform algebra. If any linear and multiplicative functional on A is of the form f 7→ f(x) for some x ∈ X, the algebra is called natural. A subset K of X is a peak set for A if there is an f ∈ A such that f ≡ 1 on K and |f(x)| < 1 for x / ∈ K; if K = {x0} , then x0 is a peak point. If no proper subset of K is a peak set we call it a minimal peak set for A. It is well known [2] that if A is a uniform algebra on a metrizable set X then any peak set contains a peak point. In [1] H. G. Dales constructed a natural Banach function algebra on a compact subset of C having a peak set without any peak point. T. G. Honary [3] provided an example in R, however his algebra is not natural. In this note we give a very simple example of a natural Banach function algebra on the unit disc with peak sets not containing any peak point. Let K be an open nonempty subset of the complex plane C. By C(K) we denote the algebra of all bounded complex valued functions on K with continuous and bounded first order partial derivatives on K. C(K) is a Banach function algebra on K if equipped with the norm ‖f‖ = ‖f‖∞ + ‖fx‖∞ + ‖fy‖∞ , where ‖·‖∞ is the sup norm. By A(K) we denote the uniform algebra of all continuous functions on K which are analytic on K. We put Dr = {z ∈ C : |z| < r} and C− = {z : Re z < 0}. Theorem 1. Let A be a subalgebra of C(D1) consisting of all functions which are analytic on D 1 2 . Then D 1 2 is a minimal peak set for A; in particular D 1 2 does not contain any peak point. Lemma 2. There is no h ∈ A(D1) such that h(1) = 0 and |1 + (z − 1)h(z)| < 1 for z 6= 1. Proof of the lemma. Assume that such a function does exist. If we compose 1 + (z − 1)h(z) with a suitable fractional linear transformation we get an f ∈ A (C−) Received by the editors November 8, 1995. 1991 Mathematics Subject Classification. Primary 46J10. c ©1997 American Mathematical Society

In the cone of nonnegative superharmonic functions on a bounded euclidean region Ω \Omega , quasi-units were introduced as those elements invariant under the infinitesimal generator of the fundamental one-parameter semigroup of operators S λ ( λ ⩾ 0 ) {S_\lambda }(\lambda \geqslant 0) . All harmonic measures and capacitary potentials are quasi-units, but the latter class has more extensive closure properties. Quasi-units arise naturally under various operations of classical potential theory and have important applications, for example in proving that the convex set of Green’s potentials u u of positive mass distributions on Ω \Omega with u ⩽ 1 u \leqslant 1 has as its extreme points precisely the capacitary potentials. Some new properties of quasi-units are developed here. In particular, it is shown that quasi-units can be characterized as limits of increasing sequences of continuous equilibrium potentials for which the equilibrium values tend to 1.

A unified formulation of Gronwall's inequality (GI) for certain Stieltjes integrals is presented. An advantage in using Stieltjes integrals lies in the fact that some of the "continuous" and "discrete" versions of the inequality may be obtained as a consequence of the same theorem. We are concerned here with an extension of the classical GI [2, p. 35] to a form which includes most of the Stieltjes formulations of the latter and a more general result of J. V. Herod [3]. We will essentially be following the treatment given in [3] wherein an abstract GI was formulated (Lemma 1 below) generalizing some previous work of Schmaedeke and Sell [10]. In the latter the mean Stieltjes integral (MSI) and the Dushnik or interior integral (DI) (see e.g. [10] and references therein) provided a more general framework than the natural Riemann-Stieltjes integral (RSI), since it allowed the possibility of discontinuous integrands. A sharp version of the GI was given by Jones [4] using RSI. Use of the Lebesgue-Stieltjes integral provided Pandit [8, p. 323] with a tool for the reformulation of GI with the scope of applying the latter to continuous and discrete phenomena (cf. also [4]). F. V. Atkinson [1, p. 455] also gave an extension of GI using the left Cauchy-Stieltjes integral (CSI) (cf. e.g. [9]). Though the result was not sharp in general it provided, as such, a useful tool for the basic study of Volterra-Stieltjes integral equations. Moreover a discrete version could again be obtained by choosing the distribution function appropriately [6], [8]. Our aim is to provide a more general GI (Theorem 1 below) the form of which was prompted by Pachpatte [7]. 1. In the sequel all functions will be real-valued, defined on an interval S [0, b], say, and a, ,I will denote, unless otherwise specified, right-continuous nondecreasing functions on S. We will mainly be following [3] in the use of notation. Thus OB will stand for the collection of functions of bounded variation on S. 1 will denote the constant function whose value is 1 on S whereas lx will be the function whose value is 1 at x and zero elsewhere. Finally Ox = 1 lx for x in S. As in [3] we let J be a function from OB to the collection of functions from S x S to the real numbers satisfying the following properties: If each of f and g is Received by the editors May 22, 1980. 1980 Mathematics Subject Classification Primary 34A40.

