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Computing eigenvalues of boundary-value problems using sinc-Gaussian method.

Abstract

In this paper we compute the eigenvalues of second order Birkhoff-regular eigenvalue problems using the sinc-Gaussian method established by Qian (2002). The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than the classical sinc-method. The error bounds on C of the sinc-Gaussian scheme is investigated by Schmeisser and Stenger (2007). To accomplish our tasks we study both error estimates and the associated amplitude errors on C of the sinc-Gaussian method. Numerical worked examples are given at the end of the paper with comparisons with the classical sinc-methods.

Key words and phrases: Sinc methods, Birkhoff-regular problems, error bounds, Gaussian convergence factor.

2000 AMS Mathematics Subject Classification--34L16, 65L15, 94A20.

1. Introduction

Let [sigma] > 0. The Paley-Wiener space [PW.sup.2.sub.[sigma]] is the space of all entire functions of exponential type [sigma] which lie in [L.sup.2](R) when restricted to R. Elements of [PW.sup.2.sub.[sigma]] are called band-limited functions with band-width [sigma]. Assume that f([lambda]) [member of] [PW.sup.2.sub.[sigma]], then f([lambda]) can be reconstructed via the classical sampling expansion

f([lambda]) = [[infinity].summation over (n=-[infinity]) f(n[pi]/[sigma]) sinc ([sigma][lambda] - [n][pi]), [lambda][member of]C, (1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Series (1) converges absolutely and uniformly on compact subsets of C, and uniformly on R, cf. [8]. Expansion (1) is used in several approximation problems which are known as sinc methods, see e.g. [16,17,25,26]. In particular, the sinc-method is used to approximate eigenvalues of boundary value problems, see for example [1,2,4,6,11]. The sinc-method has a slow rate of decay at infinity, which is as slow as O([absolute value of [[lambda].sup.-1]]). There are several attempts to relax the rate of decay. One of the interesting ways is to multiply the sinc-function in (1) by a kernel function, see e.g. [7, 15, 27]. Let h [member of] (0, [pi]/[sigma]] and [gamma] [member of] (0, [pi] - h[sigma]). Assume that [PHI]([lambda]) [member of] P[W.sup.2.sub.[gamma]] such that [PHI](0) = 1, then for f([lambda]) [member of] P[W.sup.2.sub.[sigma]] we have the expansion, [24]

f([lambda]) = [[infinity].summation over (n=-[infinity])]f(nh) sinc ([h.sup.-1][pi][lambda] - n[pi]) [PHI]([h.sup.- 1][lambda] - n). (3)

The speed of convergence of the series in (3) is determined by the decay of [absolute value of [PHI]([lambda])]. But the decay of an entire function of exponential type cannot be as fast as [e.sup.-c[absolute value of x]] as [absolute value of x] [right arrow] [infinity], for any positive number c, [24]. In [20], L. Qian has introduced the following regularized sampling formula. For h [member of] (0, [pi]/[sigma]], N [member of] N and r > 0, Qian defined the operator, [20]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [S.sub.n]([h.sup.-1][pi]x) := sinc ([h.sup.-1][pi] x - n[pi]), [Z.sub.N](x) := {n [member of] Z: [absolute value of [h.sup.-1]x] - n] [less than or equal to] N} and [x] denotes the integer part of x [member of] R, see also [21, 22]. Qian also derived the following error bound. If f(x) [member of] P[W.sup.2.sub.[sigma]], h [member of] (0, [pi]/[sigma]] and a := min{r([pi] - h[sigma]), (N - 2)/r} [greater than or equal to] 1, then [20, 21]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Here [[parallel]f[parallel].sub.2] denotes the [L.sup.2](-[infinity], [infinity])-norm. Since we aim to use the sampling formula with [lambda] [member of] C, we need to consider error analysis associated with (4) on C. In [24] Schmeisser and Stenger extended the operator (4) to the complex domain C. For [sigma] > 0, h [member of] (0, [pi]/[sigma]] and [alpha] := ([pi] - h[sigma])/2, they defined the operator, [24]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [Z.sub.N]([lambda]) := {n [member of] Z: [absolute value of [h-sup.1]R[lambda] + 1/2] - n] [less than or equal N} and N [member of] N. Note that the summation limits in (6) depend on the real part of [lambda]. Schmeisser and Stenger, [24] proved that if f is an entire function of exponential type [sigma] > 0, then for h [member of] (0, [pi]/[sigma]), [alpha] := ([pi] - h[sigma])/2, N [member of] N, [absolute value of [??][lambda]] < N, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [[parallel]f[parallel].sub.2] denotes the [L.sup.[infinity]](-[infinity], [infinity])-norm and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

It is worthy to mention that Schmeisser-Stenger's estimate is obtained for a less restrictive class of functions. For example the recovered functions are not necessarily band-limited.

