Interior scale functions

'c will now investigate the consequence of choice of interior Daubechies scale functions having entire supports within the domain of integration as the pseudo-weight function in formal steps of Gauss-type quadrature formula. It is observed that in spite of the loss of positive definiteness of the inner product on V (Аг = 0), node of the polynomial P is real and lies within the support of *p. To obtain higher order polynomials Pn(x) we recall the following classical theorems involving non-negative weight functions.

Theorem 3.5. If n — 0,1.2. • • ■ ^ is the monte orthogonal polynomial wtth respect to the

positive definite inner product <, > . then monic polynomials can be generated through the recurrence relation (Gautschi, 200f. plO),


The range of subscript n is infinite or finite (< d — 1), depending on whether the inner product is positive definite on set of all polynomials or on V((. but not on Pn, n > d.

Those polynomials can also be obtained by using an alternative formula involving moments directly (Gautschi, 2004, p52).

Theorem 3.6. If Ptl(x)(n = 0,1,2,......) is defined above, then

Here, A'n is the determinant obtained by removing the nth column and (n + l)st row of Hankel determinant An+i.

However, these two theorems cannot be used in a straightforward manner for pseudo-weight functions in Daubechies family due to Д2 = 0 or < Pi, Pi >= 0. So by Theorems 3.4 and 3.5, P-2{x) does not exist. It is observed that Ar = 0, for r > 3 but is not necessarily positive definite. As a result, orthogonal polynomials of degree > 3 is admissible. As per the notation of Gautschi (Gautschi, 2004, p36), we regard them FOPs. Such polynomials cannot be determined recursively for к > 3 by using rules of Theorem 3.4 due to non-availability of P-2{x). Thus, one has to determine the polynomials Pn(x),n > 3 either by using the formula of Theorem 3.5 or directly by using the principle suggested by Kessler et ah (Kessler et ah, 2003b) described below. Given a collection of moments pm, m = 0,l,2,---,2n — 1, construct the nth degree monic polynomial

containing n unknown coefficients ao, ai, • • •, an_i so that

This gives a system of n linear algebraic equations of n unknowns ац, a, - ■ ■, a„_i. Since //’s (г = 0,1, ■ ■ • 2n — 1) are all real, solution of these equations provides the polynomial Pn(x) with real coefficients ao, ai, • ■ ■, a„_j. By construction, for given n, the polynomial Pn(x) is orthogonal to all other polynomials of lower order except p2{x).

It is worthy to mention about the difference among the Gauss-type quadrature rules developed earlier by Kwon (Kwon, 1997) or Johnson et al. (Johnson et ah, 1999) and the Gauss-Daubechies quadrature formula derived here. In the former two approaches approximating polynomials are interpolating polynomials, whose values are equal to the values of the integrable function at preassigned nodes within the support of the scale function involved in the integral. Consequently, the error in approximation involve derivative of order r of the integrable function in case of r interpolating points. The principle of approximating integrable function in the framework of Gauss-Daubechies quadrature rule is more stringent. Here the approximating polynomial and its first derivative assume respectively same values of the integrable function and its first order derivative at the nodes of the quadrature formula which are zeros of the approximating polynomials too.

In case of positive definite inner product, zeros of the orthogonal polynomials with real coefficients can be determined either by employing standard numerical techniques or by finding eigenvalues of the Jacobi matrix (Gautschi, 2004, p.13)

But in case of inner products involving Daubechies scale function type pseudo-weight functions, method based on Jacobi matrices is not admissible due to non-availability of aj. It is found that standard library function “LinearSolvef ]” in MATHEMATICA (Wolfram, 1999) can evaluate efficiently all the roots with expected accuracy as one desires. Invoking that library function for obtaining the roots of FOPs, it is observed that for some n, some of their zeros are complex and according to the standard theorem of classical algebra, complex roots appear in pairs with their conjugates. Moreover, one or two roots of a few FOPs may lie outside the support of the pseudo-weight function p(x).

The location of zeros of FOPs for n = 3,4,......,24 for scale functions within interior

class of Daubechies family with three, six and ten vanishing moments for their wavelets, Dau3[—2,3], Dau6[—5,6], Dau 10[—9.10] respectively, are presented in Figs. 3.1a, 3.1b and 3.1c. The region between vertical dashed lines in the figures indicates support of the corresponding pseudoweight function. The location of real roots are described by single point, the same for the complex roots by pair of points placed vertically. Few zeros for some FOPs may be absent due to their location far beyond the region accommodated in the figures.

Once the roots x[ (i = 1, • ■ ■ ,n) of FOPs of degree n with their real part within supp p1 have been determined, the pseudo-weights ,i = 1, ■ ■ ■ ,n of the quadrature formula

can be obtained by solving the system of linear equations

formed by using f(x) = ж ,к = 0,1,2,n — 1 in the quadrature rule. Since a few roots of FOPs for some n are complex and appear with their conjugates, corresponding pseudo-weights are also complex and appear with their conjugates. At this point we address the question on whether use of complex zeros as the nodes of the quadrature rule may pose problems in the evaluation of the integral p1 (x)f(x)dx.

supp if1

Theorem 3.7. For the class of integrable functions satisfying f(z) = f(z), the approximate value


of the integral p1 (x)f(x)dx obtained by using the quadrature rule Q„ [f(x), p!] = ^ u>- f(x)

supp !p1 l=

involved with complex nodes having their real parts within supp p, are real.

