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CHROMATIC DERIVATIVES AND EXPANSIONS WITH WEIGHTS 6 1 A´.P.HORVA´TH 0 2 n Abstract. Chromaticderivativesandseriesexpansions ofbandlimitedfunc- a tions have recently been introduced in signal processing and they have been J showntobeusefulinpracticalapplications. Weextendthenotionofchromatic derivativeusingvaryingweights. Whenthekernelfunctionoftheintegralop- 2 erator is positive, this extension ensures chromatic expansions around every 2 points. Besides oldexamples, themodifiedmethod isdemonstrated viasome new ones as Walsh-Fourier transform, and Poisson-wavelet transform. More- ] A overthechromaticexpansionofafunctioninsomeLp-spaceisinvestigated. C . h t 1. Introduction a m A band limited signal, f(t) with finite energy can be represented by different [ series expansions. As an analytic function, it can be represented by its Taylor series, as an L2-function, it can be expanded to some Fourier series, and by the 1 v Whittaker-Shannon-Kotel’nikovsamplingtheoremitcanberepresentedintheform 5 ∞ sinπ(t n) 3 f(t) f(n) − . 1 ∼ π(t n) n= − 6 X−∞ The notion of chromatic derivatives and chromatic expansions have recently been 0 . introduced by A. Ignjatović (cf. e.g. [8] - [11]). This alternative representation 1 is possessed of some useful properties as follows. It is of local nature as Taylor 0 6 series, but it can be handled numerically much better than the Taylor expansion. 1 InsomecasesitisaFourierseriesinsomeHilbertspace,oritsfundamentalsystem : of functions forms a Riesz basis or a frame, furtheremore its fundamental system v i of functions is generated by a single function as Gabor or wavelet systems. The X chromaticseriesexpansionsofsignalsassociatedwithFouriertransformhaveseveral r practical applications, cf. e.g. [3], [6], [18] and references therein. a The extension of the method to more general integral transforms, differential operators and weighted spaces was introduced by A. Zayed (cf. e.g. [22] - [24]). This type of generalization allows to find connections of the chromatic derivatives andexpansionswiththe Mellintransformtechniquewhichwasappliedto studying the asymptotic behavior of Fourier ransforms of orthogonal polynomials and with some kinds of differential equations cf. [24] and references therein. Ouraimistoextendchromaticderivativesandchromaticexpansionsusingvary- ingweights. Insomecases(forinstanceFourier,Walsh-Fourier,Hankeltransforms) there is only a single point x for wich the kernel function Ψ(x , ) is positive, but 0 0 · 2010 Mathematics Subject Classification. Primary41A58,42C15;Secondary 44A15,42C40. Key words and phrases. Chromatic derivatives, chromatic expansions, de la Vallée Poussin means,Walshtransform,Poissonwavelet. SupportedbyHungarianNationalFoundation forScientificResearch,GrantNo. K-100461. 1 2 A´.P.HORVA´TH for instance in cases of Laplace, Bargmann and Poisson-wavelet transforms it has at most finite many sign changes for a large set of x -s. In these cases we define 0 chromatic series expansions around every x -s in question. 