In this paper we will use the sinc-Gaussian sampling formula (6) to compute eigenvalues of regular second-order eigenvalue problems. As is expected, the new method reduces the error bounds remarkably, see the examples of Section 4. To establish the method we need to estimate [absolute value of f'([lambda]) - ([G'.sub.h,N]f)([lambda])] on C. Since alternative samples will be used in our sampling formulae, amplitude error appears in our scheme. For this reason we will derive estimates for the amplitude error associated with (6). This will be done in the next section. Section three contains the technique and the associated error analysis. The last section contains conclusions and related questions.

2. Error estimates

This section is devoted to the derivation of estimates for the error bounds associated with the operator (6). We start with estimating [absolute value of f'([lambda]) - ([G'.sub.h,N]f])([lambda])]. We will use the method of Schmeisser and Stenger established in [24].

Theorem 2.1. Let [sigma], M [member of] [R.sup.+] be fixed and f be entire such that

[absolute value of ([lambda])] [less than or equal to] M [e.sup.[sigma][absolute value of [??][lambda]], [lambda] [member of] C.(9)

For h [member of] (0, [pi]/[sigma]), [alpha] := ([pi] - h[sigma])/2, N [member of] N, z [member of] C and [absolute value of [??]z] < N, we have

[absolute value of f'([lambda]) - ([G'.sub.h,N]f)([lambda])] [less than or equal to] 2 (2[alpha] + [pi] [absolute value of cos [h.sup.-1][pi][lambda]] + [absolute value of sin [h.sup.-1][pi][lambda]])

.[[parallel]f[parallel].sub.[inifinity]] [e.sup.-[alpha]N]/[square root of [pi][alpha]N [[beta].sub.N]([h.sup.- 1][??][lambda]), (10)

where the function [[beta].sub.N](*) is defined in (8).

Proof. We may assume that [sigma] < [pi] and h = 1, since the general case can be consequently deduced. Let [lambda] [member of] C and denote by R the rectangle with vertices at [+ or -](N + 1/2) + [R[lambda] + 1/2] + i(y [+ or -] N). By the residue theorem, the authors of [24] proved that

f([lambda]) - ([G.sub.1,N]f])([lambda]) = sin[pi][lambda]/2[pi]i [[integral].sub.R] f([zeta])G([square root of [omega]]([lambda]) - [zeta]))/([lambda] - [zeta]) sin[pi][zeta] (11)

where [omega] := [alpha]/N and G(z) := exp([-z.sup.2]). Differentiating (11) we have for [lambda] [member of] C

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

The second integral of (12) is estimated in [24] via

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Using the same technique of Schmeisser-Stenger, we can obtain the following estimates

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

From triangle inequality and (12)-(15), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Since f [member of] [L.sup.[infinity]](R), then M can he replaced in (16) by [[parallel]f[parallel].sub.[infinity]]. []

Now we investigate the amplitude error associated with (6). The amplitude error arises when the exact values f(nh) of (6) are replaced by approximate ones [??](nh). We assume that [??](nh) are closed to f(nh), i.e. there is [epsilon] > 0, sufficiently small such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

The amplitude error is defined to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

In the following theorem we estimate both

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 2.2. Let [sigma], M, f, N, h, [alpha] be as in the previous theorem. Assume that (17) holds. Then we have for [lambda] [member of] C, [absolute value of [??][lambda]] < N

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Proof. Since

[S.sub.n]([h.sup.-1][pi][lambda]) := h/2[pi] [[integral].sup.[pi]/h.sub.[pi]/h] [e.sup.iw]([lambda]-n[pi]/[sigma])dw, (21)

then

[absolute value of [S.sub.n]([h.sup.-1][pi][lambda]] [less than or equal to] exp ([h.sup.-1][pi][absolute value of [??][lambda]]), n [member of] Z. (22)

