Nonlinear trigonometric approximation and the Dirac delta function

Transcription

1 Journal of Computational and Applied Mathematics 9 (6) Nonlinear trigonometric approximation and the Dirac delta function Xiubin Xu Department of Information and Computational Science, Zhejiang Normal University, Jinhua, Zhejiang 34, PR China Received 8 February 6; received in revised form 8 October 6 Abstract The nonlinear approximations based on two types of trigonometric generating functions are developed. It is shown that such nonlinear approximations to the Dirac delta function on, are the corresponding Gaussian quadratures applied to some Stieltjes integrals, whose integrands contain weights and the two types of generating functions. In addition, the convergence is proved and the error terms are obtained. Some numerical tests are also shown. 6 Elsevier B.V. All rights reserved. Keywords: Nonlinear approximation; Generating function method; Dirac delta function; Gaussian quadrature. Introduction Nonlinear approximation methods with good convergence properties for formal power series, formal Fourier series and formal series of orthogonal polynomials have been studied by many mathematicians and physicists, see for example, 6,. A common feature of these nonlinear approximation methods is the use of a generating function for the basis functions of a linear series. For a detailed description of the generating function method, the reader is referred to 3,. Let v(x, t) = u k (x)t k k= be a generating function for functions {u k (x)}. Let f(x)be the function to be approximated and given by the formal series f(x)= f k u k (x). () k= () Supported in part by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Zhejiang Province, and Zhejiang Normal University Research Funds. Tel.: address: xxu@zjnu.cn /$ - see front matter 6 Elsevier B.V. All rights reserved. doi:.6/j.cam.6..

2 X. Xu / Journal of Computational and Applied Mathematics 9 (6) The sequence (moments) {f k } enables us to introduce a linear funtional T acting on the space of the polynomials, as usual, T(t k ) = f k, k =,,.... (3) Thus, one can write, at least formally f(x)= T(v(x,t)), (4) where T acts on the variable t, x being a parameter. Suppose now that the functional T is approximated by the following expression T(g) I n (g) := a j g(t j ), (5) where the parameters {a j } and {t j } are to be determined by imposing such conditions of agreement I n (P ) = T(P) for any polynomial P of degree n at most, or equivalently, I n (t k ) = T(t k ) = f k, k =,,...,n. (6) From Eqs. (5) and (6), the conditions of agreement become a j tj k = f k, k =,,...,n. (7) In order to solve Eq. (7), one can follow a standard approach. Set P(t)= n (t t j ) = c + c t + +c n t n + t n so that it holds that T(t k P(t))=, k=,,...,n, or equivalently, c f k + c f k+ + +c n f k+n + f k+n =, k =,,...,n. (8) In other words, the polynomial P(t)whose zeros provide with the parameters {t j } represents the nth formal orthogonal polynomial with respect to the functional T as shown in 3. Certainly, if {t j } have multiplicities {r j }, one should take into account the corresponding derivation conditions as shown in, I n (g) = s r j i= a ji i! i t i g(t), n=,,..., (9) t=tj where s r j = n. From Eq. (4), it seems reasonable to take as an approximation for f(x)that one arising when T(v(x,t))is approximated by I n (v(x, t)), yielding F(n; x) := a j v(x, t j ) = I n (v(x, t)) f(x), n=,,.... () Thus, Eqs. (5), (6) and Eq. () imply the following property. Proposition. If the functional T is expressed by an integral T(g(t))= b a w(t)g(t)dt,

3 36 X. Xu / Journal of Computational and Applied Mathematics 9 (6) where a,b may be an infinite interval and the weight w(x) on a,b, then the nonlinear approximations {F(n; x)} to f(x)are the corresponding Gaussian quadratures applied to the integral b a w(t)v(x,t)dt. In physics, one uses the Dirac delta function δ(x) as a source term for differential equations. The solution when the forcing term is a delta is the Green s function. If ever an expansion of the solution was required then the expansion of δ(x) is useful. In recent years, expansions of the Dirac delta function have appeared in the literature. For example, 8,,3 developed approaches for problems in Chemical Physics based on the polynomial expansions of the Dirac delta function. Charron and Small 4 studied the nonlinear approximations to the Legendre expansion of δ(x) based on the generating function of the Legendre polynomials. In this paper, we develop the nonlinear approximations based on two types of trigonometric generating functions, called geometric and exponential generating functions. we show that these two types of nonlinear approximations to the Fourier series of δ(x) on, are, respectively, the corresponding Gaussian quadratures applied to two Stieltjes integrals represented by the generating functions. In addition, the functions represented by the Stieltjes integrals have the properties that they are zero for all nonzero x, and a total integral of one on,. Furthermore, the approximations are proved to be convergent and the error terms are obtained. Two graphic illustrations and some other numerical results are also given to show the numerical experiments and numerical power of our method.. Trigonometric generating function method In this section, we develop the nonlinear approximations based on two types of trigonometric generating functions. To construct nonlinear approximations for Fourier series we need generating functions for the Fourier terms. For any analytic function (x), (xt) may be used as a generating function for powers of x provided that none of the derivatives (n) () vanishes. For such a function, (e iπx t) given by (e iπx t) = n= (n) () einπx t n n! generates all powers of e iπx and may be used to generate the terms of a Fourier series. Taking real or imaginary parts selects the cosine or sine functions. We consider two functions with differing rates of convergence. The function (x) = x = x n, x, ) n= generates powers of x with coefficients while (x) = e x = n= x n, x (, + ) n! has coefficients /n! which suppress the effects of the higher order terms considerably. We shall call the former a geometric generating function due to the geometric series and the latter an exponential generating function. Definition. Regarding s, t and x as real, we define the following generating functions, labelled with g and e for geometric and exponential function, respectively, ( vg (x, s) = Re se iπx ) ( vg (x, t) = Im te iπx ) = = + cos(nπx)s n, s <, () n= sin(nπx)t n, t <, () n=

