ON THE LAGRANGE COMPLEX INTERPOLATION

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1 U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 2, 200 ISSN ON HE LAGRANGE COMPLEX INERPOLAION Adria NEAGOE I lucrare prez uele rezultate legate de erpolarea Lagrage î domeiul complex ( cor. prop. şi prop. 2 ). Formula (6) este o extidere a formulei lui Shao de eşatioare (7), petru cazul mometelor echidistate de eşatioare. I 2 am adăugat u rezultat privid eşatioarea i domeiul frecveţelor. I 3 am dat o extidere multidimesioală a formulei de eşatioare, care u foloseşte erpolarea Lagrage, ci o abordare distribuţioală. I this work, I preset some results regardig the Lagrage erpolatio i the complex domai (cor. prop. ad prop. 2). Formula (6) is a extesio of the well-kow Shao s samplig formula (7), for samplig equidistat momets. I 2 I preset a simple result regardig the samplig i frequecy. I 3 I give a multidimesioal extesio, which does ot use the Lagrage erpolatio but a distributioal approach. Key words: Lagrage erpolatio i the complex domai, samplig theorem, multidimesioal samplig theorem.. Lagrage erpolatio i complex domai Recall that a fuctio f : C C is called etire if is holomorphic ( aalytical ) i all the complex plae. If (a ), is a give sequece of complex umbers ad z, the Lagrage erpolatio problem cosists of fidig a etire fuctio f so that f ( z ) = a,. his problem has solutios oly if we impose some supplemetary codios regardig the sequece (a ). PROPOSIION. Fix a real umber >0. Let z =, wh Z ad z the fuctio φ: C C, ϕ ( z) = si. If a sequece (a ), Z has the property M that ε > 0, M > 0 ad N( ε ) so that a < for N( ε), the the +ε series aϕ( z) =, () ϕ' ( z )( ) = z z Prof., Jea Moet heoretical High School, Bucharest, Romaia, adriaeagoe78@yahoo.com

2 64 Adria Neagoe is uiformly coverget o ay compact i C. z Proof : he series () becomes a si = Z ( z ) cos si u asa ( z ), where we have deoted sa( u) = ( u 0 ) ad sa ( 0) =. Z u If R > 0 ad z R, the for every large eough, we have M S(R) a sa( ( z )), where S ( R) = sup{ sa( u) ; u R}. +ε So, for ay N fixed, a sa( ( z )) 2M S(R). + ε > N = N Accordigly, the rest of the cosidered series of fuctios is uiformly coverget towards zero for N Applyig the fact that the sum of a series of etire fuctios, uiformly coverget o ay compact, is a etire fuctio, we get the followig: COROLLARY : Uder the hypothesis of prop., the fuctio f : C C, = asa ( z ) is etire ad, moreover f ( ) = a, for ay Z []. Z 2. A geeralizatio of the samplig Shao formula Fix > 0 ad let f : C C be a etire fuctio. Deote ρ = ( + ) 2 ad cosider the complex egral f ( u) du I ( z) =, Z (2) 2 i u z =ρ ( u z)si he residue of the fuctio uder the egral i the simple pole k is equal to f ( k) ad for z k, the residue i z is equal to. he, z z ( k )cos k si acordig to the residues theorem, results that f ( k) I ( z) = + (3) z z k ρ ( k )cos k si

3 O the Lagrage complex erpolatio 65 O the other had, we estimate the egral (2) direct by the parametrizatio of the path of egratio ; put u = ρ e, t [0,2], hece 2 f ( ρe ) ρie 2 I ( z) = dθ (4) iθ 0 ( ρe z) si( ρe ) But for ay t [ 0,2], exp( ρ si t ) < si ρe. z z Moreover, for ay z fixed, lim e = so e > for ay large ρ ρ 2 eough. From (4) oe ca obtai the estimatio 2 4 f ( ρ e ) I ( z) dt. (5) 0 exp( ρ si t ) PROPOSIION 2. Let > 0 fixed ad f : C C be a etire fuctio wh the property that there is M > 0 ad a real umber β (0, ) so that f ( x) M for ay x R ad M exp( βr s ) for ay z = re. he = f ( ) sa( ( z )). (6) Z Proof : hus, f ( ρ e ) M exp( βρ s ) ; so, accordig to (5), 2 4M I ( z) exp(( β ) ρ si t ) dt. Because β <, I ( z) 0 for 0, uiformly o ay compact, the relatio (6) follows. NOE. If the values f(), Z are previously fixed, the etire fuctio f is uiquely determiated ( by applyig the idety priciple ). CORROLARY. If f : R C belogs to L L 2 ad suppfˆ [ b, b], the f ( t) = f ( ) sa( bt ), a.e. t R. (7) Z b his is the classical formulatio of the Shao formula. he formula (6) is a geeralizatio of this. Here is a simpler argumet to get (7), by usig distributios: if u meas the u step fuctio the f ˆ( ω ) = ( u( ω + b) u( ω b)) fˆ( ω 2b), for ay ω R [ideed, if ω b, Z