0 The other direction of our investigation is introducing the chromatic expansion of a function f in certain Lp-spaces and proving the Lp-convergence of its de la Vallée Poussin means. It also turns out, that the rate of convergence of the de la ValléePoussinmeanscanbeestimatedbytheerrorofthenear-bestapproximating polynomials of fîn the weighted space in question. The article is organized as follows. In Section 2 we give a summary of the methodandsomeresultsfromthe literature. Bythe methodofsection4,weprove some convergencetheorems in sections 5 and 6. Sections 3 and 6 contain the main curiosityofthepaper,namelythemodifiedmethod,andtheexampleswithvarying weights. 2. Preliminaries In this section we summarize the method and some properties of the chromatic derivativeandchromaticexpansion. More details canbe foundin [3], [6], [8]- [12], [18], [20], [22] - [24] and references therein. Before Theorem 1 we do not take care on function spaces and convergence. Let Ψ(x,y) be a kernel function and I an integral transform on some function Ψ spaces, with a measure ν on a finite or infinite interval (c,d), say d (1) f(x):=I (fˆ)(x)= fˆ(y)Ψ(x,y)dν(y). Ψ Zc Let f(x) be defined on (a,b). Let L be a linear differential operator: m ∂kf (2) L (f)(x)= d (x) (x), x k ∂xk k=0 X where d (x) are continuous functions on (a,b). Let us suppose that Ψ(x,y) is k smooth as a function of x, and (3) L (Ψ(x,y))=yΨ(x,y). x Finally let us assume that there is an x (a,b) such that 0 ∈ (4) Ψ(x ,y) const=0 y (c,d), 0 ≡ 6 ∈ (orifΨ(x ,y) 0,thenletusassumethatforsomeα,lim Ψ(x,y) const=0 on (c,d)).0 ≡ x→0(+) xα ≡ 6 (Sometimes it is importantthat I is the inversetransformof anintegraltrans- Ψ form, that is IΨ =(IΦ)−1, where Φ(x,y) is a kernel function on a finite or infinite interval (a,b), and I (f)(y) =: fˆ(y) = bf(x)Φ(x,y)dµ(x). cf. e.g. [10] and Φ a subsection 3.1 in [24].) R Examples. There are several examples in the literature on operators like above (cf. e.g. [9], [10], [11], [22], [24], [23]), e. g. the Fourier transform on R: Ψ(x,y)= eixy, L = i ∂ , Ψ(0,y) 1; the Laplace transform on R : Ψ(x,y) = e xy, x − ∂x ≡ + − Lx = −∂∂x, Ψ(0,y) ≡ 1; the Hankel transform on R+: Ψ(x,y) = √xJα x√y , Lx =−∂∂x22 + α2x−214, limx→0+ Ψx(αx+,y21) = 2νΓy(αα2+1), that is dν(y)= ydαy2 , etc. (cid:0) (cid:1) CHROMATIC DERIVATIVES 3 Let p(x) be an arbitrary polynomial. According to (3) (5) p(L )(Ψ(x,y))=p(y)Ψ(x,y), x and so by (1) d (6) p(L )(f)(x)= fˆ(y)p(y)Ψ(x,y)dν(y). x Zc Let w be now a positive weight function on (c,d) with finite moments, and let p be the orthogonal polynomials on (c,d) with respect to w, that is let { n}∞n=0 d (7) p (y)p (y)w(y)dν(y)=δ . n m m,n Zc (Let us remark here that usually dν(y) is dy, except some cases, see for instance tha Hankel transform.) The nth chromatic derivative of f with respect to I and Ψ w is d (8) Kn(f)(x):= fˆ(y)p (y)Ψ(x,y)dν(y), n Zc whentheintegralexists. Supposingthatthereisanx (a,b)suchthatΨ(x ,y) 0 0 ∈ ≡ 1 on (c,d), we have d fˆ(y) (9) Kn(f)(x ):= p (y)w(y)dν(y). 