4 ( ve (x, s) = Re e seiπx ) ve (x, t) = Im(e teiπx ) = X. Xu / Journal of Computational and Applied Mathematics 9 (6) n= = + n= Now evaluating the real and imaginary parts we obtain vg (x, s) = ( s )/, s <, s cos(πx) + s t sin(πx) vg (x, t) =, t <, t cos(πx) + t ve (x, s) = e s cos(πx) cos(s sin(πx)), ve (x, t) = e t cos(πx) sin(t sin(πx)). cos(nπx) sn n!, (3) sin(nπx) tn n!. (4) In these latter equations the parameters s and t may now be regarded as complex. The expressions that approximate a function f(x)and use these generating functions, take the form n+ Fg(n; x) = a j vg (x, s j ) + b j vg (x, t j ), n =,,... (5) and n+ Fe(n; x) = a j ve (x, s j ) + b j ve (x, t j ), n =,,..., (6) where the 4n+ free parameters {a j }, {b j }, {s j } and {t j } are fixed by means of 4n+ conditions of agreement between f(x)and Fg(n; x) or f(x)and Fe(n; x). These conditions can be written in the form of Eq. (7): where n+ a j sj k = f k, k =,,...,n +, b j tj k = p k, k =,,...,n, (7) f k = f(x)cos(kπx)dx, p k = f(x)sin(kπx)dx for Fg(n; x), and f k = k! f(x)cos(kπx)dx, p k = k! f(x)sin(kπx)dx for Fe(n; x). The {s j } and {t j } are roots of polynomials. If any has multiplicity greater than one, then the form of Fgor Femust be modified to the form indicated in Eq. (9). The approximations {Fg} in Eq. (5) are based on the geometric generating functions defined in Eqs. () and (), and require that the magnitudes of {s j } and {t j } are less than one. When using this generating function, if any of the {s j } and {t j } have magnitudes that violate this condition, the approximations become invalid and the conditions of agreement do not express the equations as indicated above. In this case we can use the magnitude reduction method described in 4 to reduce the magnitudes of {s j } and {t j } until they are less than one.

5 38 X. Xu / Journal of Computational and Applied Mathematics 9 (6) Approximations to the Dirac delta function In this section, we apply the nonlinear approximations described in the previous section to the Dirac delta function. Let f(x)= δ(x), x,. Then its Fourier expansion is f(x) + cos kπx, k= which diverges everywhere. If it is approximated by {Fg(n; x)} as defined in Eq. (5), then the approximations take the form Fg(n; x) = a j vg (x, t j ), n =,,..., where we have modified the upper limit of summation from n + ton for convenience. In this case, the moments are f k = T(t k ) =, k =,,,.... For these moments the system (8) becomes singular since the elements of the matrix to be inverted are all s and hence we cannot solve them for the parameters {t j }. The remedy is to modify the weights in the generating function vg defined in Eq. () slightly by inserting the numbers k + into the kth power terms. Since d/dt(t t k ) = (k + )t k, this effect is obtained in the following way: vg (x, t) := t tvg (x, t) = 4t + 4t 3 cos πx t 4 ( t cos πx + t ) = + (k + ) cos kπxt k, t <. (8) k= This may also be obtained by regarding t as real in vg (x, t) = Re ( te iπx ), t <. (9) We now apply the generating function method to f(x)obtaining the approximations Fg (n; x) = a j vg (x, t j ), n =,,.... () The constant terms (moments) in the agreement conditions (7) are f k =, k =,,...,n. k + Now the parameters in Eq. () may be obtained in a straightforward way for any positive integer n. In general, such approximations may not converge, and when they do the convergence may be very difficult to prove. But in the case at hand we have Theorem. The nonlinear approximations {Fg (n; x)} to the Dirac delta function on, are the corresponding Gaussian quadratures applied to the integral δg (x) := vg (x, t) dt. () Furthermore, δg (x) is zero for all nonzero x, and δg (x) dx =.