If a function on an open set in R n {\textbf {R}^n} has the mean value property for one ball at each point of the domain, the function will be said to possess the restricted mean value property. (The ordinary or unrestricted mean value property requires that the mean value property hold for every ball in the domain.) We specify the single ball at each point x by its radius δ ( x ) \delta (x) , a function of x. Under appropriate conditions on δ \delta and the function, the restricted mean value property implies that the function is harmonic, giving a converse to the mean value theorem (see references). In the present paper a converse to the mean value theorem is proved, in which the function δ \delta is well behaved, but the function is only required to be nonnegative. A converse theorem for more general means than averages over balls is also obtained. These results extend theorems of D. Heath, W. Veech, and the author (see references). Some connections are also pointed out between converse mean value theorems and mass cancellation.

E. Atencia and A. de la Torre proved that the ergodic maximal function operator is bounded on L p ( ω ) {L^p}(\omega ) if ω \omega satisfies an appropriate analogue of Muckenhoupt’s A p {A_p} condition. An alternate proof of this result is given.

In this paper we study the range of the isometry on H p {H^p} arising from an inner function which is zero at zero by composition. The range of such an isometry is characterized as a closed subspace M \mathfrak {M} of H p {H^p} (weak- ∗ ^ \ast closed for p = ∞ p = \infty ) satisfying the following: (i) the constant function 1 is in M \mathfrak {M} ; (ii) if f ∈ M f \in \mathfrak {M} and g ∈ H ∞ ∩ M g \in {H^\infty } \cap \mathfrak {M} , then f g ∈ M fg \in \mathfrak {M} ; (iii) if f ∈ M f \in \mathfrak {M} has inner-outer factorization f = χ ⋅ F f = \chi \cdot F , then χ \chi is in M \mathfrak {M} ; (iv) if { B α : α ∈ A } \{ {B_\alpha }:\alpha \in \mathcal {A}\} is a collection of inner functions in M \mathfrak {M} , then the greatest common divisor of { B α : α ∈ A } \{ {B_\alpha }:\alpha \in \mathcal {A}\} is also in M \mathfrak {M} ; and (v) if f ∈ M , B ∈ M f \in \mathfrak {M},B \in \mathfrak {M} , where B B is inner and B ¯ ⋅ f ∈ H p \bar B \cdot f \in {H^p} , then B ¯ ⋅ f ∈ M \bar B \cdot f \in \mathfrak {M} . The proof makes use of the fact that there exists a projection onto such a subspace satisfying the axioms of an expectation operator, which for p = 2 p = 2 , is simply the orthogonal projection. This characterization is applied to give an equivalent formulation of a conjecture of Nordgren concerning reducing subspaces of analytic Toeplitz operators.

Necessary and sufficient conditions in terms of Lebesgue sets are presented for the following two insertion properties for real-valued functions defined on a topological space: (1) g ⩽ f g \leqslant f there is a continuous function h h such that g ⩽ h ⩽ f g \leqslant h \leqslant f , and for each x x for which g ( x ) > f ( x ) g(x) > f(x) then g ( x ) > h ( x ) > f ( x ) g(x) > h(x) > f(x) . (2) g > f g > f there is a continuous function h h such that g > h > f g > h > f .

It is shown that a locally integrable function f f on R n {{\mathbf {R}}^n} has bounded lower oscillation ( f ∈ BLO ) (f \in {\text {BLO}}) if and only if f = M F + h f = MF + h , where F F has bounded mean oscillation ( F ∈ BMO ) (F \in {\text {BMO}}) and M F > ∞ MF > \infty a.e., and h h is bounded. Here, M F MF is a variant of the familiar Hardy-Littlewood maximal function: M F = sup Q ∍ x Q ( F ) MF = {\text {sup}_{Q\backepsilon x}}Q(F) (no absolute values), where Q ( F ) Q(F) is the mean value of F F over the cube Q Q .

A function f : X → Y f:X \to Y is called s-continuous if for each x ∈ X x \in X and each open set V containing f ( x ) f(x) and having connected complement there is an open set U containing x such that f ( U ) ⊂ V f(U) \subset V . In this paper basic properties of s-continuous functions are studied; conditions on domain and/or range implying continuity of s-continuous functions are obtained which generalize recent theorems of Jones, Lee and Long on semiconnected functions. Improvements of recent results of Hagan, Kohli and Long concerning the continuity of certain connected functions follow as a consequence. Also characterizations of semilocally connected spaces in terms of s-continuous functions are obtained.

A condition on the Walsh-Fourier coefficients of a continuous function f f sufficient to conclude that f f is locally constant is obtained. The condition contains certain conditions identified earlier by Bočkarev, Coury, Skvorcov and Wade, and Powell and Wade.