0 w(y) n Zc Let d (10) ϕ(x)= w(y)Ψ(x,y)dν(y), Zc and d (11) ϕ (x):=Kn(ϕ)(x)= w(y)p (y)Ψ(x,y)dν(y), x (a,b). n n ∈ Zc The chromatic expansion of f is ∞ (12) f(x) Kn(f)(x )Kn(ϕ)(x). 0 ∼ n=0 X The conditions below ensure the existence of the integrals above, and the convergence of the series. By the previous notations let (c,d)=(0,d), where 0<d . ≤∞ Theorem 1. (Th. 5. in [22]) Let us assume that Ψ and fˆsatisfy the following conditions: a) w(y)Ψ(x,y) L2(R ) + ∈ b) fˆ L2(0,d) p√w ∈ c) If d= , we further assume that the integrals ∞ ∞ynfˆ(y)Ψ(x,y)dy < , n=0,1,2,... ∞ Z0 converge uniformly. 4 A´.P.HORVA´TH Consider f(x) := I (fˆ) = dfˆ(y)Ψ(x,y)dy, x (a,b), 0 < d . Then, Ψ 0 ∈ ≤ ∞ the chromatic series expansion of f converges pointwise for x (a,b), and locally R ∈ uniformly in (a,b). Moreover ∞ (13) Ψ(x,y)= ϕ (x)p (y). n n n=0 X 3. Modification of the method The main point of our modification is the following: instead of (4) we give a weaker assumption. Propery (4) was ensured by the initial condition (3.5) in [24], when the chromatic derivatives were asociated with a Sturm-Liouville differential operator. Whenthechromaticderivativeswereasociatedwithamoregeneraldiffer- entialoperator,thekernelfunctionoftheintegralhadtobeoneofthefundamental solutions of the initial value problem (cf. [24] and references therein). By (4), the coefficients of the chromatic expansion of a function are the chromatic derivatives of the function at a certain x , so the chromatic expansion is located at a certain 0 point. Originally, in the bandlimited case, by the special translation properties of the Fourier transform, the chromatic expansion had the following form ∞ f ( 1)nKn[f](u)Kn[m](t u), ∼ − − n=0 X where u is an arbitrary point of the interval, and the convergenceof the expansion was independent of u (cf. [9]). One hand the motivation of the modification is to get expansions around more points. On the other hand, substituting assumption (4)byaweakerone,namelybythefollowing: letx beanypointforwhichΨ(x ,y) 0 0 hasfinitemanysignchanges,wecanextendthenotionofchromaticderivativesand expansions to a wider family of kernel functions (cf. subsection 6.3). Finally we also modify the notations in connection with the weight, to make it more compatible to the earlier results. Let Ψ(x,y) be a kernel function which is (piecewise) smooth as a function of x and I be an integral transform on a finite or infinite interval (c,d) such that Ψ d (14) f(x):=I (fˆ)= fˆ(y)Ψ(x,y)dν(y). Ψ Zc Letusassumethattheintegralaboveexistsandf(x)isdefinedonx (a,b). Let ∈ L be a linear differential operator like (2) for which (3) is satisfied. Furthermore x letw beapositiveweightfunction,forsimplicitywithfinitemomentson(c,d). Let us denote by (15) f (x):=I fˆ√w . w Ψ Let us recall that if p is any polynomial of(cid:16)degre(cid:17)e n, then n d p (L )(f )(x)= fˆ(y) w(y)p (y)Ψ(x,y)dν(y). n x w n Zc p Let x (a,b) such that Ψ(x ,y)>0 on (c,d), let us denote by 0 0 ∈ (16) w (y)=w(y)Ψ(x ,y)2. x0 0 CHROMATIC DERIVATIVES 5 Ifw hasfinitemoments,letp =γ xn+... bethenthorthogonalpolynomial x0 x0,n n,x0 with respect to w , that is x0 d (17) p (y)p (y)w dν(y)=δ . x0,n x0,m x0 n,m Zc The nth chromatic derivative of f with respect to x and w at x is 0 d (18) Kn (f)(x):=p (L )(f )(x)= fˆ(y) w(y)p (y)Ψ(x,y)dν(y). wx0 x0,n x w Zc x0,n p If we write fˆ=gx0√wx0, we have (19) Kwnx0(f)(x0)=hfˆ,√wx0px0,nidν =hgx0√wx0,px0,n√wx0idν, that is Kn (f)(x ) is the nth Fourier coefficient of g with respect to p . wx0 0 x0 { x0,k}∞k=0 Usually px0,n√wx0 ∈ Lpd′ν for all 1 ≤ p′ ≤ ∞, Kwnx0(f)(x0) exists if fˆ∈ Lpdν for some 1 p , and similarly Kn (f)(x) also exists. Thus g has the formal ≤ ≤ ∞ wx0 x0 Fourier series ∞ g Kn (f)(x )p . x0 ∼ wx0 0 x0,n n=0 X Let (20) ϕx0,n :=IΨ(px0,n√wx0), which exists usually in the ordinary sense as above, and (21) ϕ :=I (Ψ(x , )) x0 Ψ 0 · in ordinary or in distribution sense. Also in ordinary or in distribution sense (cf. [23], section 5), (22) ϕ =Kn (ϕ ). x0,n wx0 x0 Thus f has the formal chromatic expansion ∞ (23) f(x) Kn (f)(x )Kn (ϕ )(x). ∼ wx0 0 wx0 x0 n=0 X With these notations instead of (13) we have, if Ψ(x, ) Lp((c,d)) with some · ∈ 1 p , and formally ≤ ≤∞ (24) Ψ(x,y) ϕ (x)p (y) w (y). ∼ x0,n x0,n x0 n X p When Ψ(x ,y) has finite many sign changes, according to the sign of Ψ(x ,y) 0 0 we decompose (c,d) to finite many subintervals, and proceed as above on every subintervals independently (cf. subsection 6.3 below). If I is the inverse transform of some integral transform on (a,b) I , that is Ψ Φ fˆ(x)= bf(x)Φ(x,y)dµ(x) such that, say a R Φ(x,y)=Ψ(x,y), and the operator is isometric: fˆ,gˆ = f,g , dν dµ h i h i then by the transform I one can get the formal expansion of f with respect to Ψ the orthonormalsystem ϕx0,n :=IΨ(px0,n√wx0). This is the situation for instance 6 A´.P.HORVA´TH in cases of Fourier-, Hankel-, Walsh-, Sturm-Liouville- and wavelet transforms. By (19) it is clear that if the system ϕ is complete, the Fourier expansion of an x0,n f L2(a,b) with respect to ϕ coincides with its chromatic expansion, ∈ x0,n ∞ ∞ f(x)= f,ϕ ϕ (x)= Kn (f)(x )Kn (ϕ)(x). h x0,nidµ x0,n wx0 0 wx0 n=0 n=0 X X Without an isometry we also have series expansions as above, but ϕ is no { x0,n} longer orthogonal. In the first case one can use the usual Fourier seies technics to study the convergenceproperties of the series,in the secondcase one can take into consideration the properties of the transformation. Itiseasytoseethatthemethodcitedfromtheliteratureandsummarizedinthe previous section is a subcase of this modified one. Examples with varying weights can be found in Section 6. 