From [18] and (22) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

Using the inequality,

[absolute value of exp(-[[lambda].sup.2])] [less than or equal to] exp (- [(R[lambda].sup.2])exp([([??][lambda]).sup.2]), (24)

we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

Assume that [absolute value of [??][lambda]] < N. Let = [[h.sup.-1]R[lambda] + 1/2] - n = l. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

Since x > 0, then the following inequality is valid, [19],

[[integral].sup.[infinity].sub.x] exp(-[t.sup.2])dt [less than or equal to] exp([-x.sup.2])/x + [square root of [x.sup.2] + 4/[pi] (27)

Therefore

[summation over n[member of][Z.sub.N]([lambda]) exp [(-[alpha]/N([h.sup.-1]R[lambda] - n).sup.2]) [less than or equal to] 2 (1 + [square root of N/[alpha][pi]) [e.sup.-[alpha]/4N]. (28)

Substituting from (28) in (25) we get (19). Now we prove (20). Differentiating (21), we obtain

[S'.sub.n]([h.sup.-1][pi][lambda] := ih/2[pi] [[integral].sup.[pi]/h.sub.[pi]/h w[e.sup.iw[lambda]]dw.

Therefore

[absolute value of [S'.sub.n]([h.sup.-1][pi][lambda] [less than or equal to] [pi]/h exp ([h.sup.-1][pi][absolute value of [??][lambda]], n [member of] Z. (29)

From the definition of ([G.sub.h,N]f)([lambda]), (29) and using the inequality [absolute value of sin([h.sup.-1][pi]) [lambda] - n[pi]] [less than or equal to] exp ([h.sup.-1][pi][absolute value of [??][lambda]]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

Using the inequality (24), the hypothesis [absolute value of [??][lambda]] < N and estimation (28), we end with (20).

3. The Method and its error analysis

Consider the second order eigenvalue problem

-y"(x, [mu]) + q(x)y(x, [mu]) = [[mu].sup.2]y(x, [mu]), 0 [less than or equal to] x [less than or equal to] b, (31)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

where q(*) [member of] [L.sup.1](0, b) , [mu] [member of] C and ([[alpha].sub.i1], [[alpha].sub.i2], [[beta].sub.i1], [[beta].sub.i2]) [member of] [C.sup.4], i = 1,2, are linearly independent. We assume that the boundary conditions (32) are regular in the sense of Birkhoff-Stone, see [3, 23]. In [18, [section]4.8] sufficient conditions on [[alpha].sup.ij], [[beta].sup.ij], i, j = 1, 2 are given to guarantee that (32) are regular. The regularity of the boundary conditions guarantees the existence of a countable set of eigenvalues without finite limit points, cf. [18, p. 64 ]. The eigenvalues are not necessarily real and each eigenvalue has algebraic multiplicity which is less then or equals 2. Now let [y.sub.1](*, [mu]) and [y.sub.2](*, [mu]) denote the solutions of (31) satisfying the following initial conditions

[y.sup.(i-1).sub.j](0, [mu]) = [[delta].sub.ij], i,j = 1, 2, [mu][member of]C. (33)

It is known, [13, 18], that [y.sub.i](x,[mu]) is entire in [mu] for x [member of] [0, b] and that the eigenvalues of problem (31)-(32) are the zeros of the characteristic determinant

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

The characteristic determinant [DELTA]([mu]) is also an entire function of [mu]. In [1] the authors proved that [DELTA]([mu]) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

where T and [??] are the Volterra operators defined respectively by

(Ty)(x, [mu]) := [[integral].sup.x.sub.0]sin[mu](x - t)/[mu]q(t)y(t, [mu])dt, (36)

([??]y)(x, [mu]) := [[integral].sup.x.sub.0]cos([mu](x - t))q(t)y(t, [mu])dt, (37)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (38)

Here [T.sup.0] is the identity operator and all series of (35) converge uniformly on [0, b]. As in [1] we split [DELTA]([mu]) into two parts via