Location of zeros of FOPs for Daubechies scale functions a) Dau3[—2, 3], b) Dau6[—5,6] and c) DaulO[—9,10]

Figure 3.1: Location of zeros of FOPs for Daubechies scale functions a) Dau3[—2, 3], b) Dau6[—5,6] and c) DaulO[—9,10].

Proof: For a given n, let the subscripts of nodes X{ (omitting the superscript I) involved in the quadrature formula be classified into two sets, viz., i € Г® when nodes are real and i € Гс when the nodes are complex and we denote them by xf and xf respectively together with their pseudo-weights as and ojf. Moreover, {ж®, г € Г Re xf,i € Гс} 6 supp <,?(a,’). Then

Due to the finite support of (ж), condition on f(x) assumed in the proposition holds good for most of the integrals appearing in the multiresolution approximation of L2-functions as well as multiscale or nonstandard representation of differential or integral operators. If the function f{x) has any singularity within supp ip1, such integrals f(x) ip1 (a;) have to be treated separately with the

supp ip1

aid of refinement equation for ! (x). This observation is also equally valid for integrals involving the scale functions ipB T and wavelets ф1вт. Here, В stands for L or R and, T stands for LT or RT.

Corollai'y 3.8. Using the notation xf = xBe + txjm, ojf = шВе + гшт and f(xf) = fBe{xf) + гfIm(xf) and the property of f(x) assumed in the proposition, the quadrature rule may be recast into the form

The stability constant an, defined in (Gautschi, 1997), is given by

Ti 2K ^ -

in case of interior scale functions since ^ = !(x)dx = 1. This measures the suscepti-


bility of the quadrature sum to rounding errors (the larger the sum, the larger the error caused by cancellation). Figure 3.2 depicts an against n for DauZ, DauQ and Dow 10. Results presented in this figure exhibits the fact that the stability constant crn is close to 1 for most of the quadrature rules having admissible nodes n < 24 associated with the Daubechies scale functions Dan A'(2 < К < 10) in interior class. Its value shoots up close to 3 only for quadrature rules having nodes around 15 for the scale function DaulO which again is bounded above by 1.75 till the number of nodes increases up to 24. This observation indicates that admissible Gauss-Daubechies quadrature rule is stable having nodes at least up to 24.

Stability constants for quadrature formula involving several nodes for scale functions Dau3, Dau6 and DaulO

Figure 3.2: Stability constants for quadrature formula involving several nodes for scale functions Dau3, Dau6 and DaulO. Boundai'y scale functions и’В on R+, on R )

In case of boundary scale functions in the classes Ф1*!1-, фГ1«, the Hankel determinants are nonzero. According to Gautschi (Gautschi, 2004) each of the boundary scale functions in both classes may be regarded as quasi-weight functions separately. As discussed above, nodes and quasi-weights of quadrature formula for numerical evaluation of integrals involving product of integrable function with boundary scale functions can be obtained for each member of leR and Ф’1д1и separately. As in the case of integrals involving interior scale functions some of the node and weights for each of the boundary scale functions may be complex. Even a few of them may lie outside the support of the scale functions involved in the integrals. The quadrature formula involving up to 24 nodes for each of the boundary scale functions except for К = 10 is found to be stable. To

find an admissible quadrature rule, we present in Figs. 3.3a and 3.3b the distribution of zeros over the support of boundary scale functions for several degrees of FOPs for each of the boundary scale functions having three vanishing moments of their wavelets.

Distribution of zeros of FOPs corresponding to the boundary scale functions a.i) рд, a.ii) ip}, a.iii)

Figure 3.3: Distribution of zeros of FOPs corresponding to the boundary scale functions a.i) рд, a.ii) ip}, a.iii) 2 hr R+ and b.i) <р^ b.ii) 2, b.iii)y>^3 in E_ in CDV basis (Cohen et ah, 1993) for К = 3.

The variation of stability constants a of the quadrature ride with number of nodes for each member of the families leR and фГ1дМ are presented in Figs. 3.4a and 3.4b respectively.

Stability constants a versus number of nodes n of quadrature rules for integrals associated with boundary scale functions in CDV basis (Cohen et al., 1993) a)

Figure 3.4: Stability constants a versus number of nodes n of quadrature rules for integrals associated with boundary scale functions in CDV basis (Cohen et al., 1993) a) and b) ^3 for К = 3. Truncated scale functions ЬТКТ on [0.2K - 1])

In order to compare the approximate values of the integrals of the product of integrable functions and truncated scale functions obtained by using Gauss-Daubechies quadrature rule proposed here and Gauss-type quadrature rules proposed by other researchers (Huybrechs and Vandewalle, 2005; Xiao et al., 2006), we have to find nodes and quasi-weights associated with the above integrals when the support of Daubechies scale functions overlaps partially with the domain of integration. Since the algorithm involved here for determining nodes and weights for the quadrature rule depends only upon the moments of scale functions, determination of their values for integrals involving truncated scale functions in Meyer basis (Goswami and Chan, 2011) is straightforward. Here zeros of FOPs (presented along the abscissa) and stability constants involving quasi-weights of quadrature rule for the integrals involving Daubechies (K=3) scale function having partial supports within [0, к] and [/>•. 2К — 1 ](A- = 1, • • •, 4) are calculated and described in Figs. 3.5 and 3.6.

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