a sine approximation for the indefinite integral€¦ · received april 23, 1981; revised march 10,...

MATHEMATICS OF COMPUTATIONVOLUME 41. NUMBER 164OCTOBER 1983. PAGES 559-572

A Sine Approximation for the

Indefinite Integral

By Ralph Baker Kearfott

Abstract. A method for computing fa f(t) dl, x = (0, I) is outlined, where f(t) may have

singularities at t = 0 and í = 1. The method depends on the approximation properties of

Whittaker cardinal, or sine function expansions; the general technique may be used for

semi-infinite or infinite intervals in addition to (0,1). Tables of numerical results are given.

1. Introduction and Summary. Here we present a basic method and experimental

results for approximating Fix) = /0V fit)dt, x E (0,1), where/is smooth, but may

have singularities at t = 0 or t = 1. The method is based on approximation proper-

ties of sine functions

iir/h)ix — kh)

With minor modifications, the method may also be employed to compute jx fit)dt,

t E ia,b), where a, b or both a and b are possibly infinite.

Our underlying formula is similar to formula (3.36) in [4], given there without

proof. Our formula is considered in detail here, is proven, and is backed by

numerical experiments. (See Section 4 for more discussion.)

In Section 2 we give the relevant properties of sine functions, assumptions, and

basic techniques. (Such techniques appear in more detail in [4] and elsewhere.) In

Section 3 we present and verify our integral approximation formulas. Four tables of

experimental results and conclusions appear in Section 4.

2. Underlying Properties and Assumptions. A general review of sine functions and

their uses has recently been given by Stenger in [4]. We therefore only outline

properties important to our present goals, and refer the reader to [4] for further

references.

Approximations on (0,1 ) are obtained from corresponding approximations on

R = (-co, oo) via a conformai map. To be approximable on R, /£ C°°(R) must

obey certain analyticity and boundedness conditions in a strip in the complex plane

C which contains R. Through the conformai map, we obtain a corresponding

"eye-shaped" region (cf. Figure 4.1, p. 185, of [4]) containing the interval (0,1), in

which our integrands must obey certain analyticity and boundedness conditions.

With this in mind, we first list properties over R.

Received April 23, 1981; revised March 10, 1982.

1980 Mathematics Subject Classification. Primary 65D30; Secondary 41-04, 41A25, 41A55.Key words and phrases. Indefinite integrals, singular integrals, quadrature, sine functions, Whittaker's

cardinal function, approximation theory.

^1983 American Mathematical Society

0025-5718/83 $1.00 + $.25 per page

559

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

560 RALPH BAKER KEARFOTT

Defintion 2.1. Let d > 0 and let 6Üd denote the open strip

(2.1) %={zEC:\lm(z)\<d).

Let B iaf}d) denote the family of all functions/ that are analytic in s\)d and which

satisfy

(2.2) ¡d\fix + iy)[dy-+0 asx-±oo

and such that

Wp

< oo.(2.3) Np(f,%) = ma_Uf\fix + iy)\Ddx)j + (/j/(x - iy)f dx)

We denote B^d) by Bi^d); in all results below, p = 1 for simplicity unless

otherwise stated, although most of these results generalize to arbitrary p; cf. [4,

Section 3.1].

If/ E Bi^j), we approximate/by a truncated " Whittaker cardinal series"

(2.4) f(x) = CNif,h)(x)= 2 f(kh)S(k,h)(x),k = -N

where Sik, h)ix) is as in (1.1). We state the following without proof:

Theorem 2.1. (See [2, pp. 237-238], and [4, pp. 177-178].) Suppose f E Bpi%)for

some p E [1, oo), and suppose

(2.5) \f(x)\<Ce-* forallxER(t.e.,\f(x)\= 0(e-"M))

for some C and a > 0 dependent on f. Choose h = [7rd/(aN)]]/2. Then there is a C"

dependent on f, but not on N, such that

(2.6) H/U) - CN(f, h)ix)\\x < c'A"/2e-(WaA,»'/2,

where || • H^ is the supremum norm over R.

Here we need not only to approximate functions /, but also their integrals. We

have

Theorem 2.2. (Trapezoidal Rule; cf. e.g. [2, p. 229], or [4, p. 178].) Suppose

f E Bi^Ùj), suppose f satisfies (2.5), and choose h' = [2-nd/aN]l/2. Then

(2.7)r ¿,-(2T7ctaN),/2c,efXf(t)dt-h' 2 f(kh')

-oo k = -N

for some integral approximation constant C, dependent on f but not on N.