[DELTA]([mu]) := K([mu]) + U([mu]), (39)

where K([mu]) is the known part

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

and U([mu]) is the infinite (unknown) sum

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

Since q(*) [member of] [L.sup.1](0, b), then U([mu]) is an entire function of exponential type 2b, cf. [1]. Moreover when [C.sub.5] = 0, U([mu]) is an entire function of exponential type b, [1]. Now we approximate the function U([mu]) using the operator (6) where h [member of] (0, [pi]/2b] and [alpha] := ([pi] - 2hb)/2 and then we obtain

[absolute value of U([mu]) - ([G.sub.h,N]U)([mu])] [less than or equal to][T.sub.h,N]([mu]), [absolute value of [??][mu]] < N, (42)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (43)

The samples U(nh) = [DELTA](nh) - K(nh), n [member of] [Z.sub.N]([mu]) cannot be computed explicitly in the general case. We approximate these samples numerically by solving the initial-value problems defined by (31) and (33) to obtain the approximate values [??](nh), n [member of] [Z.sub.N]([mu]), i. e. [??](nh) = [??](nh) - K(nh). Accordingly we have the explicit expansion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (44)

Therefore we get, cf. Theorem 2.2,

[absolute value of ([G.sub.h,N]U)]([mu]) - ([G.sub.h,N][??])([mu])] [less than or equal to] [A.sub.[epsilon],N]([??][mu]), [absolute value of [??][mu]] < N, (45)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (46)

Now let [[??].sub.N]([mu]) := K([mu]) + ([G.sub.h,N][??])([mu]). From (42) and (45) we obtain

[absolute value of [DELTA]([mu]) - [[??].sub.N]([mu])] [less than or equal to] [T.sub.h,N]([mu]) + [A.sub.[epsilon],N]([??][mu]), [absolute value of [??][mu]] < N. (47)

By similarly treatments we can get the estimate

[absolute value of [DELTA]'([mu]) - [[??]'.sub.N]([mu])] [less than or equal to] [T.sub.h,N]([mu]) + [A.sub.[epsilon],N]([??][mu]), [absolute value of [??][mu]] < N, (48)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (49)

To derive error bounds for [absolute value of [[mu].sup.*] - [[mu].sub.N] we used the following interesting mean value theorem, [14].

Theorem A. Assume that f([mu]) is holomorphic on an open convex set [OMEGA] [member of] C. Let a and b the distinct points in [OMEGA] and [[LAMBDA].sub.a,b] the line joining a and b in [OMEGA]. Then there exist [[xi].sub.1], [[xi].sub.2] [member of] [[LAMBDA].sub.a,b] such that

R(f(b) - f(a)/b - a) = R(f'([[xi].sub.1])), [??](f(b) - f(a)/b - a) = [??](f'([[xi].sub.2])). (50)

Let [([[mu].sup.*]).sup.2] be an eigenvalue and [([[mu].sub.N]).sub.2] be its desired approximation, i.e. [DELTA]([[mu].sup.*]) = 0 and [[??].sub.N]([[mu].sub.N]) = 0. From (47) we have [absolute value of [[??].sub.N]([[mu].sup.*])] [less than or equal to] [T.sub.h,N]([[[mu].sup.*]) + [A.sub.[epsilon],N]([??][[mu].sup.*]). Let [[PI].sub.N] to be the region

[[mu].sup.*] [member of] [[PI].sub.N] := {[mu] [member of] C: [absolute value of T[mu]] < N and [absolute value of [[??].sub.N]([mu])] [less than or equal to] [T.sub.h,N]([mu]) + [A.sub.[epsilon],N](T[mu])}. (51)

From [[PI].sub.N] we determine an enclosure annulus where [[mu].sub.N] lies in. We solve the equation [[??].sub.N]([mu]) = [+ or -]([T.sub.h,N]([mu]) + [A.sub.[epsilon],N]([??][mu])). Let [S.sub.N] be the set of all these solutions in the strip [absolute value of T[mu]] < N. Then [[mu].sup.*] [member of] [S.sub.N]. Let [[mu].sub.0], [[mu].sub.1] be

[[mu].sub.0] := inf{[absolute value of [mu] : [mu] [member of] [S.sub.N]}, [[mu].sub.1] := sup{[absolute value of [mu] : [mu] [member of] [S.sub.N]}. (52)

The enclosure annulus where PN lies in is [[mu].sub.N] := {[mu] [member of] C : [[mu].sub.0] [less than or equal to] [mu] [less than or equal to] [[mu].sub.1]}. If [[mu].sup.*] [member of] R, the enclosure annulus becomes an interval IN. The computation of the error bounds of [absolute value of [[mu].sub.*] - [[mu].sub.N]] depends on whether [[mu].sup.*] is simple or double.