4. Convergence As we have seen (cf. e.g. Th.1 and the cited references) there are L2, pointwise andlocallyuniformconvergencetheorems inconnectionwith chromaticseries expansions. In this section we give the sketch of thoughts to get Lp and (locally) uniform convergence theorems by de la Vallée Poussin means of the chromatic series,andestimatesontherateofconvergence. Thetheoremsandproofsinconcrete cases can be found in the next sections. Let us assume that hˆ is a function in some (Banach) space and I (hˆ) exists. Ψ Furthermore let us assume that (25) I (hˆ) C hˆ̺ , Ψ q,µ 1 s,ν k k ≤ k k for some 1 q,s . Let fˆ be as before, which has a formal Fourier series with respect≤to {px≤0,n∞√wx0}. Denoting by gx0 := √wfˆx0, gx0 has the expansion gx0 ∼ ∞n=0Kwnx0(f)(x0)px0,n. Sˆn := Sˆn(gx0,y) is the nth partial sum of this series. Let P n 1 σˆ :=σˆ (g ,y)= 1 − Sˆ , vˆ :=vˆ (g ,y)=2σˆ σˆ n n x0 n k n n x0 2n− n k=0 X be the nth Cesàro mean and the nth de la Vallée Poussin mean of this formal series respectively. We chose de la Vallée Poussin mean because of its reproducing property. Let̺ (y)>0(i=1,2)andmeasurablefunctionson(c,d). Letusassume i further that (26) kσˆn(gx0)√wx0̺1ks,ν ≤Ckfˆ̺2ks,ν. Denoting by vñ :=IΨ(vˆn(gx0)√wx0), by (25) and (26) we have kvñ−fkq,µ ≤Ck(vˆn(gx0)−gx0)√wx0̺1ks,ν ≤C k(vˆn(gx0 −pn))√wx0̺1ks,ν +k(gx0 −pn)√wx0̺1ks,ν ≤C(cid:0) k(gx0 −pn)√wx0̺2ks,ν +k(gx0 −pn)√wx0̺1ks,ν .(cid:1) (cid:0) (cid:1) CHROMATIC DERIVATIVES 7 That is if there is a sequence of polynomials such that the right-hand side of the previous expression tends to zero when n tends to infinity, then the de la Vallée Poussin means of the chromatic expansion tend to the function in the mean. Similarly |(vñ−f)(x)|≤ IΨ((gx0 −pn)√wx0)(x) + IΨ(vˆn(gx0 −pn)√wx0)(x) If (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (27) I (hˆ)(x) C(x) hˆ̺ , Ψ 1 s,ν ≤ k k (cid:12) (cid:12) and the polynomials are den(cid:12)se in the(cid:12)space in question, then v tends pointwise to (cid:12) (cid:12) n f. If C(x) has a uniform bound on every compact subintervals of (a,b), then the convergence is locally uniform. In order to give the estimations in a closed form, let us introduce the following notation: Notation. Let w be a positive weightfunction with finite moments on aninterval (a,b). Let fw Lp . The error of the best approximating polynomial of degree ∈ (a,b) n of f in the weighted Lp-space is E (f) = inf (f p )w . n p,w n p pn∈Pnk − k There are several results connected with estimates of E (f) , cf. e.g. [4] in the n p,w unweighted case, [15] for Laguerre weights, [5] for Jacobi weights, [14] for Freud weights, [17] for analytic functions, etc. Collecting the calculations above and supposing that p is a near-best approxi- n mating polynomial, we can state Theorem 2. With the notations above, if (25) and (26) fulfil fˆ v˜ f CmaxE , n q,µ n k − k ≤ i=1,2 √wx0!s,√wx0̺i,ν and if (27) fulfils (v˜ f)(x) C(x), n | − |≤ which ensures pointwise, local uniform or uniform convergence according to the properties of C(x). 