To obtain approximations over (0, 1), we make use of the following conformai

map:

(2-8) v(,) = kjg(T4_), 9'(z)=^~y

For any d such that 0 < d < ir/2, q> maps the region

(2.9) %={z:\arg[z/(<-z)]\<d}


A SINC APPROXIMATION FOR THE INDEFINITE INTEGRAL 561

onto ^j where 6ffd is as in (2.1). (Note that 9^ consists of two circular arcs

intersecting symmetrically with an angle of 2d at 0 and 1); (cf. [4, p. 185].) Define

(2.10) z = xpiw) = no-\w) = j^, xP'iw)=xPiw)[l-xPiw)].

Suppose/is defined and analytic in the interior of ®ùd, and define

(2.11) f(^)=f(^))xp(w)[l-xpiw)]

for w E %. Then/ G Bi^d) provided

(2.12) lim inf f \f(z) dz\< oo,

where Q is some curve; we obtain (2.12) from (2.2) and (2.3) by applying the change

of variables z = xpiw) to the integrals in (2.2) and (2.3).

Suppose, for example,/has singularities at z = 0 and at z = 1, but/is analytic in

°0d, and / is continuous on 3'?)</\{0,1). Then, by examining the integrals over

portions of the curves S near z = 0 and z = 1 and the integrals over portions of G

away from z = 0 and z = 1 separately, one can show (2.12) holds if and only if

f \f(z)dz[< oo.

Suppose further that/is such that we may take d = it/2, so 9^ = [z:\z — {\= {).

Then xp maps the line (z : Im(z) = tt/2} to the semicircle (|z — {-1= {, Im(z) > 0}

and the line {z : Im(z) = -m/2) to the semicircle {\z — \\= \, Im(z) < 0). Condi-

tion (2.12) then becomes

(2.12)' Se\f(z)dz\=\j2;\f(\ + \e«)\dt

= C"/4 \f(eiucos(u)) | du + r/A |/(1 - eivcosiv)) \dv<*o.Jir/4 J-tt/4

Condition (2.12)' then holds for those/which satisfy

(2.13) f(z) = O(|z|°o) asz-^0, and f(z) = 0(\1 - zf<) asz-*l,

where a0 > -1 and a, > -1, and which are continuous elsewhere on the closure of

tf)d and which are analytic in tf)d.

Corresponding to Theorem 2.1, we have

Theorem 2.3. Suppose f is analytic in some tf)d as in (2.9), suppose f satisfies (2.12)

ior (2.12)'), suppose

(2.5)' |/(jc)|<Cx"(1 -x)p, 0<x<l,

for some ß > -1 and some C dependent on f and set h = [Trd/aN]x/2, where

a = ß + 1. Then

(2.6)' /(*)" 1 f(Hkh))xP(kh)[l-xP(kh)]S(k,h)(<p(x))k = -N

< C'Nx/2e'(,rdaN)'/2

for some C dependent on f but not on N.



Similarly, we have

Theorem 2.4. Suppose f is analytic in tf)dfor some d, suppose f satisfies (2.12) (or

(2.12)'), suppose f satisfies (2.5)' for some ß > -1 and some C dependent on f, and set

h' = [2ird/aN]l/2. Then

(2.7)' Cf(t)dt-h' 2 f(Hkh'))xP(kh')[l - xp(kh')]•'n ,k = -N

for some C, dependent on f but not on N.

^C,e -(27TÍ/OV)''

3. The Indefinite Integral Approximation Formulas. Analogously to Section 2, we

will first obtain an approximation procedure for integrals of the form

(3.1) F(w) = f f(t)dt,

where / E Bi fyd ) (cf. Definition 2.1). We will then use the conformai mapping

procedure to obtain approximation rules for integrals of the form

(3.1)' F(x) = f X)fi<pit))dt,

where <p and xp can be as in formulas (2.8) and (2.10), respectively. We begin with the

following lemma, which is a direct consequence of Theorem 2.1.

Lemma 3.1. Define Fiw) as in (3.1), suppose F E Bpi0l)d), for some p E [1, oo),

suppose |F(w)|s£ Ce""w for some C and a > 0 and all w E R, and set h =

[■ïïd/iaN)]]/2. Then

(3.2) /(Odtí N

1(k = -N

f/0>di Sik,h)iw) < C'A/l/V'"/o,/V)'

for all w E R and for some indefinite integral approximation constant C¡ which does

not depend on N.