Theorem 3.3. Let [([[mu].sup.*]).sup.2] be a simple eigenvalue of (31)-(32) and [[mu].sub.N] be its approximation. Then, for [absolute value of T[mu]] < N, we have the following estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (53)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the line joining [[mu].sup.*], [[mu].sub.N] in the strip [absolute value of T[mu]] < N.

Proof. Replacing [mu] by [[mu].sub.N] in (47) we obtain

[absolute value of [DELTA]([[mu].sub.N]) - [DELTA]([[mu].sup.*]] < [T.sub.h,N]([[micro].sub.N]) + [A.sub.[epsilon],N]([??][[mu].sub.N]), (54)

where we have used [[??].sub.N]([[mu].sub.N]) = [DELTA]([[mu].sup.*]) = 0. From the complex mean value theorem, cf. Theorem A above, there exist [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (55)

Combining (54) and (55) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (56)

Since [[mu].sup.*] is a simple zero of [DELTA]([mu]) and N is sufficiently large, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57)

and hence dividing on the left hand side of (57), we obtain (53).

Now we estimate the error when [([[mu].sup.*]).sup.2] is a double zeros of [DELTA]([mu]), i.e. [DELTA]([[mu].sup.*]) = 0, [DELTA]'([[mu].sup.*]) = 0 and A"([[mu].sup.*]) [not equal to] 0.

Theorem 3.4. Let [([[mu].sup.*]).sup.2] be a double eigenvalue of (31)-(32) and [[mu].sub.N] be its approximation. Then, for [absolute value of [??][mu]] < N, we have the following estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (58)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the line joining [[mu].sup.*], [[mu].sub.N] in the strip [absolute value of [??][mu]] < N.

Proof. Since [[??]'.sub.N]([[mu].sub.N]) = [DELTA]'([[mu].sup.*]) = 0, then

[absolute value of [DELTA]'([[mu].sub.N]) - [DELTA]'([[mu].sup.*])] [less than or equal to] [T.sub.h,N]([[mu].sub.N]) + [A.sub.[epsilon],N](T[mu].sub.N]).

By similarly treatments of (55) and using the complex mean value theorem, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (59)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [[mu].sup.*] is a double zeros of [DELTA]([mu]) and N is sufficiently large, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (60)

and hence dividing on the left hand side of (60), we obtain (58).

Remark 3.5. If [[mu].sup.*] is real and simple, then estimate (53) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (61)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an interval of R. Also when [[mu].sup.*] is real and double, the estimate (58) turns out to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (62)

Finally we compute an estimate for the norm [[parallel]U[parallel].sub.[infinity]] which appears in the quantities [T.sub.h,N]([mu]) and [T.sub.h,N]([mu]).

Lemma 3.6. The norm [[parallel]U[parallel].sub.[infinity]] may be estimated by

[[parallel]U[parallel].sub.[infinity]] [less than or equal to] A [e.sup.cb[tau]] + B [e.sup.2cb[tau]], (63)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. For [mu], [member of] R, [10], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (64)

where

[gamma]([mu]) := cb[tau]/1 + [absolute value of [mu]]b.

Similarly for [mu] [member of] R, cf. [4],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (65)

where

[delta]([mu]) := c [[integral].sup.b.sub.0] t [[absolute q(t)]/1 + [absolute value of [mu]]t] dt.

Applying the triangle inequality to (41) and using (64), (65) and that 0 < [delta]([mu]) < [gamma]([mu]) < cb[tau], we get the formula (63).

It is worth mentioning that the derivation of the estimate (7) of Schmeisser-Stenger and estimates (10), (19) and (20) for entire functions of exponential type allows us to derive the method without deriving lengthy estimates to prove that U([mu]) lies in a Paley-Wiener space as in [1,2,4,6,11].