Let f ∈ L 1 ( 0 , ∞ ) , δ > 0 f \in {L^1}(0,\infty ),\delta > 0 and ( G δ f ) ( t ) = δ − 1 ∫ t ∞ e ( t − s ) / δ f ( s ) d s ({G_\delta }f)(t) = {\delta ^{ - 1}}\smallint _t^\infty {e^{(t - s)/\delta }}f(s)ds . Given a partition P = { 0 = t 0 > t 1 > ⋯ > t i > t i + 1 > ⋯ } P = \{ 0 = {t_0} > {t_1} > \cdots > {t_i} > {t_{i + 1}} > \cdots \} of [ 0 , ∞ ) [0,\infty ) where t i → ∞ {t_i} \to \infty , we approximate f by the step function A P f {A_P}f defined by \[ A P f ( t ) = ( G δ i G δ i − 1 ⋯ G δ i f ) ( 0 ) for t i − 1 ⩽ t > t i , {A_P}f(t) = ({G_{{\delta _i}}}{G_{{\delta _{i - 1}}}} \cdots {G_{{\delta _i}}}f)(0)\quad {\text {for}}\;{t_{i - 1}} \leqslant t > {t_i}, \] where δ i = t i − t i − 1 {\delta _i} = {t_i} - {t_{i - 1}} . The main results concern several properties of this process, with the most important one being that A P f → f {A_P}f \to f in L 1 ( 0 , ∞ ) {L^1}(0,\infty ) as μ ( P ) = sup i δ i → 0 \mu (P) = {\sup _i}{\delta _i} \to 0 . An application to difference approximations of evolution problems is sketched.

measure on G will be denoted by jt, and we normalize jt so that ti(G)= 1. By a representation of G we will mean a finite-dimensional unitary representation of G. Let U be a faithful representation of G and let V= U 03 U be the direct sum of U and its complex conjugate representation. Let To(U) be the space of constant functions on G, let T1(U) be the linear space spanned by To(U) and the coordinate functions of V, and for each integer n ? 1 let Tn(U) be the space spanned by all functions ff,2-fn where J E T1(U) for 1 ? i? n. We will call Tn(U) the space of U-trigonometric polynomials of degree ? n. For each nonnegative integer n let Un be the orthogonal projection of L2(G) onto Tn(U). There is a natural way to extend Un to a projection from L1(G) onto Tn(U) (see (2.1)) and we will denote this extension by Un. Iff is a function in L1(G) we will call the sequence {Unf} the U-Fourier series for f, and we will call Unf the nth partial sum of the U-Fourier series for f If f E L2(G, y), then Unf -?f in L2(G, [u) as n -s o. If x E G then the U-Fourier series for f at x is defined to be the sequence {Unf(x)}. If G= T is the group of complex numbers of absolute value 1, and U is the 1-dimensional representation of T on C defined by

Here * denotes complex conjugation. This work is concerned with the concrete spectral theory of Toeplitz operators that are associated with functions W that satisfy the following two hypotheses: (i) W is real, bounded below, and absolutely integrable on r, but it is not equivalent to a constant function. (ii) For each real X the set rx= {O: W(O) L2(dp) such that UT U-' = M, where M is the multiplication operator on L2(dp) which sends g(X) into Xg(X). Hypothesis (i) implies that T is bounded below, so its Friedrichs extension (again named T) is self-adjoint. I am deeply indebted to the referee, who simplified the proof and helped clarify the exposition. The results obtained are conveniently described in terms of the set of vectors k(u) C12 defined for each uGA by

semigroup, is left amenable, then K contains a common fixed point of the fanmily S. Proof. Since S is left amenable, then by Theorem 3, there exists a net, {T1} where T. E P, such that for any fe En(S), {T,,f} converges pointwise to a constant function. For any T,, there exists 0, E 'D such that T, = E, s4,(s)r.. Let the map J,: KK be given by J,,k = Y, E s4,(s)s(k), for k E K. For the remainder of the proof, let y be a specific point in K. By compactness of K, there exists a subnet, {J,} of {J,}, such that {J,6y} converges to some yo E K. And the associated subnet, {T,} of {T}, satisfies that for anyfe rm(S), {Tjf} converges pointwise to a constant function, since {TJf} is a subnet of {Tj}. We will show that for any so E S, that soyo is the required common fixed point. (It can be shown by counterexample that yo itself need not be one.) For each tE X*, define as in [2, p. 587], a real valued function, f,E on S byf,E(s) = ,(sy), for s E S. Since ,u is continuous over the compact set K, then f,E e m(S). So for s' eS, (TJ ,1)s' = OAS) ((rsf) ) = O ?As)f(s 's) = E A(s)p(s 'sy) s e S s e S (s ( S SY) =l(S (sS(S)s)) = 1(s'J6y), by linearity of ,u and affineness of s', in steps 4 and 5, respectively. Then lim (T,6f,)s' = lim (r(s'(J6y))) = ,t(s' ( lim (J,6Y)))