Lemma 3.1 does not apply to general integrals F as in (3.1), since in general

fHc00fit)dt ¥= 0, and formula (2.5) is not valid. For such F, we consider the related

function

(3.3) G(w) = f_ M*- f f(t)dt

-c f(t)

?-—■'' -H e

ae~'Uf) dt i\(t)dt.

where

(3.4) *(')=/(')ae

(e"'+l)

(ea'+ iy

Uf), Uf) = ff(t)dt.

We then have

Lemma 3.2. Suppose |/(w)|< Ce "M for some C and a > 0 and all w E R, and

suppose G is as in (3.3). Then there is a CG such that | C7(w) |< Cce"aW for all w G R.



Proof. We have

|G(W)|.

,aw/2

,aw/2 _|_ £-aw/2f f(t)dt~Ix(f)

+-aw/2

/w f(t)dteaw/2 + e-

We will consider each factor in the two terms on the right as w -> -co and as

W -» +00.

Case 1, w -» -co. The second factor in the first term is bounded and approaches

-Ixif), while the first factor in the first term is bounded by e"a|M'1. The first factor in

the second term is bounded by 1, while the second factor in the second term is

bounded by Ce-aW/a.

Case 2,w-> oo. The first factor in the first term is bounded by 1, while the second

factor in the first term is equal to -J™fit)dt and hence is bounded by Ce'^/a.

The first factor in the second term is bounded by e"aW, while the second factor in

the second term is bounded and approaches Ixi f ).

Combining the above, we may take

Cc = C/a + max sup / /(/) dt - Ijf) , sup / fit) dt\ w«0 l-oo w>0 l-oo

~C/a + \IJf)\.

Combining Lemma 3.1 and Lemma 3.2 immediately gives

Theorem 3.1. Suppose /G Bi%), suppose |/(w)|< Ce"aW for some C and all

w E R, and suppose G E Bpi^d) for some p, where G is as in (3.3). Define

(3-5) FN(w) fit)ae

,aw/2

+

(e°" + 1) J

/CO

f(t)dt,

/OOf(u) du dt\S(k,h)(w)

e-aw/2 _|_ e«w/2

where h = [<nd/iaN)]x/2. Then there is a C'„ independent of N such that

/w fit)dt-FNiw) (J']yW2e-(irdaN)

for all w E R.

In practice the quantities

rkh

f /(<)-ae

ie°" + 1) -«C f(u)du dt= fkhg(t)dt = G(kh),

and the quantity /^/(u) du = Ixif) must be approximated. Below, we show how

to do this so that the order of approximation in (3.5) remains OiN2e'^daN) ).

First observe that linear combinations of functions in Bitf)d) are also in Bi^d);

also linear combinations of functions satisfying (2.5) satisfy (2.5) (with a different

C). Thus, we deduce



Lemma 3.3. Suppose f E Bi^), suppose |/(w) |< Ce~aM for some C and a > 0 and

all w E R, and suppose g is as in (3.4). Then g E Bi^), and there is a Cg such that

\giw) |< Ce"* for all w E R. Thus, g and jfx git) dt may be approximated as in

(2.6) and (2.7), respectively.

Let CNig, h)it) = 2j=.NgiJh)Sij, h)it) be the approximation to g. We use

CNig, h)it) to approximate the quantities Gikh) in (3.5).

Lemma 3.4. Let f G Bi%), suppose ¡fit) \< Ce"""1, t G R, and set h =

[■7rd/iaN)]l/2. Suppose g and G are as in (3.3) andi3.4), and define

N

(3.6) GNiw)= f CNig,h)it)dt= 2 g(Jh)f S(j,h)(t)dt,J-Ml, ., J-Wl,-Nh j = -N -Nh

where -Nh < w < Nh. Then there is a constant C dependent on f such that | GNiw) —

G(w)|< CNe-l"daN)>/2 for all w E R.

Proof.

