. To study wavelets and filter banks of high order, we begin with the zeros of Bp (y). This is the binomial series for (1 \Gamma y) \Gammap , truncated after p terms. Its zeros give the p \Gamma 1 zeros of the Daubechies filter inside the unit circle, by z + z \Gamma1 = 2 \Gamma 4y. The filter has p additional zeros at z = \Gamma1, and this construction makes it orthogonal and maximally flat. The dilation equation leads to orthogonal wavelets with p vanishing moments. Symmetric biorthogonal wavelets (generally better in image compression) come similarly from a subset of the zeros of Bp (y). We study the asymptotic behavior of these zeros. Matlab shows a remarkable plot for p = 70. The zeros approach a limiting curve j4y(1 \Gamma y)j = 1 in the complex plane, which is jz \Gamma z \Gamma1 j = 2. All zeros have jyj 1=2, and the rightmost zeros approach y = 1=2 (corresponding to z = \Sigmai ) with speed p \Gamma1=2 . The curve j4y(1 \Gamma y)j = (4ßp) 1=2p j1 \Gamma 2yj 1=p...

The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.

We give asymptotic approximations of the zeros of certain high degree polynomials. The zeros can be used to compute the filter coefficients in the dilation equations which define the compactly supported orthogonal Daubechies wavelets. Computational schemes are presented to obtain the numerical values of the zeros within high precision. 1991 Mathematics Subject Classification: 33B20, 41A60, 65D20. Keywords & Phrases: Daubechies wavelets, zeros of high degree polynomials, incomplete beta functions, uniform asymptotic expansion. 1. Daubechies wavelets The polynomial formed by the first N terms of the binomial expansion (1 \Gamma y) \GammaN = 1 X k=0 ` k +N \Gamma 1 k ' y k = 1 X k=0 (N) k k! y k that is, the polynomial PN (y) = N \Gamma1 X k=0 ` k +N \Gamma 1 k ' y k = 1 +Ny + N(N + 1) 2 y 2 + : : : + ` 2N \Gamma 2 N \Gamma 1 ' y N \Gamma1 ; (1:1) where (a) k = \Gamma(a+k)=\Gamma(a), plays an important role in the construction of the compactly supp...

We consider the complex zeros with respect to z of the incomplete gamma functions fl(a; z) and \Gamma(a; z), with a real and positive. In particular we are interested in the case that a is large. The zeros are obtained from approximations that are computed by using uniform asymptotic expansions of the incomplete gamma functions. The complex zeros of the complementary error function are used as a first approximations. Applications are discussed for the zeros of the partial sums s n (z) = P n j=0 z j =j! of exp(z). 1991 Mathematics Subject Classification: 33B15, 33B20, 41A60, 65U05. Keywords & Phrases: incomplete gamma functions, zeros of incomplete gamma functions, error function, uniform asymptotic expansion. 1.

this paper is to obtain the analogs of (1.6) and (1.8) in this more general Pad'e setting, thereby generalizing the results of [4] and [9]. In the remainder of this section, we introduce needed background and known results for this study of Pad'e rational approximation to e

We give asymptotic approximations of the zeros of certain high degree polynomials. The zeros can be used to compute the filter coefficients in the dilation equations which define the compactly supported orthogonal Daubechies wavelets. Computational schemes are presented to obtain the numerical values of the zeros within high precision. 1991 Mathematics Subject Classification: 33B20, 41A60, 65D20. Keywords & Phrases: Daubechies wavelets, zeros of high degree polynomials, incomplete beta functions, uniform asymptotic expansion. 1. Daubechies wavelets The polynomial formed by the first N terms of the binomial expansion (1 \Gamma y) \GammaN = 1 X k=0 ` k +N \Gamma 1 k ' y k = 1 X k=0 (N) k k! y k that is, the polynomial PN (y) = N \Gamma1 X k=0 ` k +N \Gamma 1 k ' y k = 1 +Ny + N(N + 1) 2 y 2 + : : : + ` 2N \Gamma 2 N \Gamma 1 ' y N \Gamma1 ; (1:1) where (a) k = \Gamma(a+k)=\Gamma(a), plays an important role in the construction of the compactly supported Daubechies wavelets. There is a close connection between the zeros of PN (y) and the 2N filter coefficients h(n) of the Daubechies wavelet D 2N . For a complete account of the theory we refer to Chapter 6 of Daubechies (1992). In this section we give the details that are relevant to this paper. In later sections we describe the asymptotic methods for obtaining the zeros of PN (y) and discuss methods how to obtain the coefficients h(n) numerically. This paper is an extension of work done by Shen & Strang (1995). 2 1.1. Some properties of Daubechies wavelets We recall that the filter coefficients h(n) define the dilation equation OE(x) = p 2 2N \Gamma1 X n=0 h(n)OE(2x \Gamma n); (1:2) the solution of which is called the scaling function. We take the following normalization of ...

Dedicated to Percy Deift with gratitude and admiration. Abstract. In this paper we derive uniform asymptotic expansions for the partial sums of the exponential series. We indicate how this information will be used in a later publication to obtain full and explicitly computable asymptotic expansions with error bounds for all zeros of the Taylor polynomials pn−1(z) = Pn−1 k=0 zk / k!. Our proof is based on a representation of pn−1(nz) in terms of an integral of the form R e γ nφ(s) ds. We demonstrate how to derive uniform s−z expansions for such integrals using a Riemann-Hilbert approach. A comparison with classical steepest descent analysis shows the advantages of the Riemann-Hilbert analysis in particular for points z that are close to the critical points of φ. 1.