We say that the function F is obtained by general recursion from A, B, M, and N if (i) FA = M, FB = NF and (ii) every natural number n belongs to (R(BkA) for some k >0. If F satisfies (i), then F(BkAt) =NkMt. If A and B satisfy (ii), then every natural number n is BkAt for some k and t. Hence F is uniquely determined. Furthermore if A, B, M, and N are computable functions, then given n, we can obtain k and t effectively so F is computable. Condition (ii) is satisfied if A is the zero function 0 and B is tlle successor function S. There is a function obtained by general recursion from 0, S, M, and N if and only if M is a constant function, say M=SmO. Then Ft=Nem. Thus, iteration is included in general recursion. If G is a permutation, then F=G-1 can be obtained by general recursion from GS, GO, S, and 0 since FGS = S, FGO = OF, and every n belongs to (R((GO)kGS) for k = 0 or k =1. More generally, if F is determined by FG = H where G assumes all values, then F is obtained by general recursion from GS, GO, HS, and HO since FGS = HS, FGO =HOF, and the side condition holds as before.

Let X be a compact Hausdorff space and C(X) the Banach space of all real-valued continuous functions on X under the sup norm. If H is a linear subspace of C(X) which separates the points of X and contains the constant functions, then it is well known that there exists a smallest closed subset of X, called the Silov boundary of X relative to H and denoted by OHX, with the property that each hCH attains its maximum value on afHX. A point xeX is called an H-extremal point if the only positive linear functional u on C(X) such that u(h) = h(x) for all hCH is the evaluation functional 4. where O.,(f) =f(x) for all fE C(X). The set of H-extremal points of X is called the Choquet boundary of X relative to H and will be denoted by VHX. H. Bauer has shown [1, 2.1 ] that the Choquet boundary of X relative to H is nonempty and the Silov boundary of X relative to H is equal to the closure of the Choquet boundary. Bauer has also introduced (see [1 ]) an abstract Dirichlet problem for the above setting. If S:DHX is closed and xCX, then MS denotes the set of positive linear functionals u on C(S) such that u(h I S) =h(x) for all hCH. MS is always nonempty. The measures in MX'HX are called the H-harmonic measures belonging to x. A function f in C(X) is said to be H-harmonic if for every xEX and every u M we have u(f) =f(x). The set of H-harmonic functions is denoted by H. The Dirichlet problem is said to be solvable for X relative to H if 1^ j aHX = c(aHX). The Dirichlet problem is solvable if and only if to each x in X belongs exactly one H-harmonic measure (see [1, Satz 9]). Let X1, X2 be compact Hausdorff spaces and H1, H2 separating linear subspaces of C(X1), C(X2) respectively, which contain the constant functions. Equipped with the usual product topology, the cartesian product X1 X X2 is a compact Hausdorff space. In this paper we show that a subfamily H1+H2 of C(XIXX2) can be defined, in a natural way, for which VH1+H2Xl X X2 exists and VH1+H2Xl X X2 = VH1X1X VH2X2. We then give two theorems on the relation of the

Denote by S a topological space. All functions considered are real valued. Convergence means pointwise convergence unless otherwise stated. Suppose f is a function defined on S, xCS, and f is not continuous at x. The statement that (x,f(x)) is a removable point of discontinuity means that there exists a function g which agrees with f on S- {x} and which is continuous at x. The statement that the function u defined on S is upper semicontinuous means that, if xeS and d >u(x), then there exists a neighborhood V of x such that, if yE V, then d>u(y). The function I is lower semicontinuous if -I is upper semicontinuous. 1.2. Statement of Theorems. THEOREM 1. Suppose M is a linear space of real valued functions defined on S which contains a nonzero constant function and which is closed under the operation of absolute value, and U is the set to which u belongs only in case u is the greatest lower bound of a countable subset of M. Then, if the function f defined on S is the limit of a sequence of functions in M, it is the uniform limit of a sequence each term of which is the difference of two members of U, each of which is bounded above. THEOREM 2. Suppose S is perfectly normal and f is a function defined on S. Each two of the following three statements are equivalent: (1) the function f is the limit of a sequence of functions, each of which has at most a finite number of points of discontinuity, each of which is removable; (2) the function f is the limit of a sequence of functions, each of which has at most countably many points of discontinuity; and (3) there exist a function g which is the limit of a sequence of continuous functions and a countable subset T of S such that, if xeS- T, f(x) = g(x).

notes the Banach algebra of all complex valued continuous functions on X with the supremum norm. A subalgebra A contained in C(X) is called a function algebra on X if it satisfies the following conditions. (i) A separates points on X. (ii) The constant functions are in A, i.e., 1 belongs to A. (iii) A is closed in C(X). For a function algebra A the space of maximal ideals is denoted by MA and the Silov boundary by SA. Both SA and MA carry natural compact Hausdorff topologies and SA and X can be imbedded in MA so that, identifying SA and X with their images in MA, we have SACXCMA. Moreover A, via restriction, is a function algebra on SA and, via the Gelfand representation, extends to a function algebra on MA. SA and MA are respectively the "smallest" and "largest" compact Hausdorff spaces on which A can be realized as a function algebra.