\GNiw) - G(w)\*z\G(-Nh)\+ f \g(t)-CNig,h)(t)\dt.J-Nh

But | G(-/YA)|< Cce'aNh = CGe~(,näaN)i/\ from Lemma 3.2. Also

f [git) - CN(g, h)(t)\dt < (NH + w)C'Nx/2e-("dttN)W1 < 2N^2hC'e^daN)W1J-Nh

= CNe-^daN)>/\

from Lemma 3.3. Thus

|Gw(w) - G(w)\ < CGe-("daN>'/2 + CNe-("daN)l/1 < CNe^daN)W1

for some appropriate C.

We now replace Gikh) by Gikh) in Theorem 3.1.

Lemma 3.5. Suppose the assumptions of Theorem 3.1 hold, but

rkh

}(kh) = f fit)ae"

/OOJW) du dt(ea' + 1) J~<*

in (3.5) is replaced by GNikh) as in (3.6), to obtain

(3.5)' FNiw)= 2 GNikh)Sik,h)(w) + - J™'2 aw/2C fit)k = -N e

Then there is a CF independent of N such that

dt.

\f fit) dt - FN(w) CFN2e-(m/aN)'/2.

Proof.

/W A I fW Afit) dt - FNiw)U\j fit)dt-FNiw) +\FNiw)-FNiw)\,



where the first term on the right is bounded by C',Nx/2e (7"JaN>'/\ by Theorem 3.1.

On the other hand,

\F„(w)-FN(w)\ 2 [G(kh)-GNikh)]sik,h)iw)k = -N

N

< 2 \Gikh) - GNikh)\^2NiCNe-^daN)>/1)k = -N

< C N2e-<«daN)'/2

for some constant CF dependent on/but not on N.

For the main result, we must replace Ixif) = ffx fiu)du in (3.5)' without

changing the order of approximation.

Lemma 3.6. Suppose the assumptions of Theorem 3.1 hold, but in (3.5)' we replace

U/) = /-»/(')* by INif) = hl^^.fikh), where h = [t7¿/(«A')]i/2. Then theconclusion of Lemma 3.5 is still true.

Proof. If there are a C and an a > 0 such that |/(/)|< C<?~a|1, / G R, then, if

a' = a/2, |/(r)|< Ce-"'n,t E R. Thus from Theorem 2.2,

(3.7)

Define

(3.8)

and define

and

lU/)-'v(/)l<c/e-<^">'/\

lv(0= 2i = -N

f(jh)ae

a ¡h

Mf)(eajh + l)2

Gfl(») = f C^g^AKí)^

S(j,h)(t),

-Ml

,aw/2

(3.9) FNiw)= 2 GN(kh)S(k,h)(w)+ -' ** ' /v/ _-aH>/2 i .aw/2

a- = -a/ e 7 -t- e 7/*(/)•

Then

(3.10) \f f\t) dt - FN(w)\ *¿f f(t) - FN(w)+ \FN(w)-FN(w)\,

where FN is as in (3.5)'. By Lemma 3.5, the first term on the right of (3.10) is

bounded by CFN2e-^daN)i/\ while

(3.11) \FN(w)-FN(w)\k = -N

+

2 [GN(kh)-GN(kh)]s(k,h)(w)

Uf)-INif)\,,aw/2

e-aw/2 _j_ £aw/2



where GN is as in (3.6). The first term on the rieht of (3.11) equals

(3.12)

aea jhN N

2 2k=.N [j=.N (e«i* + l)2

[/*(/) - Uf)]fk"s(j, h)(t)dt\s(k, h)(w)

while the inner sum in (3.12) equals

aeajh k-j SÍn(7T?)(3-13) Ak,N = h 2 -^-—2\iAf)-Uf)\r^P-dt.

/ = _.v ie'fh + j) N-j

But AkN is a trapezoidal rule approximation, as in Theorem 2.2, to J^x Xk Nit)dt,

where

(3.14) \k,Aw)=[lN(f)-Uf)]ae

(eaw + l)2 Ik-w/h SÍn(77-í)

N-w/h 77/dt

Sk-w/h sin(w/)

N-w/h <!ttdt sech

[(!)-]■

where h' = i<nd/aN)x/1 = i2-nd/a'N)x/1 for a' = a/2. But the magnitude of the

integral in (3.13) is bounded by

2 r"sin(i)

thus

2 fsm(t) .-sup / ——dt<2;

(3.15) |A*,„(hO|< 2a|/y(/) 7°°(/)l <2a\INif)-IJf)\e-W.i£aw/2 _|_ e-aw/2y

It is also possible to verify that Xk Niw) E Bityj). Thus, as in Theorem 2.2,

(3.16) |/°V*(odt- Ak.N C-eA 0-(TidaN)wl

dt

for some constant C,A dependent on /but not on N. But, by (3.7)

(3.17) \fykMdt\<±Mf)-iM)\[fy&[(%y

= \IN(f)-Uf)\<C,e^d°^/2.

