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Register a new userLULU operators for functions of continuous argument
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Authors
Roumen Anguelov Carl Rohwer
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The LULU operators, well known in the nonlinear multiresolution analysis of sequences, are extended to functions defined on a continuous domain, namely, a real interval. We show that the extended operators replicate the essential properties of their discrete counterparts. More precisely, they form a fully ordered semi-group of four elements, preserve the local trend and the total variation.
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We examine the sum of modified Bessel functions with argument depending quadratically on the summation index given by [S_ u(a)=sum_{ngeq 1} (frac{1}{2} an^2)^{- u} K_ u(an^2)qquad (|arg,a|<pi/2)] as the parameter $|a|to 0$. It is shown that the positive real $a$-axis is a Stokes line, where an infinite number of increasingly subdominant exponentially small terms present in the asymptotic expansion undergo a smooth, but rapid, transition as this ray is crossed. Particular attention is devoted to the details of the expansion on the Stokes line as $ato 0$ through positive values. Numerical results are presented to support the asymptotic theory.
Let $Pin Bbb Q_p[x,y]$, $sin Bbb C$ with sufficiently large real part, and consider the integral operator $ (A_{P,s}f)(y):=frac{1}{1-p^{-1}}int_{Bbb Z_p}|P(x,y)|^sf(x) |dx| $ on $L^2(Bbb Z_p)$. We show that if $P$ is homogeneous then for each character $chi$ of $Bbb Z_p^times$ the characteristic function $det(1-uA_{P,s,chi})$ of the restriction $A_{P,s,chi}$ of $A_{P,s}$ to the eigenspace $L^2(Bbb Z_p)_chi$ is the $q$-Wronskian of a set of solutions of a (possibly confluent) $q$-hypergeometric equation. In particular, the nonzero eigenvalues of $A_{P,s,chi}$ are the reciprocals of the zeros of such $q$-Wronskian.
Asymptotic expansions for the Bateman and Havelock functions defined respectively by the integrals [frac{2}{pi}int_0^{pi/2} !!!begin{array}{c} cossinend{array}!(xtan u- u u),du] are obtained for large real $x$ and large order $ u>0$ when $ u=O(|x|)$. The expansions are obtained by application of the method of steepest descents combined with an inversion process to determine the coefficients. Numerical results are presented to illustrate the accuracy of the different expansions obtained.
We construct, under the assumption that union of less than continuum many meager subsets of R is meager in R, an additive connectivity function f:R-->R with Cantor intermediate value property which is not almost continuous. This gives a partial answer to a question of D. Banaszewski. We also show that every extendable function g:R-->R with a dense graph satisfies the following stronger version of the SCIVP property: for every a<b and every perfect set K between g(a) and g(b) there is a perfect subset C of (a,b) such that g[C] subset K and g|C is continuous strictly increasing. This property is used to construct a ZFC example of an additive almost continuous function f:R-->R which has the strong Cantor intermediate value property but is not extendable. This answers a question of H. Rosen. This also generalizes Rosens result that a similar (but not additive) function exists under the assumption of the continuum hypothesis.
In this paper we study the large linear and algebraic size of the family of unbounded continuous and integrable functions in $[0,+infty)$ and of the family of sequences of these functions converging to zero uniformly on compacta and in $L^1$-norm. In addition, we concentrate on the speed at which these functions grow, their smoothness and the strength of their convergence to zero.
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