It is known that if p ( z ) p\left ( z \right ) is an analytic polynomial which has no zeros in the open unit disc and distinct zeros in the unit circle, then p ( z ) / ‖ p ( z ) ‖ 1 p\left ( z \right )/{\left \| {p\left ( z \right )} \right \|_1} is an exposed point of the unit ball of the Hardy space H 1 {H^1} . In this paper, it is proved that for a bounded analytic function f f with ‖ f ‖ ∞ ⩽ 1 {\left \| f \right \|_\infty } \leqslant 1 , p ( f ) / ‖ p ( f ) ‖ 1 p\left ( f \right )/{\left \| {p\left ( f \right )} \right \|_1} is also an exposed point.

For the examined functions, we have obtained a structure formula and estimates for | f ( z ) / ( 1 − z ) | |f(z)/(1 - z)|{\text { }} and | arg ⁡ ( f ( z ) / ( 1 − z ) ) | |\arg (f(z)/(1 - z))| , the moduli of the partial sums of the coefficient series and the moduli of the coefficients.

It is shown there are precisely 2 n {2^n} topologically distinct continua of weight n n and power m m where p ≦ n ≦ m p \leqq n \leqq m and p p is the smallest cardinal for which there is a continuum of power m m and weight p p . In particular, there are precisely 2 m {2^m} topologically distinct continua of power m m .

Let X be a compact Hausdorff space and C(X) the Banach algebra of all complex-valued continuous functions on X under the sup-norm. By a function algebra A we mean a closed subalgebra of C(X) which separates points and contains the constant functions. Let S(A) denote the space of maxinmal ideals of A and let E(A) denote the essential set of A as defined by Bear [1], i.e. E(A) is the hull of the largest closed ideal of C(X) which is contained in A. We will obtain another characterization of the essential set in the case when S(A) =X and thereby obtain some results of a global nature from local hypotheses.

We provide new proofs of the theorems of Choquet and Deny and of Foguel concerning iterates of convolutions of a measure on a locally compact abelian group. Let G be a locally compact abelian group, F its dual group and m its Haar measure. ,u denotes a bounded, complex, Borel measure on G with Fourier-Stieltjes transform Ai on 1F. The ergodic and mixing properties of probability measures ,u were characterized by Choquet and Deny [1] and by Foguel [2]. In this note we provide new proofs which are transparent from the point of view of harmonic analysis. In the concluding remark, the proof for the theorem of Choquet and Deny is extended naturally to compact nonabelian groups. Let Io denote {f E L1 (G): f(e) = 0}, where e is the identity of F and 1ttn is the n-times convolution of ,u. THEOREM 1. Let 1u be a bounded, complex Borel measure on G such that I,unI1 < c for all n. Then limnoo II,un * fj1j = 0 for f E Io if and only if jA(-y)j < 1 for all REMARK 1. If , is a probability measure, then the condition that jft(-y)j < 1 for e F \{e} is equivalent to Foguel's condition that the support of , is not contained in any set of the form {x E G: (x, ^y) c} for any constant c and for ey E F\{e}. PROOF. Let I ={ fE Io: Iljn * fjJ1 0 as n -oo}. I is clearly a closed ideal in L1 (G). Because points of F obey synthesis, to prove I = Io it suffices to prove that Io is the only regular maximal ideal of L1 (G) that contains I [3, 39.29]. Stated more simply, for each ey E F\{e} it suffices to produce f E I such that f(^y) # O. Let ey E F\{e}. Choose K, a compact neighborhood of ^y excluding e. Choose h E L1(G) such that 0 < h < 1 hIK = 1, and the support of h is a compact set excluding e. Let f be any function in L1 (G) such that f is supported in K and f(ty) ? 0. Then (1) jjjt1 * fll = 1,tn * hn * flll ? IflflfjjI( * h)njI1. By the spectral radius theorem lim 11(,u * h)n II (l/n) SUpft I(A)h(y) C e F}. This last is less than 1 and therefore limn,oo jj(jt * h)jnj1 0. By (1) this implies that f E I, which completes the proof. Received by the editors November 23, 1983. 1980 Mathematics Subiect Classification. Primary 43A25, 60J15. (?1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page 519 This content downloaded from 207.46.13.52 on Fri, 21 Oct 2016 05:43:18 UTC All use subject to http://about.jstor.org/terms 520 THOMAS RAMSEY AND YITZHAK WEIT For the ergodic properties we have the following simple proof of the theorem of Choquet and Deny. THEOREM 2. Let ,u be a bounded, complex, Borel measure on G. Then the following are equivalent: (i) for f E L.o(G), ,t * f = f implies f = constant. (ii) #(-i) i 1 for -y E F\{e}. REMARK 2. For ,tu a probability measure, the condition ,A(-y) i 1 for -y E F\{e} is equivalent to the fact that the support of 1u generates a dense subgroup of G; i.e. that 1u is adapted. PROOF. Let W = {f E Loo(G): ,u * f = f }. W is a w*-closed, translation invariant subspace of Loc(G). For x E F we have ,u * x = A(x)x = x. Therefore, x E F n W if and only if ,A(x) 1. Suppose that W consists only of constant functions (W Ce or W = {O}). Then there is at most one continuous character in W, the identity e. For all other elements of F we have --/(-) i 1. Conversely, suppose that (u(-y) i 1 for -y i e, -y E F. Then W can contain at most one continuous character, namely e. If e is not in W we may apply an L,,(G)-form of Wiener's Theorem to conclude that W = {0} [3, 40.7]. If e is in W we must use again the fact that e, as a point of F, obeys synthesis. By [3, 40.23(b)], since W n r is exactly {e}, W = Ce. REMARK 3. Using the Peter-Weyl Theorem for compact (nonabelian) groups G, one may simply characterize minimal one-sided translation-invariant, w*-closed subspaces of Loc(G). By modifying the proof of Theorem 2 one can show that adapted probability measures are ergodic [4].

