factorization and non factorization theorems for pseudocontinuable functions abstract let be an inner function on the unit disk and let k p h p h 0 p be the associated star invariant subspace of the hardy space h p with p 1 while a nontrivial function f k p is never divisible by it may have a factor h which is not too different from in the sense that the ratio h or just the anti analytic part thereof is smooth on the circle in this case f is shown to have additional integrability and or smoothness properties much in the spirit of the hardy littlewood sobolev embedding theorem the appropriate norm estimates are established and their sharpness is discussed

zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

a note on nonregular likelihood functions in heteroscedastic regression models summary the existence of maximum likelihood estimates for a class of heteroscedastic regression models is considered for a given dispersion function we show that under a weak condition the likelihood is singular at points corresponding to nonreplicated observations causing unrestricted maximum likelihood estimation to break down whilst for an alternative class of dispersion functions we obtain a much stronger linear independence condition for the likelihood to be unbounded

level spacings distribution for large random matrices gaussian fluctuations we study the level spacings distribution for eigenvalues of large n x n matrices from the classical compact groups in the scaling limit when the mean distance between nearest eigenvalues equals 1 defining by 77n s the number of nearest neighbors spacings greater than s 0 smaller than s 0 we prove functional limit theorem for the process 77n s e n s n1 2 giving weak convergence of this distribution to some gaussian random process on 0 x the limiting gaussian random process is universal for all classical compact groups it is holder continuous with any exponent less than 1 2 similar results can be obtained for the n level spacings distribution

structural properties of the wiener filter stability smoothness properties and fir approximation behavior any wiener filter can be interpreted as a cascade of a whitening and estimation filter the whitening filter is determined due to the spectral factorization of the spectral density of the input signal for the calculation of the estimation filter the spectral factorization as well as the so called plus operator is needed this correspondence investigates in detail the behavior of these two operations and studies the corresponding properties of both filters then the practical consequences for the overall wiener filter are discussed it is shown that if the given spectral densities are smooth houmllder continuous functions the resulting wiener filter will always be stable and can be approximated arbitrarily well by a finite impulse response fir filter moreover the smoothness of the spectral densities characterizes how fast the fir filter approximates the desired filter characteristic and the correspondence gives a class of approximation polynomials which actually achieves the optimal approximation behavior on the other hand if the spectral densities are continuous but not houmllder continuous the resulting wiener filter may not be stable

on the distribution of imaginary parts of zeros of the riemann zeta function ii we continue our investigation of the distribution of the fractional parts of where is a fixed non zero real number and runs over the imaginary parts of the non trivial zeros of the riemann zeta function we establish some connections to montgomery s pair correlation function and the distribution of primes in short intervals we also discuss analogous results for a more general l function

holomorphic functions and quasiconformal mappings with smooth moduli abstract we discuss the implication f g f g where f is a holomorphic function resp a quasiconformal mapping on a domain g c n resp g r n and g is the lipschitz space associated with a majorant

the pair correlation of homotetic images of zeros of the riemann zeta function we investigate the pair correlation of zeros of the riemann zeta function s and zeros of the function k s 1 2 1 2 for any fixed positive integer k

determinantal processes with number variance saturation consider dyson s hermitian brownian motion model after a finite time s where the process is started at n equidistant points on the real line these n points after time s form a determinantal process and has a limit as n this limting determinantal process has the interesting feature that it shows number variance saturation the variance of the number of particles in an interval converges to a limiting value as the length of the interval goes to infinity number variance saturation is also seen for example in the zeros of the riemann function 21 2 the process can also be constructed using non intersecting paths and we consider several variants of this construction one construction leads to a model which shows a transition from a non universal behaviour with number variance saturation to a universal sine kernel behaviour as we go up the line

semiclassical form factor for chaotic systems with spin we study the properties of the two point spectral form factor for classically chaotic systems with spin in the semiclassical limit with a suitable semiclassical trace formula as our principal tool to this end we introduce a regularized form factor and discuss the limit in which the so called diagonal approximation can be recovered the incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics provided this skew product is mixing we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory

differentiation in star invariant subspaces i boundedness and compactness abstract given a discrete set c r let r p denote the subspace of lp r generated by rational functions with poles in we determine for which s the differentiation operator d dx r p lp r is bounded or compact

ramanujan hardy littlewood riesz phenomena for hecke forms abstract we generalize a result of ramanujan hardy and littlewood to the setting of primitive hecke forms which interestingly exhibits faster convergence than in the initial case of the riemann zeta function we also provide a criterion in the spirit of riesz for the riemann hypothesis for the associated l functions

division and multiplication by inner functions and embedding theorems for star invariant subspaces in this paper the full answer is obtained for the following classes x the space bmo of functions of bounded mean oscillation the algebra qc of quasidef continuous functions the space 9 mult bmo of bmo multipliers and finally the lipschitz classes a o ae 1 see g for the definitions and the basic properties of these spaces the most important things will be recalled below the corresponding analytic subspaces are bmoa qa 9ca mult bmoa and aa respectively in order to formulate the results fix a number 0 1 and consider