4.1. Lemmas. In this subsection we collect from the literature all the lemmas which will be useful to prove Theorem 2 in actual weighted spaces. The lemmas are reformulated to our purpose. Letw beapositive,measurableweightwithfinite momentsonaninterval(c,d), p are the orthonormal polynomials on (c,d) with respect to w, γ > 0 is the w,n n leading coefficient of p , ̺ is positive and measurable on (c,d). w,n n γ Γ =max k,w , Λ = w̺ p2 . n,w k n γk+1,w n,w̺ (cid:13) w,k(cid:13) ≤ (cid:13) kX=0 (cid:13) (cid:13) (cid:13)∞ (cid:13) (cid:13) (cid:13) (cid:13) 8 A´.P.HORVA´TH Lemma 1. (Lemma 1 in [7]) Let us suppose that Γ Λ n,w n,w̺ (28) <K n for somesuitableK. Let0<̺<M acontinuousfunction on (c,d). Ifh w L , ̺ ∈ p then for 1<p< q ∞ w (29) kσn(h)√w̺kp ≤C(p)kh ̺kp, r (30) σ (h)√w̺ C h√w , n k k∞ ≤ k k∞ w (31) σ (h)√w C h . n 1 ≤ k ̺k1 r (cid:13) (cid:13) (cid:13) (cid:13) Lemma 2. (cf. Theorem 1 in [16]) Let 1 p , α 0, a,b 0, W(x) = e−x22xα2 1+xx a(1 + x)b. σn(f,x) is the n≤th (≤C,∞1)-mea≥n of the ≥Laguerre-(α)- Fourier s(cid:16)eries(cid:17)of a function f. Then σ (f,x)W(x) ) C f(x)W(x) ), n p,(0, p,(0, k k ∞ ≤ k k ∞ where 1 α 1 a< +min , ; p 2 4 ′ (cid:18) (cid:19) 1 3 1 1 b< + , a+b<2 + , if 1 p 4, p 4 p 2 ≤ ≤ ′ ′ 1 5 4 b< + , a+b< +1, if 4<p , 3p 4 3p ≤∞ ′ ′ Corollary. Let α>0, w(x)=xαe x, f√w Lp and let σ (f) as above. Then − n ∈ σ (f)√w C f√w . n p p k k ≤ k k In the following Lemma w and v are nonnegative functions, the Laplace transform of a function f is defined on (0, ) Lf is also defined on (0, ), the Hardy x ∞ ∞ operatorHf is givenby Hf(x)= f(t)dt, andthe adjointoperatorofH is given 0 by H∗f(x)= x∞f(t)dt. Let us deRnote by g∗ the decreasing rearrangemantof g. Lemma 3. (cRf. Theorems 1 and 2 of [2])Let w and v be as above. In order for the Laplace transform L:Lp(v) Lq(w) boundedly, set u(x)=x 2w 1 . → − x If 1<p q < , it is sufficient that ≤ ∞ (cid:0) (cid:1) 1 (a) H 1 ∗pp′ p′ (H∗(u))q1 C, or (b) H v−pp′ p1′ (Lw)q1 C,   v  ≤ ≤ (cid:18) (cid:19) (cid:16) (cid:16) (cid:17)(cid:17) and if1<q<p< with 1 = 1 1, it is sufficient that ∞ r q − p 1 r ∞ 1 ∗pp′ 1 ∗pp′ q′ 1 (c) I1 = v H v  (H∗(u))q ≤∞, Z0 (cid:18) (cid:19) (cid:18) (cid:19)       CHROMATIC DERIVATIVES 9 or r 1 (d) I2 = ∞v−pp′ H v−pp′ q′ (Lw)1q . Z0 (cid:16) (cid:16) (cid:17)(cid:17) ! ≤∞ 5. Examples with a fixed weight 5.1. Fourier transform. Chromatic derivatives and expansions with respect to theFouriertransformwereinvestigatedbyseveralauthors,cf. e.g. [3],[6], [18],[9], [18], [22]. Here we examine the Lp′-convergence of the de la Vallée Poussin means ofthetransformedFourierseries. Ourstartingpointisafunctionfˆ Lp 1 p 2 ∈ ≤ ≤ instead of L2, and the rate of convergence is also estimated. Let us recall that (c,d) = R, dν(y) = dy Ψ(x,y) = 1 e ixy, that is Ψ(0,y) √2π − ≡ 1 thus w =w = 1 w, p =√2πp . According to (22) √2π x0 0 2π x0,n w,n 1 ϕ0(x)=F−1 =δ(x). √2π (cid:18) (cid:19) L = i ∂ , L Ψ(x,y) = yΨ(x,y). We have to mention here that it is easy to x − ∂x x see that ϕ = Kn (ϕ ) in distribution sense. Indeed let ξ be infinitely many differentiab0l,enand cwom0 pa0ctly supported on R and let p (y) = n a yk. Then 0,n k=0 k by definition and finally by changing the order of integrals we have P n ∂k √w Kwn0(ϕ0) ξ = √2πk=0ak(−i)kxk!