If M M is a closed A A -submodule of C ( X ) C\left ( X \right ) where A A is a uniform algebra on X X which contains a separating family of unimodular functions, and if M M is a quotient space of some C ( Y ) C\left ( Y \right ) , then M M is an ideal in C ( X ) C\left ( X \right ) . If there is an example of a uniform algebra A A on some X X such that A ≠ C ( X ) A \ne C\left ( X \right ) but A A is complemented in C ( X ) C\left ( X \right ) , then there is such an example with A A separable.

It is shown that the set { e int : n ⩾ 1 } \{ {e^{\operatorname {int} }}:n \geqslant 1\} spans L 2 ( σ ) {\mathfrak {L}^2}(\sigma ) if σ \sigma is a singular measure on the unit circle. The proof makes no appeal either to the F. and M. Riesz theorem on measures or to Hilbert space methods.

Recently E. P. Lane proved that if space X has the weak C-insertion property and satisfies another condition, then X has the strong C-insertion property. This paper establishes the converse of this result. In a recent paper, E. P. Lane [1] established two main results about the insertion of continuous functions. The second of these is: THEOREM 3.1. Let P, and P2 be C-properties and consider the following condition: (a) If g and f are functions on X such that g O} = Un .I F,, and (ii) for each n the sets A(f g, 2-) and Fn are completely separated. If X satisfies the weak C-insertion property for (P1, P2) and if X satisfies (a), then X satisfies the strong C-insertion property for (P1, P2). Conversely, if X satisfies the strong C-insertion property for (P1, P2) and f g satisfies the property P1 (actually P1 should have read PF), then X satisfies (a). This paper shows that the last sentence of the above theorem can be strengthened to read: Conversely, if X satisfies the strong C-insertion property for (P1, ,,P2), then X satisfies (a). Thus (a) is a necessary and sufficient condition for a space with the weak C-insertion property to have the strong C-insertion property. A property P defined relative to a real-valued function on a topological space is a C-property provided any constant function has property P and provided the sum of a function with property P and any continuous function also has property P. Let P1 and P2 be C-properties. A space X is said to have the weak C-insertion property for (P1, ,,P2) iff for any functions g and f on X such that g < f, g has property P1 and f has property P2, then there exists a continuous function h on X such that g < h < f. A space X is said to have the strong C-insertion property for (P1, P2) iff for any functions g andf on X such that g < f, g satisfies P1 andf satisfies P2, then there exists a continuous function h on X such that g < h < f and such that if g(x) < f(x) for any x in X, then g(x) < h(x) < f(x). If f is a real-valued function defined on a space X and if {x/f(x) < r) C A(f, r) c {x/f(x) < r}, Received by the editors October 14, 1979 and, in revised form, January 31, 1980. AMS (MOS) subject classifications (1970). Primary 54C30; Secondary 54C05.