6 X. Xu / Journal of Computational and Applied Mathematics 9 (6) Proof. As discussed above, our conditions of agreement are a j tj k = f k, k =,,...,n, where f k =, k =,,,...,n. k + Defining T(g(t))= g(t)dt, then f k = T(t k ) = t k dt, k =,,,.... By Proposition, the first part of the Theorem follows. As to the second part, if x, \{}, then it follows from Eq. (9) that δg dt (x) = Re ( te iπx ) ( ) = Re e iπx =. To deal with the integral of the integral representation, we have u δg (x) dx = lim Re u ( te iπx ) dt dx u = lim Re u ( te iπx ) dx dt. Here we have used the fact, Eq. (9) that the limit of any sequence h γ (x) of generalized functions is defined by lim h γ(x) g(x)dx lim h γ (x)g(x) dx γ γ for any regular function g(x). Putting z = e iπx, then dx ( te iπx ) = z iπ C (z t) dz, where the contour C is centered at t forafixedt, ) and in the clockwise sense. By using the residue theorem we have dx ( te iπx ) =, and hence δg (x) dx =, which completes the proof. The comparison of the geometric generating function approximation and linear approximation using the same data is illustrated in Fig.. To demonstrate the convergence and error terms, we need the following definition and lemmas. Definition. Consider a sequence of functions {F n (x)} with finite integrals F n (x) dx <, n=,,....

7 4 X. Xu / Journal of Computational and Applied Mathematics 9 (6) b a x Fig.. Approximation of the Delta function by Fourier series: (a) term linear approximation; (b) geometric nonlinear approximation, n = 6. If {F n (x)} satisfy lim F n(x) = for x, \{} and lim F n (x) dx =, n n then we say that {F n (x)} converges to δ(x) on,. Lemma (Szëgo, p. 5). If w(t) on a,b (which may be an infinite interval), then a Gaussian formula b a w(t)f(t)dt A j f(t j ) which has degree n is a Riemann Stieltjes sum. Lemma (Kincaid and Cheney 7, p. 497). Consider a Gaussian formula with error term: b a w(t)f(t)dt = A j f(t j ) + E. For an f C n a,b, we have E = f (n) (ξ) (n)! b a w(t)q (t) dt, where a<ξ <band Q(t) = n (t t j ). The following theorem gives the convergence and error terms. Theorem. The sequence {Fg (n; x)} converges to δ(x) for x,. For each positive integer n and each x, \{}, there exists a point ξ (, ), such that E n (x) = δ(x) Fg (n; x) = (n!)4 (n)! Re e iπx (ξ e iπx ) n+.

8 X. Xu / Journal of Computational and Applied Mathematics 9 (6) Proof. From the second part of Theorem, when x, \{}, the Riemann Stieltjes integral with w(x) in Eq. () is equal to, so the first part of Theorem and Lemma give lim n Fg (n; x) =. While Fg (n; x)dx = a j vg (x, t j ) dx = = a j Re ( t j e iπx ) dx a j = f =, which gives the convergence. For each positive integer n and each x, \{}, since the function n e iπx t n vg (x, t) = (n + )!Re (t e iπx ) n+ is continuous on, with respect to t, by Theorem and Lemma, there exists a point ξ (, ), such that E n (x) = δg (x) Fg (n; x) = n (n)! t n vg (x, t) Q n (t) dt t=ξ e iπx = (n + )Re (ξ e iπx ) n+ Q n (t) dt, () where Q n (t) = n (t t j ). Let P n (x) be the Legendre polynomial of order n. Then {t j } are the n roots of the polynomial P n (t) = P n (t ), t,. Assume that x,x,...,x n are the roots of P n (x) in,. Then t j = (x j + )/, j=,,...,n, and Q n (t) dt = ( ) x + Q n dx = n ( x + x ) j + dx = n n+ (x x j ) = (n!) 4 n + (n)!. Substituting this expression into Eq. () yields E n (x) = (n!)4 (n)! Re e iπx (ξ e iπx ) n+. The proof is complete. dx = n (n!) n+ Pn (x) dx (n)! We now consider the approximations {Fe(n; x)} defined in Eq. (6) to the Dirac delta function. In this case, {Fe(n; x)} take the form Fe(n; x) = a j ve (x, t j ), n =,,...,