Combining (3.17) and (3.16), we see that the quantity in (3.12) is bounded by

N

(3.18) 2 (C, + cf)e-<"daN)'/2 = 2(C, + C,A)Ne^daN)'/2.

k = -N

Combining the bounds (3.18), (3.7), and (3.11) now gives

(3.19) \FN(w) - FN(w)[< 2(C, + CfA)Ne-("daN)'/2 + C,e^daN)W2.

We obtain the conclusion in Lemma 3.6 from Lemma 3.5, 3.10, and 3.19.



We summarize the above in the following

Theorem 3.2 (Main Result). Suppose f G Bi%), and |/(w) |« Ce"aM for some

a > 0 and C, and all w ER. Suppose further G E Bi^j), where G is defined in terms

°ff by (3.3). Define aq, q an integer, by

(3.20) aq-^——dt.

Set h = [trd/iaN)]]/2, and approximate f_wxfit) dt by

(3.21)

FA») f(jh)~ae a i h

(eajh + I)2Mf) (ar,+J + ak_J)\S(k,h)(w)

,aw/2

e-aw/2 + £aw/2 nU ),

wheretNif) = h%%=_Nfikh). Then there is a constant CF independent of N such that

\f fit) dt - F„iw) < CFN2e-^daN)

for all w G R.

Proof. Theorem 3.2 is a restatement of Lemma 3.6, once one observes

rkh/kh Sij,h)it)dt.

-Nh

A table of aq, q = 1.100, is found on p. 175 of [4]. Also, aq = Si(gîr)/•&■, an

asymptotic expansion for which is given on p. 244 of [1]; we are told this expansion

gives 20 significant figures for q > 20.

Theorem 3.2 is used in conjunction with conformai mapping techniques (ex-

plained in [4, Section 4], and the references therein) to evaluate singular integrals of

the form f¡¡ fit)dt, where a or A or both a and b are finite, and / possibly has

singularities at a or b. We consider here (a, b) = (0,1), and the conformai maps <p

and xp defined in (2.8) and (2.10). From Theorem 3.2 and considerations in Section 2

of this paper, we obtain

Theorem 3.3. Suppose (as in (2.8)-(2.13)) that f E 5(<t)¿), and suppose

(3.22) \f(x)\< C\xf (1 -xf all xEiO, 1),

for some C and ß > 1. Set a = ß + 1, and set h = iird/aN)x/2. Define

(3.21)'

,ajh

FNi*) f(Xj)(Xj)(l - Xj)ae

(eaJh + 1)

S(k,h)(cp(x)) +

Mf) i°N +j >k-j)

xa + (1 - x)a!N(f)



where INif) = hlNk=_Nfixk)ixk)il - xk) and where Xj = xpijh) = eJh/il + e'h).

Then, for x G (0,1),

(3-23) \ffit)dt-FNix) CFNle20-(vdaN),/2

for some C dependent on f but independent of N. If /(z) = 0(\zf) as z -» 0, z G C,

then f E Bi%), with d = v/2.

Note that (3.22) is equivalent to

|/(^(w))^(w)[l -xp(w)]\ = \f(x)x(l + x) |« ce"* for w GR.

Theorem 3.3. is otherwise obtained from Theorem 3.2 by making the transformation

(3.24) f fit) dt = rx)fixpiu))xpiu)[l - xPiu)] du,J0 J-x

then applying Theorem 3.2 to the right member of (3.24).

To comment on the power of the approximations in Theorem 3.2 and Theorem

3.3, we note that it is unnecessary to know precisely what the order a is; in

particular, if ¡fit) |< ce~°" and 0 < a' < a, then we may replace a by a' in any of the

formulas. Note that the convergence order is not preserved in this way when we use,

say, Newton-Cotes or Gauss type formulas derived with the method of moments for

weighted integrands of the form xapix), p a polynomial. Also, sine function

approximations are optimal, as explained in [3].