It is well known that any linear functional L on C(X) (real or complex) such that J|L|J = 1 =L(1) is positive, i.e., Lf _ 0 whenever f _ 0. More generally, this can be used to show that an operator T from C(X) into C(Y) is positive provided fj TII = 1 and Tl = 1. The positivity of T is equivalent of course, to the assertion that if the range of f is contained in the nonnegative real axis, then so is the range of Tf. The theorem we prove below says more than this, in that it gives a description of the range of Tf in terms of the range of f, for arbitrary f. Furthermore, the proof is very simple and elementary. Let A and B be linear spaces of bounded complex-valued functions on the sets X and Y, respectively, with the supremum norm. We assume that both A and B contain the constant functions. If Z is a subset of the complex plane, conv Z denotes the closed convex hull of Z.

It is shown that the linear manifold of H2 consisting of all noncyclic vectors of the backward shift operator S* is not a countable union of operator ranges. Let S denote the shift operator acting on the Hardy space H2, and S* jOa i0 a 2. its adjoint (S*/)(e = e (f(e") /(O)). A function J e H is noncyclic (for S*) if the closed linear span of /, S*/, S2 , * is not all of H2. The set N of noncyclic vectors is known to be a dense, first category, linear manifold in H2 [2], [3], [12]. Our main results are that a closed subspace K of N must be contained in (qH 2)1 for some inner function q, and that N is not a countable union of operator ranges. This settles strongly in the negative a question raised by Douglas, Shapiro and Shields [3, p. 74]. The proof requires the following theorem (proved in [9]) on invariant subspaces of HH{-the Hilbert space of all functions defined on the disk with values in the separable Hilbert space { and which are weakly in H2 (for details about H2 see [5]). Theorem 1. Let V1 C Hy be a closed subspace invariant under the shift operator such that for each e e J{ (thought of as a constant function in H) there exists an inner function qe such that qee e V. Then there exists a fixed inner function q such that qe e )1 for all e e 3{ (and therefore XR D qH2). The connection between N and invariant subspaces of HH is made via a particular class of invariant subspaces as follows: let E E H2 and define )1E = E HH2: (F, E) e H1 , where (-, .) is the inner product of R. 1E is clearly closed and invariant. Subspaces of this type were first considered by Helson, who observed that maximal invariant subspaces (if Received by the editors April 12, 1971. AMS (MOS) subject classifications (1970). Primary 46E20, 47A15; Secondary 30A78, 47B120.

If S * is the family of subrings of C*(X) such that if S E S *, S contains the constant functions and is closed under uniform convergence, then the following are equivalent for a space (X, 5). (a) (X, 5) is pseudocompact. (b) If S E S * functionally separates points and zero sets, S generates (X, 5). (c) If S E S * functionally separates zero sets, S = C*(X). (d) If S E S * generates the zero sets on (X, 5), S = C*(X). (e) If f E S E S * and Z(f) = 4 then I/f E S (even when it is required that S generate the topology). (f) If f E S E S then lfl E S (even when it is required that S generate the topology).

The Stone-Weierstrass theorem is used to prove two general results about algebras of continuous functions, and each of these yields a necessary and sufficient condition for the planar function algebras R(K) and C(K) to be coincident. We begin with a result which slightly extends the Stone-Weierstrass theorem. THEOREM 1. (Almost selfadjoint algebras are selfadjoint.) Let X be a compact Hausdorff space, and let f E C(X ). Suppose m and n are relatively prime positive integers. Let A be the closed subalgebra of C(X) generated by fam, fnl, and the constant functions. Then A = { g o f: g E C (f (X ))}. In particular, iff separates the points of X, then A = C(X). PROOF. Let K = f (X) and let B be the closed subalgebra of C(K) generated by zm, zn, and the constant functions. We claim B = C(K). By the usual Stone-Weierstrass theorem, it suffices to show that z E B and z E B. Now m B and n E B imply (Zm)n(zn)m E B, or rz2mn E B. Since the function h(u) = ul/2mn is uniformly approximable by polynomials on the interval [0, ||zj|2Kmn] and B is closed, it follows that Izl E B. Since (m, n) = 1, we can find integers a and b with am + bn = 1. Without loss of generality, we can assume that a > 0, b < 0. Then on K {0} we have Z = zam+bn = (zm)a/(zn)-b = (zm)a(zn)-b/lzl-2bn

