ERROR ESTIMATES FOR APPROXIMATING THE FOURIER TRANSFORM OF FUNCTIONS OF BOUNDED VARIATION

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1 ERROR ESTIMATES FOR APPROXIMATING THE FOURIER TRANSFORM OF FUNCTIONS OF BOUNDED VARIATION N.S. BARNETT, S.S. DRAGOMIR, AND G. HANNA Absrc. I his pper we poi ou pproximio for he Fourier rsform for fucios of bouded vriio d sudy he pproximio error of ceri ssocied qudrure rules. 1. Iroducio The Fourier Trsform hs pplicios i wide vriey of fields i sciece d egieerig [1, p. xi. I his pper, by use of some iegrl ideiies d iequliies developed i [4(see lso [5, we poi ou some pproximios of he Fourier rsform i erms of he complex expoeil me E (z, w (see Secio 2 d sudy he error of pproximio for differe clsses of mppigs of bouded vriio defied o fiie iervls. Le g : [, b R be Lebesgue iegrble mppig defied o he fiie iervl [, b d F (g is Fourier rsform, i.e., F (g ( := g (s e 2πis ds. The iverse Fourier rsform of g will lso be cosidered, d will be defied by F 1 (g ( := The followig iequliy ws obied i [2: g (s e 2πis ds. Theorem 1. Le g : [, b R be bsoluely coiuous mppig o [, b. The we hve he iequliy F (g (x E ( 2πix, 2πixb g ( d 1 3 g (b 2, if g L [, b, 2 q 1 (b 1 q g p, if g L p [, b ; 1 p q = 1, p > 1, ; [(q+1(q+2 1 q (b g 1 De: My 21, Mhemics Subjec Clssificio. Primry 42Xxx; 26D15; Secodry, 41A55. Key words d phrses. Fourier Trsform, Alyic Iequliies, Adpive Qudrure Formule. 1

2 2 N.S. BARNETT, S.S. DRAGOMIR, AND G. HANNA for ll x [, b, x 0, where E is he expoeil me of wo complex umbers, h is, e z e w z w if z w E (z, w :=, z, w C. exp (w if z = w The followig iequliy for more geerl clss of fucios ws poied ou i [3. Theorem 2. Le g : [, b R be mesurble mppig o [, b, he we hve he iequliy: F (g (x E ( 2πix, 2πixb g (s ds 2π 3 x (b 2 g if g L [, b ; 2 1+ q 1 π(b 1+ q 1 1 x g p if g L p [, b, p > 1, p q = 1; for ll x [, b, x 0. [(q+1(q+2 1 q 2π x (b g 1 if g L 1 [, b ; I is he mi im of his pper o poi ou some ew iequliies for he Fourier rsform of fucios of bouded vriio. Error bouds for some ssocied qudrure formule re lso meioed. The followig iequliy holds: 2. Some Iequliies Theorem 3. Le g : [, b R be mppig of bouded vriio o [, b, he we hve he iequliy (2.1 F (g (x E ( 2πix, 2πixb g (s ds 3 4 (b (g for ll x [, b, x 0, where b (g is he ol vriio of g o [, b. Proof. Usig he iegrio by prs formul for he Riem-Sieljes iegrl, we hve (see lso [4 h (2.2 d (2.3 (s dg (s = ( g ( = (b g ( g (s ds g (s ds, for ll [, b. Addig (2.2 d (2.3 d dividig by (b, we deduce he represeio [4: (2.4 g ( = 1 b g (s ds b (s dg (s b,

3 ERROR ESTIMATES 3 for ll [, b, which is iself of ieres. Assume h x [, b, x 0, he, (2.5 s F (g (x = = = 1 b g ( e 2πix d [ 1 b b g (s ds b b g (s ds b = E ( 2πix, 2πixb b b e 2πix d ( (s dg (s e 2πix d (s dg (s e 2πix d ( ( g (s ds e 2πix d (s dg (s e 2πix d ( e 2πix d = (b E ( 2πix, 2πixb. e 2πix d, Usig he properies of modulus, we hve, by (2.5, h (2.6 F (g (x E ( 2πix, 2πixb g (s ds 1 ( (s dg (s e 2πix d b ( b b e 2πix d b 1 b (s dg (s e 2πix d e 2πix d b = 1 b (s dg (s d b d.