4. Experimental Results. To empirically test formula (3.21)', we tried it for various

TV on the following functions:

(i)/,(x) = |x"2/3;

(ii)/2(x) = f*'/3;

(iii)/3(x) = ¿[x~2/3 + (1 - *r2/3]; and

(iv)/4(*) = ¿[*-9 + (i7*r7]-In each case, the difference FNix) — /0A /(/) dt was computed at x = Ai, i = 1,..., 10,

for TV = 4,8,16, and 32, corresponding to 9,17,33, and 65 evaluations of /,

respectively. These results were compared to the errors Ffiix) — ¡q fit)dt, where

Ffiix) is the result of applying a 2TV-point Gauss-Legendre formula to compute

fo f(t)dt, for 27V = 8,16, and 32; F^ix) was defined to be the result of applying

the composite 32-point Gauss-Legendre formula with 2 subdivisions. These particu-

lar formulas were chosen for comparison since the Gauss formulas are commonly

thought to be generally accurate, and since the 8-, 16-, and 32-point formulas were

closest in number of points to 9,17, and 33 of those formulas already programmed

on our system.

The functions/, / = 1,... ,4, are normalized so that j0x fit)dt = 1. The functions

have singularities at x = 0 and x = 1 of varying orders; the singularities of/3 are the

same order at x = 0 and x = 1, while the singularities of /, /' ¥= 3, differ in order at

x = 0 and x = 1.

Four tables appear, corresponding to each of the four functions. Table 3 has three

parts, corresponding to various estimated orders ß (cf. (3.22)).

The sine function approximations compare favorably to the corresponding Gauss

formulas, except for/2(x). We postulate that the worse the singularity, the better the

merit of the sine function approximation.



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For Table 1, ß = -2/3 (a = 1/3) was used, while ß = 1/3 in Table 2. The values

ß = -2/3, ß = -.467, and ß = -.867 were used in Tables 3a, 3b, and 3c, respec-

tively, while the value ß = -.9 was used in Table 4. From these and other

experiments, it appears it is better to underestimate ß than to overestimate it.

In all cases, d was taken to be 7r/2.

Stenger has proven a formula analogous to (3.5)', namely

(4.1) f git)dt = h 2 2 ok..,gilh)Sik, h)ix) + OiNe-^d^/2),-°° k = -N l=-N

using formulas (2.17), (3.7), and (3.11) in [4] and the uniqueness of the coefficients in

the sine function expansion [5]. (This formula appears without proof as formula

(3.36) in [4]*.) It is then possible, using the same argument as in the proof of Lemma

3.6, to obtain an analogue to Lemma 3.6 from (4.1). This analogue was also tried in

the same cases as above; in all cases, the tables of errors were the same to at least

two significant digits. Since (4.1) is more elegant and requires less additions, it is

perhaps to be preferred.

The formula resulting from (3.21) when IN is replaced by Ix was also tried. From

the resulting error tables, it appeared that the approximation for Ix accounted for

approximately half of the total error. Thus, it seems unnecessary to use a separate

stepsize h in computing 1N, as suggested by (2.7). Also, taking d other than 77/2 in no

case improved the performance.

The computations were done on a Honeywell 6880 (Multics system) in double

precision (63-bit mantissas). To get values at x = 1, x = 1 — 2"62 was used.

Acknowledgements. I wish to thank Professor Frank Stenger for introducing me to

these approximation techniques. I also wish to thank the referee for his careful

reading and for his suggestions in making the paper more comprehensible and

useful.

Department of Mathematics and Statistics

University of Southwestern Louisiana

Lafayette, Louisiana 70504

1. M. Abramowitz & I. A. Stegun, Editors, Handbook of Mathematical Functions, Dover, New York,

1970.2. F. Stenger, "Approximations via Whittaker's cardinal function," J. Approx. Theory, v. 17, 1976,

pp. 222-240.3. F. Stenger, "Optimal convergence of minimum norm approximations in Hp", Numer. Math., v. 29,

1978, pp. 345-362.4. F. Stenger, "Numerical methods based on Whittaker cardinal, or sine functions," SI A M Rev., v.

23, 1981, pp. 165-224.5. F. Stenger, private communication.

* Formula (3.36) in [4] was attributed to Kearfott, Sikorski, and Stenger; that paper never existed. Also,

the derived formula (4.58) in [4] was mistakenly attributed to John Lund; that formula corresponds to

(3.23) of this paper.


a sine approximation for the indefinite integral€¦ · received april 23, 1981; revised march 10,...

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