Formal orthogonal polynomials, nodes, weights of wavelets

Table of Contents:

In multiresolution approximation of a function, the approximation is decomposed into several pieces (Goswami and Chan, 2011). The first one is the projection into the approximation space Vj spanned by an unconditional basis formed by ipjk- The remaining pieces are projections into the detail spaces 4j>./ which are linear spans of other unconditional bases formed by фц- (j > J), wavelets of several resolutions. Actually, coefficients of the wavelet bases provide subtle behaviour, viz., singularity, periodicity, etc. of the function. Thus, getting the knowledge of wavelet coefficients without having recourse to the coefficients of scale function t hrough pyramid algorit hm may sometimes be beneficial. Such goal can be attained if one has any quadrature rule for evaluating integrals involving product of integrable functions and wavelets directly.

In spite of vanishing of first few moments of wavelets (vanishing moment condition f xn ф]к{х) dx = 0, n = 0,1, ■ • • К — 1 instead of fi° = f = 1 for Daubechies

supp 4>jk supp

family), algorithms for getting nodes and weights for Gauss-Daubechies quadrature rules can be executed even for the integrals involving product of integrable functions and wavelets. In our study, it is found that nodes as depicted in Fig. 3.7 and pseudo-weights for quadrature rules associated

Location of zeros of FOPs associated with the quadrature rules for integrals involving Daubechies truncated scale functions (K = 3) in Meyer basis (Goswami and Chan, 2011) a.i)

Figure 3.5: Location of zeros of FOPs associated with the quadrature rules for integrals involving Daubechies truncated scale functions (K = 3) in Meyer basis (Goswami and Chan, 2011) a.i) T, a.ii) a.iii) , a.iv) tp%T and b.i) , b.ii) tp§T, b.iii) , b.iv) .

with the integrals containing wavelets have almost similar qualitative behaviour as in the case of integrals involving interior scale functions. However, the stability of the quadrature rule cannot formally be analysed due to vanishing of denominator in the definition of the stability constant a. We

Stability constants for quadrature rules associated with the integrals involving a) 2 3 4!

Figure 3.6: Stability constants for quadrature rules associated with the integrals involving a) 2 3 4!

b) ^1^2,3a-

Position of zeros of FOPs involved with wavelets in Daubechies family having a) three, b) six and c) ten vanishing moments

Figure 3.7: Position of zeros of FOPs involved with wavelets in Daubechies family having a) three, b) six and c) ten vanishing moments. Notations for identification of real and complex roots are the same as described in Fig. 3.1.

thus compute only the sum of the absolute values of pseudo-weights to get an idea of the stability of the quadrature rule. The sum is found within the range of (0.5,1.5) for admissible quadrature rules with number of nodes n lying in [K + 3,24], К being the number of vanishing moments of the wavelets (here, 2 < К < 10). Nodes and quasi-weights of the quadrature rule for integrals involving wavelets belonging to the boundary and truncated classes can also be evaluated following the same algorithm as in the case of scale functions discussed in the next section.

Algorithm

The requisite steps for obtaining zeros of FOPs and pseudo- or quasi-weights for Gauss-Daubechies quadrature rule may be summarized as follows: (Here, /tm denotes )

  • • Calculate moments of scale functions and wavelets whenever those are in interior, truncated or orthogonal class by using the principle mentioned in section 3.3.
  • • Use moments up to order 2n — 1 into the formula (3.4.1.3) to form a system of linear equations

П — 1

for n unknown coefficients ao, ai, ■ ■ ■, a„ i of FOP aix% + x" °f degree n.

i=о

• Find the zeros aq (i = 1, • • •, n) of FOP.

• Evaluate pseudo- or quasi-weights uit (i = 1, • • •, n) of the quadrature rule by solving the system of equations

by employing the zeros Xi obtained in the previous step and moments obtained in the initial step.