9 4 X. Xu / Journal of Computational and Applied Mathematics 9 (6) where we have modified the upper limit of summation from n + ton for convenience. In order to discuss the approximate errors, we write the well-known error expression of a Gaussian Laguerre quadrature here. Lemma 3 (Ma et al. 9, p. 94). Consider a Gaussian Laguerre quadrature with error term: e t f(t)dt = A j f(t j ) + E. For an f C n, ), we have E = (n!) (n)! f (n) (ξ), where ξ, ). Our results for {Fe(n; x)} are Theorem 3. The nonlinear approximations {Fe(n; x)} to the Dirac delta function on, are the corresponding Gaussian Laguerre quadratures applied to the integral δe(x) := e t ve (x, t) dt. (3) In addition, δe(x) is zero for all nonzero x, and δe(x) dx =. Furthermore, the sequence {Fe(n; x)} converges to δ(x) for x,. For each positive integer n and each x, \{}, there exists a point ξ, ), such that E n (x) = δ(x) Fe(n; x) = (n!) (n)! Re(eξeiπx +inπx ). Proof. Our conditions of agreement are where a j tj k = f k, k =,,...,n, f k = k!, k =,,,...,n. Defining T(g(t))= e t g(t)dt, then f k = T(t k ) = e t t k dt, k =,,,.... By Proposition, the first part of the theorem follows. As to the second part, if x, \{}, then from Eq. (3), δe(x) = Re e t(eiπx) dt = sin πx/ et(cos πx) sin t sin πx πx t= =.

10 X. Xu / Journal of Computational and Applied Mathematics 9 (6) Similar to the proof of Theorem, we have u δe(x) dx = lim Re e teiπx dt dx u + u = Re e t e teiπx dx dt lim u + u = Re lim e t e tz dz dt, u + iπ C z where the contour C is the unit circle in the counterclockwise sense. Using the Cauchy theorem, we have δe(x) dx = e t dt =. The convergence can be proved as in Theorem by noticing that Fe(n; x)dx = = = a j e t ve (x, t j ) dx a j a j = f =. ( Re e t j e iπx ) dx We now turn to prove the approximate errors. For each positive integer n and each x, \{}, since the function n t n ve (x, t) = Re (e ) teiπx +inπx is continuous on, ) with respect to t, by the first part of this theorem and Lemma 3, there exists a point ξ, ), such that E n (x) = δe(x) Fe(n; x) = (n!) n (n)! t n ve (x, t) = (n!) ( ) (n)! Re e ξeiπx +inπx. The proof is complete. t=ξ The comparison of the exponential generating function method and linear approximation using the same data is illustrated in Fig.. Remark. Although Theorems and 3 provide us beautiful error formulas, we should not use them to estimate error bounds. Because in our case, {Fg (n; x)} and {Fe(n; x)} are approximations to the Dirac delta function, {Fg (n; x) δ(x)} and {Fe(n; x) δ(x)} are expected to be not uniformly bounded on (, ) and (, ). We can check this numerically, For example, for n=, 3,...,8, the values of Fg (n; x)at x = are, respectively, , , , , ,.49977, ; and those of Fe(n; x) are, respectively, , , , , , , But, by noticing that δ(x) = for any x, \{}, we can numerically compute the approximation errors at any x, \{} via a computer. For example, again for n =, 3,...,8, the errors Fg (n; x) δ(x) at x =.5 are, respectively, ,.75768,.759,.356 5, ,

11 44 X. Xu / Journal of Computational and Applied Mathematics 9 (6) b a x Fig.. Approximation of the Delta function by Fourier series: (a) term linear approximation; (b) exponential nonlinear approximation, n = ,. 9 ; and the corresponding ones for Fe(n; x) δ(x) are, respectively, , , , , , , Remark. The following is an example to show the numerical power of our proposed method. Let {, <x<, s(x) =, <x<,, x =,,. It is an odd function and its Fourier expansion is s(x) = π k= k ()k sin kπx, x,, which converges to s(x) at any x,. Consider the following nonlinear approximations to s(x): where F g (n; x) = a j v g (x, t j ), n =,,..., v g (x, t) = π t vg (x, t) = sin πx π t cos πx + t = sin(n + )πxt n, t <, π and vg (x, t) is defined in Eq. (). (The reason why we use v g (x, t) instead of vg (x, t) is that we may obtain t j with t j at the case of vg (x, t).) If we use the linear approximation method to approximate s(x), the convergence rate is very slow. For example, at x =.5, the series requires the first terms to yield an approximate value to s(.5) =. While our nonlinear approximations need only the first 8 nonzero terms of the series to get a better approximate value, F g (8,.5) Remark 3. All the numerical results in Remarks and were carried out in Maple 9.5 on a 3-bit PC. The data in Remark and the nonlinear result in Remark were computed by using the default floating point representation of decimal digits of mantissa in Maple 9.5, while the linear approximation in Remark was computed by extending the length of the mantissa in Maple 9.5 to decimal digits. n=

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