4. Examples and comparisons

This section includes some examples illustrating the sinc-Gaussian method. All four examples are computed in [1] with the classical sinc method. We also compare our results with higher order approximations derived by the sinc-methods as in [5,12]. In this higher order method more terms are taken from U([mu]) toK([mu]). However as we will see in all cases the sinc-Gaussian method gives remarkably better results. Let E, [E.sub.h] and [E.sub.G] denote the absolute errors associated with the results of classical sinc method, higher order sinc method and sinc-Gaussian method respectively. We have used Mathematica to derive these examples.

Example 4.7. Consider the self-adjoint boundary value problem

-y"(x) + q(x)y(x) = [[mu].sup.2]y(x), 0 [less than or equal to] x [less than or equal to] 1, (66)

[U.sub.1](y) := y(0) + y'(0) + 2y(1) = 0, (67)

[U.sub.2](y) := 2y'(0) + y(1) + y'(1) = 0, (68)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (69)

This problem is self adjoint. The characteristic determinant is

[DELTA]([mu]) := 4 + 4 cos [square root of 1 + [[mu].sup.2]] + [square root of 1 + [[mu].sup.2]] sin [square root of 1 + [[mu].sup.2]] + sin [square root of 1 + [[mu].sup.2]]/[square root of 1 + [[mu].sup.2]]. (70)

The boundary conditions are strongly regular, cf. [18], and the eigenvalues are simple.

N = 20, h = 0.5
[mu]             Exact                    Sinc

[[mu].sub.1]     2.9781881070693568       2.9781814265432742
[[mu].sub.2]     4.863906114201897        4.8639142423916715
[[mu].sub.3]     9.371576153977740        9.371573550445417
[[mu].sub.4]     11.881215042897248       11.88124190997209

[mu]             Higher sinc (k = 4)      Sinc-Gaussian

[[mu].sub.1]     2.978188876730206        2.978188107069362
[[mu].sub.2]     4.863906019543501        4.863906114201865
[[mu].sub.3]     9.371575729138579        9.371576153977733
[[mu].sub.4]     11.881220331160241       11.881215042897251

    [mu]                 E                   [E.sub.h]

[[mu].sub.1]    6.680 x [10.sub.-6]     7.697 x [10.sub.-7]
[[mu].sub.2]    8.128 x [10.sub.-6]     9.466 x [10.sub.-8]
[[mu].sub.3]    2.604 x [10.sub.-6]     4.248 x [10.sub.-7]
[[mu].sub.4]    2.687 x [10.sub.-6]     5.288 x [10.sub.-6]

    [mu]             [E.sub.G]

[[mu].sub.1]    5.329 x [10.sub.-15]
[[mu].sub.2]    3.197 x [10.sub.-14]
[[mu].sub.3]    7.105 x [10.sub.-15]
[[mu].sub.4]    3.553 x [10.sub.-15]


Example 4.8. The anti-periodic boundary value problem

y"(x) + y(x) = -[[mu].sup.2]y(x), 0 [less than or equal to] x [less than or equal to] 1, (71)

[U.sub.1](y) := y(0) + y(1) = 0, (72)

[U.sub.2](y) := y'(0) + y'(1) = 0, (73)

is self adjoint. The characteristic determinant is

[DELTA]([mu]) := 4 [cos.sup.2] [square root of 1 + [[mu].sup.2]/2]. (74)

Thus the eigenvalues are [[mu].sup.2.sub.k] = [((2k - 1)[pi]).sup.2] - 1, k [member of] Z. All eigenvalues are double from geometric and algebraic points of view.

N = 20, h = 0.5
[mu]             Exact                   Sinc

[[mu].sub.1]     2.97818810706936568     2.978185258432546
[[mu].sub.2]     9.37157615397774        9.371566647265318
[[mu].sub.3]     15.676099962274863      15.676082063009392
[[mu].sub.4]     21.96840038904468       21.968368992899627

[mu]             Higher sinc (k = 4)     Sinc-Gaussian

[[mu].sub.1]     2.9781901187288893      2.978188107069267
[[mu].sub.2]     9.371577293028581       9.371576153977685
[[mu].sub.3]     15.676089772118555      15.676099962274904
[[mu].sub.4]     21.968377143119735      21.968400389044643

[mu]                      E                   [E.sub.h]