Once the moments fim(in = 0, • • •, 2n — 1) for given n become known, coefficients a, and zeros aq of FOP and pseudo- or quasi-weights w, (г = 1 • • •, n) can be evaluated immediately by using the set (m = 0, • • ■ ,2n — 1) separately for each pseudo- or quasi-weight function of the interior,

orthogonal boundary or truncated class into the following program in MATHEMATICA.

Return[{node, wt}];

In order to avoid ill-conditioned matrices in the system of equations for weights w, for higher n we have maintained working precision as 74 throughout our calculation from evaluation of moments to finding nodes and weights as mentioned in the program nodewt[-}. It may be noted that {x^,x^ i = 1, • • •, n) and corresponding weights wf i = 1, • ■ ■, n) involved in the quadrature rule of Barinka et al. (Barinka et ah, 2001, Eq. (25)), can be evaluated by using inputs {/./)], //.'], • • ■, g'in-1} f°r appropriate lift c and {/(q , /if, ■ • ■, |} as ft-List separately into the same program mentioned

above.

After evaluation of zeros of FOPs and pseudo- or quasi-weights for given n, it is necessary to verify their admissibility as the node of the quadrature rule by checking whether real part of each of the zeros lies within the support of the corresponding pseudo- or quasi-weight function involved in the integral. This can be verified automatically by supplying low-pass filter h, lower and upper limit of integration as k 1. k2 respectively and degree of FOP. nn as input into the program in MATHE- MATICA described below.

Before running of the program ndrc[.] one has to store

Given the inputs mentioned above, output of the program is expected to pair R or C and a number, or N and x. The number, whenever it appears, indicates the stability constant of the quadrature rule correct up to three significant digits. The symbol К or € convey that all the zeros are real or few of them are complex. The symbols N and x imply that few zeros of FOP lie outside the support of pseudo- or quasi-weight function so that the quadrature rule with the prescribed number of nodes is not admissible.

Error estimates

In approximating a function by a 2n — 1 degree polynomial with the aid of n-interpolating points Xi, x%, ■ ■ ■ ,xn, the interpolation error (Hildebrand, 1987; Gautschi, 2012, p. 177) is

The error in the Gauss-quadrature rule involving the positive semi-definite weight function ш(х) or the positive definite inner product < /, g >ш can then be estimated as

Certainly, x, X2, • ■ •, xn are zeros of the orthogonal polynomial P„ . In case of Daubechies scale functions like pseudo-weight functions, the polynomial part of the integrand in the above integral is always positive semi definite. This is due to the fact that the nodes of FOPs appear with their conjugates whenever they are complex. As the scale functions and wavelets are both L2 functions, by first mean value theorem of integral calculus, we can find а ц € (m, M) (different from the notation used in section 3.3) such that

Here, m = inf -—and M = sup -—т^Щ^ш(.г’). Assuming the function /(2тг) (ж) to be xeT> ' >' X£-p ' '

varying slowly and using the fact that f(x) extracts average behavior of f(x) within supp

ip, the estimate of //. for uj(x) = can be found as • A At this point it is observed that for

11

f(x) = x2n, the error < x2n ^ u>i x2n in admissible n-point quadrature rule is close to the

t=i

integral < P2 >v. Thus, an empirical estimate for En[f, p] is found as

However, in case of uj(x) = ф(х), // in (3.4.1.9) cannot be estimated as the average, since ■ф(х) extracts local variation (equivalent to (K + l)th order derivative of f(x) for DauK wavelets) of the function f(x) appearing in the integral f(x)ip(x)dx. So, we retain the estimate of /г in the

supp ip

error for evaluation of integrals for wavelet coefficients by Gauss-Daubechies quadrature rule as

so that the expression for error becomes

It may be observed that the estimate of the error

given in (Huybrechs and Vandewalle, 2005, Eq. (11), p. 125) for quadrature rules with n nodes involving interpolating polynomials is far more complicated than the estimate derived here.

 
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