4 4 N.S. BARNETT, S.S. DRAGOMIR, AND G. HANNA I is well kow h if p : [c, d R is coiuous d v : [c, d R is of bouded vriio o [c, d, he he Riem-Sieljes iegrl d p (x dv (x exiss d c d d (2.7 p (x dv (x sup p (x (v. x [c,d c Applyig (2.7 o he iervls [, d [, b, we deduce h (s dg (s ( (g, (b (g d furher h, (s dg (s + Usig (2.6, ( (g + (b (g [ mx {, b } (g + (g [,b [ 1 = (g (b b 2. F (g (x E ( 2πix, 2πixb g (s ds 1 [ 1 (g (b + b 2 + b 2 d = 3 4 (b (g, s simple clculio shows h + b (b 2 2 d =, 4 d he heorem is compleely proved. Remrk 1. If we cosider he iverse Fourier rsform F 1 (g (x = he, by similr rgume, we c prove h (2.8 F 1 (g (x E (2πix, 2πixb c g ( e 2πix d, g (s ds 3 4 (b (g, x [, b, x 0.

5 ERROR ESTIMATES 5 The followig corollries re url cosequece. Corollry 1. Le g : [, b R be moooic mppig o [, b. The we hve he iequliy (2.9 F (g (x E ( 2πix, 2πixb g (s ds 3 (b g (b g (, 4 for ll x [, b, x 0. The proof is obvious by Theorem 3, kig io ccou h every moooic mppig is of bouded vriio d b (g = g (b g (. Corollry 2. Le g : [, b R be L Lipschizi mppig o [, b, i.e., (L g ( g (s L s for ll, s [, b. The we hve he iequliy (2.10 F (g (x E ( 2πix, 2πixb g (s ds 3 4 L (b 2. The proof is obvious by Theorem 3, kig io ccou h if g : [, b R is L Lipschizi, he L is of bouded vriio o [, b d b (g L (b. 3. A Numericl Qudrure Formul Le I : = x 0 < x 1 <... < x 1 < x = b be divisio of he iervl [, b, pu h k := +1 (k = 0,..., 1 d ν (h := mx {h k k = 0,..., 1}. Defie he sum (see lso [2 d [3 1 xk+1 (3.1 E (g, I, x := E ( 2πix, 2πix+1 g ( d, where x [, b, x 0. The followig pproximio heorem holds. Theorem 4. Le g : [, b R be mppig of bouded vriio o [, b. The we hve he qudrure rule (3.2 F (g (x = E (g, I, x + R (g, I, x ; where E (g, I, x is s defied i (3.1 d he remider R (g, I, x sisfies he esime (3.3 R (g, I, x 3 4 ν (h (g. Proof. Applyig Theorem 3 o every subiervl [, +1, we c se h xk+1 xk+1 g ( e 2πix d E ( 2πix, 2πix+1 g ( d 3 4 h k x k+1 (g,