A new upper bound for the John constant is established. Let F denote the set of functions f(z) which are nonconstant and analytic in the unit disk Izi I 1 and let G denote the subset of F consisting of those f(z) in F which are not univalent. For f(z) in F we define M(f) = sup If'(z)I and m(f) = inf If'(z)1. John [1] defined the constant y by =f CG m(f) This constant has come to be known as the John constant. In the same paper John showed that y > exp( 0 instead of in the disk. In what follows, H and H will denote the open and closed upper half-plane, respectively. Let f(z) be analytic and not univalent in H. If a and / denote the infimum and supremum of If'(z)l in H, respectively, then y ra and consequently y < 8/3a. For the nonunivalent function f(z) which we study here If'(z)l is piecewise constant on the real line. What is more, it is the simplest kind of nontrivial piecewise constant function, namely one that takes on one constant value on some interval and another outside that interval. Our choice of f(z) is motivated by the following consideration: Let z1 # Z2 be two points in IzI < 1. One can show using standard variational arguments that if ho(z) is a function for which ih(zl) h(z2)1 attains a positive minimum for the class of functions h(z) which satisfy m < jh'(z)l < M in lzl < 1, then Iho(z)l is piecewise constant on lzl = 1. (This fact is solely Received by the editors November 3, 1980 and, in revised form, March 24, 1981. 1980 Mathematics Subject Classifjcation. Primary 30A40.

It is well known that locally constant functions on symbolic spaces have unique equilibrium states. In this paper we investigate the nature of equilibrium states for a type of continuous function which need not have a finite range. Although most of these functions have a unique equilibrium state, phase transitions or multiple equilibrium states do occur and can be analyzed. Introduction. The study of equilibrium states and phase transitions is a major theme of modern statistical physics. This paper examines the equilibrium states for a class of functions which have their roots in Fisher's work on condensation (1). The nature of our paper, however, is entirely mathematical. At each integer (site) we allow p different possible behaviors, e.g. spin values or

An example is constructed of a function which is almost continuous in the sense of Singal and Singal but not in the sense of Stallings. Let X be the set of real numbers with the topology 7 consisting of the usual open sets together with the sets of the form UnD, where U is an open set in the usual topology and D the set of all irrational numbers. Letf: [0, 1]-+(X, 7) be defined byf(x)=x. Thenf is almost continuous in the sense of Singal and Singal (and also in the sense of Husain3). Since the only continuous functions on [0, 1]-+(X, 7) are the constant functions (Steen and Seeback [3, p. 89]), f is not almost continuous in the sense of Stallings. This answers an open problem recently posed by Long and Carnahan r21. REFERENCES 1. H. Blumberg, New properties of all real functions, Trans. Amer. Math. Soc. 24 (1922), 113-128. 2. P. E. Long and D. A. Carnahan, Comparing almost continuous functions, Proc. Amer. Math. Soc. 38 (1973), 413-418. 3. L. A. Steen and J. A. Seeback, Jr., Counterexamples in topology, Holt, Rinehart and Winston, New York, 1970. MR 42 #1040. UNIVERSITY OF ARKANSAS, FAYETTEVILLE, ARKANSAS 72701 SOUTHERN ILLINOIS UNIVERSITY, CARBONDALE, ILLINOIS 62901 SLIPPERY ROCK STATE COLLEGE, SLIPPERY ROCK, PENNSYLVANIA 16057 LAKEHEAD UNIVERSITY, THUNDER BAY, ONTARIO P7B SEI, CANADA Received by the editors July 12, 1973. AMS (MOS) subject classifications (1970). Primary 54C10.

General number-theoretic identities which involve various multiplications are given. The Tatuzawa-Iseki identity which occurs in certain elementary proofs of the Prime Number Theorem is included as a special case. For three different multiplications which are defined on the set of number-theoretic (arithmetical) functions we will use the notations f g if (f g) (n) =f(n)g(n), f *g if (f *g)(n) = ,(f(k)g(m):km = n), f #g if (f# g)(x) = , (f(k)g(x/k); 1 < k < x). It can be shown that the #-operation is not associative and not commutative, but the other two operations have both of these properties. Properties of the ring of number-theoretic functions with the *-operation are discussed extensively in [1I]. We shall use the following notations for functions: ,t for the M6bius function, vo for the constant function for which vo(n) = 1, e for the identity function with respect to the *-operation, I for the logarithm function, and A for the von Mangoldt function for which A(n) = log p if n= pm and A(n) =0 otherwise. In this notation we have , * vo=e and,u * I=A. IDENTITY A. f#(g#h) = (f * g)#h. Direct rearrangement of the sums gives us (f # (g # h))(x) = , (f(k) , (g(m)h(x/km) :1 < m < x/k) :1 < k < x) = EI (f(k)g(m)h(x/n):km = n, 1 < n < x) = ((f*g)#1h)(x). The left inversion formulas for the #-operation can be obtained from Identity A; these follow from the Mobius inversion formulas for the *-operation. In Identity A if we letf=vo, g=,u a=,u # h, then h=vo # a; if we letf=,u, g=vo, b=vo # h, then h=,u # b. Two examples of other consequences are Received by the editors July 21, 1969. A MS Subject Classifications. Primary 1043; Secondary 1008.