on certain quasiconformal and elliptic mappings abstract let d be the closure of the unit disk d in the complex plane c and g be a continuous function in d in this paper we discuss some characterizations of elliptic mappings f satisfying the poisson s equation f g in d and then establish some sharp distortion theorems on elliptic mappings with the finite perimeter and the finite radial length respectively the obtained results are the extension of the corresponding classical results

derivative free characterizations of bounded composition operators between lipschitz spaces we prove that c phi maps h varlambda alpha into h varlambda beta 0 alpha beta le 1 if and only if 1 phi alpha belongs to varlambda beta mathbb d in the case 1 we give another two equivalent conditions

a quasiregular analogue of a theorem of hardy and littlewood suppose that f is analytic in the unit disk a theorem of hardy and littlewood relates the holder continuity of f over the unit disk to the growth of the derivative we prove here a quasiregular analogue of this result in certain domains in n dimensional space we replace values of the derivatives with a local integral average in the process we generalize a result on the continuity of quasiconformal mappings due to nakki and palka

on characterizations of bloch type hardy type and lipschitz type spaces in this paper we establish a bloch type growth theorem for generalized bloch type spaces and discuss relationships between dirichlet type spaces and hardy type spaces on certain classes of complex valued functions then we present some applications to non homogeneous yukawa pdes we also consider some properties of the lipschitz type spaces on certain classes of complex valued functions finally we will study a class of composition operators on these spaces

lengths areas and lipschitz type spaces of planar harmonic mappings abstract in this paper we give bounds for length and area distortion for harmonic k quasiconformal mappings and investigate certain lipschitz type spaces on harmonic mappings

the role of the ramanujan conjecture in analytic number theory we discuss progress towards the ramanujan conjecture for the group gln and its relation to various other topics in analytic number theory

zeros of analytic functions with or without multiplicities the classical mason stothers theorem deals with nontrivial polynomial solutions to the equation a b c it provides a lower bound on the number of distinct zeros of the polynomial abc in terms of deg a deg b and deg c we extend this to general analytic functions living on a reasonable bounded domain omega subset mathbb c rather than on the whole of mathbb c the estimates obtained are sharp for any and a generalization of the original result on polynomials can be recovered from them by a limiting argument

generalized bloch spaces integral means of hyperbolic harmonic mappings in the unit ball abstract in this paper we investigate the properties of hyperbolic harmonic mappings in the unit ball b n in r n n 2 firstly we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the bloch space b b n and the generalized bloch space l b a 0 b n respectively secondly we discuss the relationship between the integral means of hyperbolic harmonic mappings and that of their gradients the obtained results are the generalizations of hardy and littlewood s related ones in the setting of hyperbolic harmonic mappings finally we characterize the weak uniform boundedness property of hyperbolic harmonic mappings in terms of the quasihyperbolic metric

variance function estimation abstract heteroscedastic regression models are used in fields including economics engineering and the biological and physical sciences often the heteroscedasticity is modeled as a function of the covariates or the regression and other structural parameters standard asymptotic theory implies that how one estimates the variance function in particular the structural parameters has no effect on the first order properties of the regression parameter estimates there is evidence however both in practice and higher order theory to suggest that how one estimates the variance function does matter further in some settings estimation of the variance function is of independent interest or plays an important role in estimation of other quantities in this article we study variance function estimation in a unified way focusing on common methods proposed in the statistical and other literature to make both general observations and compare different estimation schemes we show that there are significant di

on the distribution of spacings between zeros of the zeta function etude numerique de la distribution des espacements des zeros de la fonction zeta de riemann observation de certains phenomenes inatendus

strong hardy littlewood theorems for analytic functions and mappings of finite distortion a formula is pointed out that explains why an analytic function often enjoys the same smoothness properties as its modulus this is extended to quasiregular mappings and mutatis mutandis to mappings of finite distortion

weak subconvexity for central values of l functions we describe a general method to obtain weak subconvexity bounds for many classes of l functions we give several examples of our bound and our work has applications to a conjecture of rudnick and sarnak for the mass equidistribution of hecke eigenforms

inner functions as improving multipliers if x y are two classes of analytic functions in the unit disk d and is an inner function is said to be x y improving if every function f x satisfying f y must actually satisfy f x this notion has been recently introduced by k m dyakonov in this paper we study the x y improving inner functions for several pairs of spaces x y in particular we prove that for any p 0 1 the qp bmoa improving inner functions and the qp b improving inner functions are precisely the inner functions which belong to the space qp here b is the bloch space we also improve some results of dyakonov on the subject regarding lipschitz spaces and besov spaces

weak subconvexity without a ramanujan hypothesis we describe a new method to obtain weak subconvexity bounds for l functions with mild hypotheses on the size of the dirichlet coefficients we verify these hypotheses for all automorphic l functions and with mild restrictions the rankin selberg l functions attached to two automorphic representations the proof relies on a new unconditional log free zero density estimate for rankin selberg l functions