4 66 Adria Neagoe the f ˆ ( ω) = 0 ad u( ω + b) u( ω b) = 0 ; ad if ω < b, the ω 2b ( b, b) oly for = 0 ad for 0, f ˆ ( ω 2b) = 0 ]. From the previous relatio, oe gets fˆ( ω ) = ( u( ω + b) u( ω b)) fˆ( ω) δ ( ω 2b) ad is eough to apply the Fourier iversio formula. z NOE. ) he fuctio ϕ ( z) = sa( ) is a good erpolator, i the sese that wheever = ϕ ( z ) f ( ), follows for ay δ>0, that = ϕ ( z + δ ) f ( δ ) [3]. 2) he sigals which do appear i ature are radom ad oe ca exted the relatio (7) to radom sigals: amely if ( ξ t ) is a statioary radom sigal wh a ull mea ad a limed bad of frequecy, the ξt = ξ sa( bt ) [2], [4]. he above corollary suggest a dual result, regardig the samplig i frequecy of the spectral fuctio fˆ of f : qualatively, the value of fˆ i ay frequecy is well determied from the values i some discrete frequecies. Z Z b PROPOSIION 3. Suppose that f L L 2 has a bouded duratio ( that is, τ > 0 such that f ( t) = 0 for t τ ). he ω R, f ˆ( ω ) = fˆ( ) sa ( τω ) (8) Z τ τ τ Proof. We have f ( t) = ( u( t + ) u( t )) ( f ( t) δ ( t τ )) ad 2 2 Z siωτ apply the Fourier trasform : fˆ( ω ) = [ fˆ( ω) δ ( ω )] = ω τ Z τ siωτ = δ ( ω ) [ fˆ( ω)], whece (8). τ ωτ Z 3. A multidimesioal extesio of the Shao formula I the case of the multidimesioal sigals, the Lagrage type erpolatio caot be directly applied. Istead, i this case, oe ca use the previous argumet, by makig use the distributios.

5 O the Lagrage complex erpolatio 67 For the multidimesioal case, the Lagrage erpolatio from, 2 is ot directly possible, but could be applied the previous argumet, by makig use the distributios. Namely, we prove: PROPOSIION 4. Let f : R C be a fuctio from L 2 (R ) wh supp f bouded. he there are vectors v,.,v R so that a.e. x R, f(x) is well determied by the values of f i the set of values Ω = kivi / k,..., k Z. i= Proof. Choose v,.,v liear idepedet ad let u,..,u their reciprocals ( such that takes place the followig relatio betwee euclidea ier products : ui, v j = 2δij for every i,j). We shall ote k v = k i v i ad i= we shall defie a samplig fuctio s, similar wh sa, so that the followig formula holds: f ( x) = f ( k v) s( x k v) a.e. x R. (9) k Z If such a fuctio really exists, the f ( k v) s( x k v) = f ( y) s( x y) δ( y kv) dy R i< y, k u > But, f ( x) = f ( y) e s( x y) dy, where W is the volume of the W k Z parallelipiped built o the vectors u,..,u. Applyig the Fourier -dimesioal operator, results f ( ω) = s( ω) f ( ω + k u) for ay ω R. he is W k Z eough to choose vectors v,.,v so that the supports of f ( ω + k u) are disjo ad cosider a fuctio s so that s = W, costat o supp f ad ull for the values ω where f ( ω + k u) 0, for k 0. he formula (9) is the obtaied by applyig the Fourier iversio formula. 4. Coclusios I this work, oe presets some results regardig the Lagrage erpolatio i the complex plae, which have close coectios wh the Shao samplig theorem ( proposios, 2 ). Oe ca also apply the adopted

6 68 Adria Neagoe method for the case of the radom sigals ad that of oequidistat momets. I the paragraph 2 I have added a formula (8) for the samplig i frequecy. Oe asserts ad proves a extesio of the samplig theorem to the case of fuctios of several variables. R E F E R E N C E S [] Ia. I. Hurghi, V.P. Iakovlev, Metode ale teoriei fuctiilor regi i radiotehica (Methods of egral fuctios theory i radio egieerig) (i Russia), Moskva, Scv. Radio, 962. [2] D. Staomir, O. Stăăşilă, Metode matematice î teoria semalelor (Mathematical methods i sigals theory), Ed. ehică, 980 (i Romaia). [3] H. akahasi, Complex fuctio theory ad umerical aalysis, Publ. Rims, Kyoto Uiv., 4, , [4] P.P. Vaidyaatha, Geeralizatios of the samplig theorem, IEE ras. Circus ad Systems, 48, , sept. 200.

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of