[[mu].sub.1]     2.849 x [10.sup.-6]     2.012 x [10.sup.-6]
[[mu].sub.2]     9.507 x [10.sup.-6]     1.139 x [10.sup.-6]
[[mu].sub.3]     1.790 x [10.sup.-5]     1.019 x [10.sup.-5]
[[mu].sub.4]     3.140 x [10.sup.-5]     2.325 x [10.sup.-5]

[mu]                  [E.sub.G]

[[mu].sub.1]     1.421 x [10.sup.-14]
[[mu].sub.2]     5.507 x [10.sup.-14]
[[mu].sub.3]     4.086 x [10.sup.-14]
[[mu].sub.4]     3.908 x [10.sup.-14]


Example 4.9. The non self-adjoint problem

-y"(x) + q(x)y(x) = [[mu].sup.2]y(x), 0 [less than or equal to] x [less than or equal to] 1, (75)

[U.sub.1](y) := 2y(0) + y(1) = 0, (76)

[U.sub.1](y) := y'(0) + y'(1) = 0. (77)

is a special case of problem (31)-(32) when [[alpha].sub.12] = [[alpha].sub.21] = [[beta].sub.12] = [[beta].sub.21] = 0, [[beta].sub.11] = [[beta].sub.22] = [[alpha].sub.22] = 1, [[alpha].sub.11] = 2, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (78)

Its characteristic determinant is

[DELTA]([mu]) := 3 + 3 cos [square root of [[mu].sup.2] + 1]. (79)

Thus the exact eigenvalues are [[mu].sup.2.sub.k] = [((2k - 1)[pi]).sup.2] - 1, [kappa] [member of] Z. Obviously all eigenvalues are algebraically double.

N = 20, h = 0.5
[mu]            Exact                   Sinc

[[mu].sub.1]    2.9781881070693568      2.978215802963121
[[mu].sub.2]    9.37157615397774        9.371667074002096
[[mu].sub.3]    15.676099962274863      15.676264157738332
[[mu].sub.4]    21.96840038904468       21.96866161298621

[mu]            Higher sinc (k = 4)     Sinc-Gaussian

[[mu].sub.1]    2.9781996803447037      2.9781881070699154
[[mu].sub.2]    9.371593195297647       9.37157615398148
[[mu].sub.3]    15.676123563351434      15.67609996227684
[[mu].sub.4]    21.96843727582219       21.96840038904169

[mu]                      E                   [E.sub.h]

[[mu].sub.1]     2.770 x [10.sup.-5]     1.157 x [10.sup.-5]
[[mu].sub.2]     9.092 x [10.sup.-5]     1.704 x [10.sup.-5]
[[mu].sub.3]     1.642 x [10.sup.-4]     2.360 x [10.sup.-5]
[[mu].sub.4]     2.612 x [10.sup.-4]     3.689 x [10.sup.-5]

[mu]                  [E.sub.G]

[[mu].sub.1]     5.587 x [10.sup.-13]
[[mu].sub.2]     3.739 x [10.sup.-12]
[[mu].sub.3]     1.977 x [10.sup.-12]
[[mu].sub.4]     2.991 x [10.sup.-12]


Example 4.10. Consider the non self-adjoint boundary value problem

y"(x) + (1 + i)y(x) = -[[mu].sup.2]y(x), 0 [less than or equal to] x [less than or equal to] 1, (80)

[U.sub.1](y) := y(0) + y(1) = 0, (81)

[U.sub.2](y) := 2y(0) + y(1) = 0. (82)

The characteristic determinant is

[DELTA]([mu]) := sin [square root of 1 + i + [[mu].sup.2]]/[square root of 1 + i + [[mu].sup.2]], (83)

The exact eigenvalues are [[mu].sup.2.sub.k] = [(k[pi]).sup.2] -1 -i, k [member of] Z\{0). Here also all eigenvalues are simple. The exact first four eigenvalues are
[[mu].sub.1]    2.982901531212887- 0.167622026663647i
[[mu].sub.2]    6.2036210116631825- 0.08059808925464174i
[[mu].sub.3]    9.371728017861798- 0.0533519537749109i
[[mu].sub.4]    12.52658228065468- 0.03991511721215217i


N = 20, h = 0.5
[mu]            Sinc

[[mu].sub.1]    2.9829015680859707- 0.16761945435822637i
[[mu].sub.2]    6.203620968394717- 0.08060326582022719i
[[mu].sub.3]    9.37172807759573- 0.053343847904264656i
[[mu].sub.4]    12.526582201352475- 0.03992668762239516i