estimation of parameters in heteroscedastic linear models this paper deals with linear regression models with non homogeneous error variances the common situation where the error variance is a smooth function of the values of the regressor variables is considred the error variance function is represented by a function of known form but its expression involves the vector of unknown regression coeffcients and an additional vector parameter estimation of by iterative weighted least squares iwls therefore also requires updating the estimate of at each iteration when the error variance function is misspecified different methods of updating would yield iwls estimators of varying degrees of efficiency under such situations a method of updating is proposed resulting in an iterative procedure for estimating that is asymptotically optimal in the class of iwls estimators

quantities equivalent to the norm of a weighted bergman space let 0 2 if f is holomorphic in the unit disc d and if is a radial weight function of secure type then the followings are equivalent d f z p z da z d f z p f z z da z 01 02 f rei pd 1 p 02 f rei pd p r rdr here f z 1 z 2 f z furthermore if f 0 0 and is monotone then three quantities on the left sides are mutually equivalent this generalizes a classical result of hardy littlewood

remarks on lp oscillation of the modulus of a holomorphic function abstract we extend dyakonov s theorem on the moduli of holomorphic functions to the case of l p norms

inner function in dirichlet type spaces abstract let x and y be two spaces of analytic functions in the unit disc d with x y for an inner function is said to be x y improving if f x whenever f x and f y in this note for a certain nonnegative and increasing function k defined in 0 we consider the space of dirichlet type d k consisting of those functions f analytic in d such that d f z 2 k 1 z 2 d a z we prove that an inner function belongs to d k if and only if it is d k bmoa bmoa improving we discuss also inner functions in d k as multipliers of d k bmoa

self improving behaviour of inner functions as multipliers abstract let x and y be two spaces of analytic functions in the disk with x y for an inner function it is sometimes true that whenever f x and f y the latter product must actually be in x we discuss this phenomenon for various pairs of analytic smoothness classes x and y

spectral form factors of rectangle billiards the berry tabor conjecture asserts that local statistical measures of the eigenvalues j of a generic integrable quantum system coincide with those of a poisson process we prove that in the case of a rectangle billiard with random ratio of sides the sum behaves for random and n large like a random walk in the complex plane with a non gaussian limit distribution the expectation value of the distribution is zero its variance which is essentially the average pair correlation function is one in accordance with the berry tabor conjecture but all higher moments 4 diverge the proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces the berry tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space

symmetry transitions in random matrix theory l functions we compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices defined by averages with respect to haar measure on so 2n and usp 2n to leading order as n on the unit circle as functions of the angle measured from one of the two symmetry points in the eigenvalue spectrum exp pm i theta n 1 le n le n our results extend previous formulae that relate just to the symmetry points i e to 0 local spectral statistics are expected to converge to those of random unitary matrices in the limit as n when is fixed and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing if y n they become functions of y in the limit when n we verify that this is true for the spectral two point correlation function but show that it is not true for the moments of the characteristic polynomials for which the leading order asymptotic approximation is a function of rather than y symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing we also investigate the moments of the logarithms of the characteristic polynomials in the same context the moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of l functions previously moments at the symmetry point 0 have been related to the moments of families of l functions evaluated at the centre of the critical strip our results motivate general conjectures for the moments of orthogonal and symplectic families of l functions evaluated at a fixed height t up the critical line these conjectures suggest that the symmetry of the non trivial zeros of the l functions influences the moments asymptotically far on the scale of the mean zero spacing from the centre of the critical strip we verify that the second moments of real quadratic dirichlet l functions and a family of automorphic l functions are consistent with our conjectures

toeplitz operators and arguments of analytic functions for a toeplitz operator t we study the interrelationship between smoothness properties of the symbol and those of the functions annihilated by t for instance it follows from our results that if is a unimodular function on the circle lying in some lipschitz or zygmund space with 0 and if f is an hp function p 1 with t f 0 then f and f lambda a le c varphi d lambda a f p for some c c p and d d p an explicit formula for the optimal exponent d is provided similar and more general results for various smoothness classes are obtained and several approaches are discussed furthermore since a given non null function f hp lies in the kernel of t psi with psi bar z bar f f we derive information on the smoothness of hp functions with smooth arguments this can be viewed as a natural counterpart to the existing theory of analytic functions with smooth moduli

numerical computations concerning the erh this paper describes a computation which established the erh to height 10000 for all primitive dirichlet l series with conductor q 13 and to height 2500 for all q 72 all composite q 112 and other moduli the computations were based on euler maclaurin summation care was taken to obtain mathematically rigorous results the zeros were first located within 10 12 then rigorously separated using an interval arithmetic package a generalized turing criterion was used to show there were no zeros off the critical line statistics about the spacings between zeros were compiled to test the pair correlation conjecture and gue hypothesis

generalized poincare inequalities for solutions to the a harmonic equation in certain domains abstract we first prove local versions of the poincare inequality for solutions to the a harmonic equation then as applications of the local results we obtain the global versions of the poincare inequality for solutions to the a harmonic equation in l s 0 averaging domains and l s averaging domains respectively these results can be considered as generalizations of the classical poincare theorem

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12