6 6 N.S. BARNETT, S.S. DRAGOMIR, AND G. HANNA for ll k {0,..., 1} d x [, b, x 0. Summig over i from 0 o 1 d usig he geerlized rigle iequliy, we c se h d he heorem is proved. R (g, I, x = F (g (x E (g, I, x x k+1 h k = 3 4 ν (h (g, (g ν (h I prcicl pplicios, i is more coveie o cosider he equidis priioig of he iervl [, b. Thus, le I : x j = + j b, j = 0,..., ; x k+1 be equidis priio of [, b, d defie he sum (see lso [2 d [3 1 [ ( (3.4 E (g, x := E 2πix + k b (, 2πix + (k b +(k+1 b g ( d. +k b The followig corollry of Theorem 4 holds. Corollry 3. Le g be s defied i Theorem 4. The we hve (3.5 F (g (x = E (g, x + R (g, x, where E (g, x pproximes he Fourier rsform y poi x [, b, x 0. The error of pproximio R (g, x sisfies he boud (3.6 R (g, x 3 4 (b (g, for ll x [, b, x 0. Remrk 2. If we kow he ol vriio b (g of g o [, b d would like o pproxime he Fourier rsform F (g (x by he dpive qudrure formul E (g, x wih error less h give ε > 0, we hve o divide he iervl [, b io les ε N pois, where [ 3 (b ε := (g, 4ε d [r deoes he ieger pr of r R. The followig corollries of Theorem 4 lso hold. (g

7 ERROR ESTIMATES 7 Corollry 4. Le g : [, b R be moooic mppig o [, b. The we hve he qudrure formul (3.2 where he remider is such h i sisfies he esime (3.7 R (g, I, x 3 ν (h g (b g (, x [, b, x 0. 4 I priculr, if I is ke o be equidis, he we hve he formul (3.5, where he remider R (g, x sisfies he esime (3.8 R (g, x 3 (b 4 A similr resul holds for Lipschizi mppigs. g (b g (, x [, b, x 0. Corollry 5. Le g : [, b R be Lipschizi mppig wih he cos L > 0. The we hve he qudrure formul (3.2 where he remider sisfies he boud (3.9 R (g, I, x L h 2 i 3 L (b ν (h. 4 i=0 I priculr, if I is chose o be equidis, he we hve he formul (3.5 where he remider R (g, x is such h i sisfies he iequliy (3.10 R (g, x 3L (b Some Numericl Experimes I he followig we umericlly illusre he pproximio for he Fourier rsform provided by,

8 8 N.S. BARNETT, S.S. DRAGOMIR, AND G. HANNA, (4.1 E (g, I, x 1 : = E ( 2πix +(k+1 b +k b ( + k b (, 2π i + (k b g ( d. If we cosider, for isce, he expoeil fucio g ( = exp(, [ 1, 1, he he plos of he error r (x := R (g, x, x [ 1, 1 for = 1, = 10 d = 100, respecively, re depiced i Figure 1, 2 d 3. Refereces [1 P.L. BUTZER d R.J. NESSEL, Fourier Alysis d Approximio Theory, I, Acdemic Press, New York d Lodo, [2 N.S. BARNETT d S.S. DRAGOMIR, A Approximio for he Fourier Trsform of Absoluely Coiuous Mppigs, RGMIA Res. Rep. Coll., 5(2002, Suppleme, Aricle 33. [ON LINE: hp://rgmi.vu.edu.u/v5(e.hml,proc. 4h I. Cof. o Modellig d Simulio, Vicori Uiversiy, Melboure, 2002, [3 S.S. DRAGOMIR, Y.J. CHO d S.S. KIM, A pproximio for he Fourier rsform of Lebesgue iegrble mppigs, Iequliy Theory & Applicios, Volume 2, (Eds. Y.J. Cho, J.K. Kim d S.S. Drgomir, Nov Sciece Publishers, New York, o pper. [4 S.S. DRAGOMIR, O he Osrowski s iegrl iequliy for mppigs wih bouded vriio d pplicios, Mhemicl Iequliies d Applicios, 4(1 (2001, [5 S.S. DRAGOMIR d TH. M. RASSIAS (Eds., Osrowski Type Iequliies d Applicios i Numericl Iegrio, Kluwer Acdemic Publishers, Dordrech/Boso/Lodo, School of Compuer Sciece d Mhemics, Vicori Uiversiy of Techology, PO Box 14428, Melboure Ciy, MC 8001 Ausrli E-mil ddress: {eil,sever, george}@mild.vu.edu.u

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