[mu]            Sinc-Gaussian

[[mu].sub.1]    2.9829015312128813 - 0.16762202666385873i
[[mu].sub.2]    6.20362101166314 - 0.0805980892546575i
[[mu].sub.3]    9.371728017861642 - 0.05335195377534977i
[[mu].sub.4]    12.526582280654676 - 0.03991511721215072i

[mu]             Higher sinc (k = 4)

[[mu].sub.1]     2.9829015545857884 - 0.16761944053493602i
[[mu].sub.2]     6.203620995800119 - 0.08060329354514285i
[[mu].sub.3]     9.371728034447068 - 0.05334380433266186i
[[mu].sub.4]     12.526582263460536 - 0.03992675024110139i

[mu]             E                      [E.sub.h]

[[mu].sub.1]     2.573 x [10.sup.-6]    2.586 x [10.sup.-6]
[[mu].sub.2]     5.177 x [10.sup.-6]    5.204 x [10.sup.-6]
[[mu].sub.3]     8.106 x [10.sup.-6]    8.149 x [10.sup.-6]
[[mu].sub.4]     1.157 x [10.sup.-5]    1.163 x [10.sup.-5]

[mu]             [E.sub.G]

[[mu].sub.1]     2.118 x [10.sup.-13]
[[mu].sub.2]     4.545 x [10.sup.-14]
[[mu].sub.3]     4.659 x [10.sup.-13]
[[mu].sub.4]     7.837 x [10.sup.-15]


5. Conclusions

The sinc-Gaussian method applied above gives results more accurate than the classical sinc-technique. If we write the exact errors in the normal decimal representation, we find that the absolute errors ranges between [10.sup.-11] and [10.sup.-14]. However, we should notice that the round-off and other types of errors occur in many occasions. First due to rounding [pi] and the existing radicals. Second, round-off error appears when solving transcendental equations by Mathematica. The last situation arises when solving initial value problems by numerical techniques. In this setting, it is worthy to study the effects of these types of errors on this study as well as previously established studies.

To summarize the technique when we have an eigenvalue problem of the type (31)-(32), we do the following.

1. We compute the functions [kappa]([mu]) by computing the Volterra operators. If one wants to apply higher order approximations, one should compute more terms.

2. Then we solve numerically initial value problems (31) and (33) to find [??](nh), n [member of] [Z.sub.N]([mu]) and [??}(nh) = [??] (nh) - K(nh). Here we should study the effects of different methods and find the best one. At the moment we trusted the choices of Mathematica.

3. Finally we calculate ([G.sub.h],[sup.N][??])([mu]) and find the roots of [[??].sub.N]([mu]) = K([mu]) + ([G.sub.h],[sup.N][??])([mu]) on [absolute value of [??][mu] < N, which are the desired approximations.

We end this section and the paper as well by justifying why we call the previous technique sinc-Gaussian. First of all the name Gaussian kernel came from the paper [21]. Second, although the Gaussian kernel lies under the title of sampling by generalized kernels, cf. [27], it has very interesting properties that led to the derivation of the truncation error formula of [20]. In particular the relationship between the Gaussian kernel exp(-[z.sup.2]) and the Hermite polynomials via Fourier transforms.

References

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[21] L. Qian and D.B. Creamer, A Modification of the sampling Series with a Gaussian multiplie, Sampl. Theory Signal Image Process., 5 , 1-20, 2006.

[22] L. Qian and D.B. Creamer, Localized sampling in the presence of noise, Appl. Math. Lett., 19, 351-355, 2006.

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[27] R. L. Stens, Sampling by generalized kernels, Sampling Theory in Fourier and Signal Analysis: Advanced Topics (J. R. Higgins and R. L. Stens, eds.), Oxford University Press, Oxford, 130-157, 1999.

M.H. Annaby

Department of Mathematics and Physics, Qatar University, P.O. Box 2713 Doha, Qatar

On leave from Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

e-mail address: mannaby@qu.edu.qa and mhannaby@yahoo.com

R.M. Asharabi

Department of Mathematics, Faculty of Science

Sana'a University, Yemen

e-mail address: rashad1974@hotmail.com
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