where , ; and let be the Bergman kernel function of . Then there exist two positive constants and and a function such that
holds for every . Here
and is the defining function for . The constants and depend only on and , not on .
where is an open subset of and is a positive function of one real variable which is continuously differentiable. We prove the well-posedness in the Hadamard sense (existence, uniqueness and continuous dependence of the local solution upon the initial data) in Sobolev spaces of low order.
A random variable satisfying the random variable dilation equation , where is a discrete random variable independent of with values in a lattice and weights and is an expanding and -preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density which will satisfy a dilation equation
We have obtained necessary and sufficient conditions for the existence of the density and a simple sufficient condition for 's existence in terms of the weights Wavelets in can be generated in several ways. One is through a multiresolution analysis of generated by a compactly supported prescale function . The prescale function will satisfy a dilation equation and its lattice translates will form a Riesz basis for the closed linear span of the translates. The sufficient condition for the existence of allows a tractable method for designing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient condition is necessary in the case when is a prescale function.
is considered subject to the boundary conditions
We assume that is positive and that is piecewise continuous and changes sign at its discontinuities. We give asymptotic approximations up to for , or equivalently up to for , the eigenvalues of the above boundary value problem.
where , and . In 1990 the first and second author showed that in the case and the polynomials are eigenfunctions of a unique differential operator of the form
where are independent of . This differential operator is of order if is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form
where the coefficients , and are independent of and the coefficients , and are independent of , satisfied by the Sobolev-type Laguerre polynomials . Further, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise. Finally, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise.
Using modern methods of weighted potential theory, these zero distribution results of Szeg\H{o} can be essentially recovered, along with an asymptotic formula for the weighted partial sums . We show that is the largest universal domain such that the weighted polynomials are dense in the set of functions analytic in . As an example of such results, it is shown that if is analytic in and continuous on with , then there is a sequence of polynomials , with , such that
where denotes the supremum norm on . Similar results are also derived for disks.
where 0,$"> and space dimensions . Assume that the initial data
where \frac{n}{2},$"> weighted Sobolev spaces are
Also we suppose that
where
Then we prove that there exists a positive such that the Cauchy problem above has a unique global solution satisfying the time decay property
for all 0,$"> where
where is the Lebesgue measure of , and the surface area of the unit ball in . We prove in this paper that there are extremal functions for this inequality. In other words, we show that the
is attained. Earlier results include Carleson-Chang (1986, is a ball in any dimension) and Flucher (1992, is any domain in 2-dimensions).
It is known that sometimes this number can be expressed in a natural way using the Newton polygon of . We provide necessary and sufficient conditions for this Newton polygon characterization to hold. The behavior of the integral at the supremal is also analyzed.
where is the Legendre symbol. For example for an odd prime,
where is the class number of . Similar explicit formulas are given for various polynomials including an example of Turyn's that is constructed by cyclically permuting the first quarter of the coefficients of . This is the sequence that has the largest known asymptotic merit factor. Explicitly,
where denotes the nearest integer, satisfies
where
Indeed we derive a closed form for the norm of all shifted Fekete polynomials
Namely
and if .
we let such that
for all finitely supported and all , where is the sequence of Haar functions. We construct an operator , where is superreflexive and of type 2, with such that there is no constant with
In particular it turns out that the decoupling constants , where is the identity of a Banach space , fail to be equivalent up to absolute multiplicative constants to the usual -constants. As a by-product we extend the characterization of the non-superreflexive Banach spaces by the finite tree property using lower 2-estimates of sums of martingale differences.
acting on functions in . We prove uniqueness of the martingale problem for this degenerate operator under suitable nonnegativity and regularity conditions on and . In contrast to previous work, the need only be nonnegative on the boundary rather than strictly positive, at the expense of the and being Hölder continuous. Applications to super-Markov chains are given. The proof follows Stroock and Varadhan's perturbation argument, but the underlying function space is now a weighted Hölder space and each component of the constant coefficient process being perturbed is the square of a Bessel process.
where solves the equation . The method can be applied to the Laplacian on . It also allows us to obtain similar estimates when we replace the Laplacian by an infinite-dimensional Ornstein-Uhlenbeck operator or other elliptic operators. These operators arise naturally in martingale problems arising from measure-valued branching diffusions and from stochastic partial differential equations.
as , for each in ; in particular, if is uniformly continuous. This result is similar to a Tauberian theorem of Arendt and Batty. It is obtained by applying a result of the authors concerning local stability of bounded semigroups to the translation semigroup on , and it implies several results concerning stability of solutions of Cauchy problems.
: Higher-order recurrenceswith . Finally, let be an element of . In this paper we are giving fairly general conditions depending only on on , and on under which the Diophantine equation
has only finitely many solutions . Moreover, we are giving an upper bound for the number of solutions, which depends only on . This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.
where is continuous, -periodic in and satisfies a superlinearity assumption when . For every , we prove the existence of a -periodic solution having exactly zeroes in .