(cid:18)F−1(cid:18)√2π(cid:19)(x)(cid:19)!ξ(x) (cid:0) (cid:1) X n n =(√w) ak(i)kF−1 ξ(k) =( w(y)) akykF−1(ξ)(y) ! ! kX=0 (cid:16) (cid:17) p kX=0 1 = w(y)p (y) ξ(x)eixydxdy =(ϕ )ξ. 0,n 0,n R √2π R Z Z TheintegraloperatorphasaninverseonL2,say,dµ(x)=dx,andParseval’sformula R holds true. 5.1.1. Hermite weight. (Preliminary results can be found e.g. in [20].) w(y) = e y2, p = √2πh , where h -s are the Hermite polynomials. As it − 0,n n n is shown in [7], in Lemma 1 ̺ 1 with this weight, and all the assumptions are ≡ satisfied. Thatisin(26)̺ =̺ 1. ItisalsowellknownthattheFourieroperators 1 2 ≡ mapLp toLp′,when1 p 2and 1+ 1 =1. MoreovertheHermitefunctionsare ≤ ≤ p p′ the eigenfunctions of the integral operator IΨ hn(y)e−y22 =F−1 hn(y)e−y22 = inhn(y)e−y22. Thus denoting by g = √fˆw, we h(cid:16)ave vñ(x) =(cid:17)F−1(vˆn(cid:16)(g,y) w(y)(cid:17)) = vn(f,x) w(x), where vn(f,x) is the nth de la Vallée Poussinmean of thpe Hermite expansion of f. That is the computations above give back the result p Proposition 1. Let 1 p 2, fˆ Lp(R). Then recalling the notation f = ≤ ≤ ∈ F 1 fˆ − (cid:16) (cid:17) fˆ (32) f v √w C fˆ vˆ √w CE , − n p′ ≤ − n p ≤ n √w! (cid:13)(cid:13) (cid:13)(cid:13) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) p,√w 10 A´.P.HORVA´TH where E (h) is the error of the best approximating polynomial of h of degree n, n p,w in the weigted L space. Moreover if fˆ L1, then p ∈ fˆ (33) f(x) v (x) w(x) CE , n n | − |≤ √w! p 1,√w which ensures uniform convergence. 5.1.2. Freud weights. Firstofallletusrecall,thatw(x)=e Q(x) isaFreudweight − on R, if Q is positive, even and continuous on R, and twice continuously dif- ferentiable and Q′ > 0 on (0, ) and there are constants A,B > 1 such that ∞ A (cid:16)xQ′(x)(cid:17)′ B on (0, ). Since in Lemma 1 ̺ 1 for Freud weights ≤ Q′(x) ≤ ∞ ≡ as well (cf. [7]), all the computations above are valid. The only difference is that p √w are not the eigenfunctions of the Fourier transform, so although w,n ϕ = F 1(p √w) isanorthonormalsystem,buttheyarenotweightedpoly- n − w,n { } { } nomials with respect to w. v˜ is the de la Vallée Poussinmean of the expansion of n f with respect to ϕ . Thus by (25), (26) and Lemma 1 n Statement 1. If fˆ Lp(R) 1 p 2, then ∈ ≤ ≤ fˆ f v˜ CE , n p n k − k ′ ≤ √w! p,√w and if fˆ L1, then ∈ fˆ f(x) v˜ (x) CE . n n | − |≤ √w! 1,√w 5.1.3. Jacobi weights. (Preliminary results can be found e.g. in [3], [6], [8].) Let us denote by w(x) := w(α,β)(x) = (1 x)α(1+x)β, α,β > 1. According − − to (25) and (26) we have to apply Lemma 1 again. In case of Jacobi weights ̺(x)=√1 x2, (cf. [7]). − Statement 2. Let us assume that 1 p 2, and fˆ, fˆ Lp(( 1,1)). Denot- ≤ ≤ √1 x2 ∈ − ing by − fˆ v˜ñ =F−1 vn √w̺ , √w̺! ! we have fˆ v˜˜ f CE , n p n k − k ′ ≤ √w̺! p,√w √̺ and if fˆ L1 and α,β 1, √̺ ∈ ≥ 2 fˆ vñ =F−1 vn √w , √w ! ! fˆ v˜ (x) f(x) CE . n n | − |≤ √w! 1,√w √̺