Approximation of Besov vectors by Paley-Wiener vectors in Hilbert spaces
aa r X i v : . [ m a t h . F A ] A p r APPROXIMATION OF BESOV VECTORS BY PALEY-WIENERVECTORS IN HILBERT SPACES
Isaac Z. Pesenson Meyer Z. Pesenson Abstract.
We develop an approximation theory in Hilbert spaces that gener-alizes the classical theory of approximation by entire functions of exponentialtype. The results advance harmonic analysis on manifolds and graphs, thus fa-cilitating data representation, compression, denoising and visualization. Thesetasks are of great importance to machine learning, complex data analysis andcomputer vision. Introduction
One of the main themes in Analysis is correlation between frequency contentof a function and its smoothness. In the classical approach the frequency is un-derstood in terms of the Fourier transform (or Fourier series) and smoothness isdescribed in terms of the Sobolev or Lipshitz and Besov norms. For these notionsit is well understood [1], [10] that there exists a perfect balance between the rate ofapproximation by bandlimited functions ( trigonometric polynomials) and smooth-ness described by Besov norms. For more recent results of approximations by entirefunctions of exponential type we refer to [5]-[7].The classical concepts and result were generalized to Riemannian manifolds,graphs, unitary representations of Lie groups and integral transforms in our work[11]-[21], [8].The goal of the present article is to develop a form of a Harmonic Analysiswhich holds true in general Hilbert spaces. In Introduction we formulate mainresults obtained in the paper. The exact definitions of all notions are given in thetext.We start with a self-adjoint positive definite operator L in a Hilbert space H andconsider its positive root D = L / . For the operator D one can introduce notionof the Spectral Transform F D which is an isomorphism between H and a directintegral of Hilbert spaces over R .A Paley-Wiener space P W ω ( D ) , ω >
0, is introduced as the set of all f ∈ H whose image F D f has support in [0 , ω ]. In the case when H = L ( R ) d and D is Mathematics Subject Classification.
Key words and phrases.
Self-adjoint operator, Paley-Wiener vectors, K -functor, Schrodingergoup of operators, Besov norms. Department of Mathematics, Temple University, Philadelphia, PA 19122; [email protected]. The author was supported in part by the National Geospatial-IntelligenceAgency University Research Initiative (NURI), grant HM1582-08-1-0019. CMS Department, California Institute of Technology, MC 305-16, Pasadena, CA 91125;[email protected]. The author was supported in part by the National Geospatial-IntelligenceAgency University Research Initiative (NURI), grant HM1582-08-1-0019 and by AFOSR, MURI,Award FA9550-09-1-0643 a positive square root from the Laplace operator our definition produces regularPaley-Wiener spaces of spherical exponential type.The domain D s , s ∈ R , of the operator D s , s ∈ R , plays the role of the Sobolevspace and we introduce Besov spaces B α ,q = B α ,q ( D ) , α > , ≤ q ≤ ∞ , by usingPeetre’s interpolation K -functor [2], [4], [8], [9], [23].(1.1) B α ,q ( D ) = (cid:0) H , D r/ (cid:1) Kα/r,q , where r can be any natural such that 0 < α < r, ≤ q < ∞ , or 0 ≤ α ≤ r, q = ∞ .It is crucial for us that Besov norms can be described in terms of a modulus ofcontinuity constructed in terms of the Schrodinger group e itD , wave semigroup e itD , or the heat semigroup e − tD . In what follows the notation k · k bellow means k · k H . We introduce a notion of best approximation(1.2) E ( f, ω ) = inf g ∈ P W ω ( D ) k f − g k , f ∈ H . We also consider the following family of functionals which describe a rate of decayof the Spectral transform F D (1.3) R ( f, ω ) = (cid:18)Z ∞ ω kF D ( f )( λ ) k X ( λ ) dm ( λ ) (cid:19) / , ω > . The Plancherel Theorem for F D implies that every such functional is exactly thebest approximation of f by Paley-Wiener functions from P W ω ( D ):(1.4) R ( f, ω ) = E ( f, ω ) = inf g ∈ P W ω ( D ) k f − g k . Our main results are the following.
Theorem 1.1.
The norm of the Besov space B α ,q ( D ) , α > , ≤ q ≤ ∞ is equiv-alent to the following norms (1.5) k f k + (cid:18)Z ∞ ( s α E ( f, s )) q dss (cid:19) /q , (1.6) k f k + ∞ X k =0 (cid:0) a kα E ( f, a k ) (cid:1) q ! /q , a > . (1.7) k f k + (cid:18)Z ∞ ( s α R ( f, s )) q dss (cid:19) /q , and (1.8) k f k + ∞ X k =0 (cid:0) a kα R ( f, a k ) (cid:1) q ! /q , a > . Theorem 1.2.
A vector f ∈ H belongs to B α ,q ( D ) , α > , ≤ q ≤ ∞ , if and onlyif there exists a sequence of vectors f k = f k ( f ) ∈ P W a k ( D ) , a > , k ∈ N such thatthe series P k f k converges to f in H and the following inequalities hold for some c > , c > which are independent on f ∈ B α ,q ( D )(1.9) c k f k B α ,q ( D ) ≤ ∞ X k =0 (cid:0) a kα k f k k (cid:1) q ! /q ≤ c k f k B α ,q ( D ) , a > . ESOV VECTORS AND PALEY-WIENER VECTORS IN HILBERT SPACES 3
In the case when α > , q = ∞ one has to make appropriate modifications in theabove formulas.According to (1.4) the functional E ( f, ω ) is a measure of decay of the SpectralTransform F D and the Theorems 1.1 and 1.2 show that Besov spaces on a manifold M describe decay of the Spectral transform F D associated with any appropriateoperator D .In the case H = L ( R d ) the Theorems 1.1 and 1.2 are classical and can be foundin [1], [10] and [22]. In the case when H is L -space on a Riemannian manifoldor a graph and D is the square root from the corresponding Laplace operator theTheorems 1 and 2 were proved in our papers [11]-[20].2. Paley-Wiener subspaces generated by a self-adjoint operator in aHilbert space
Now we describe Paley-Wiener functions for a self-adjoint positive definite oper-ator D in H . According to the spectral theory [3] for any self-adjoint operator D ina Hilbert space H there exist a direct integral of Hilbert spaces X = R X ( λ ) dm ( λ )and a unitary operator F D from H onto X , which transforms domain of D k , k ∈ N , onto X k = { x ∈ X | λ k x ∈ X } with norm(2.1) k x ( λ ) k X k = (cid:18)Z ∞ λ k k x ( λ ) k X ( λ ) dm ( λ ) (cid:19) / besides F D ( D k f ) = λ k ( F D f ) , if f belongs to the domain of D k . As it is known, X is the set of all m -measurable functions λ → x ( λ ) ∈ X ( λ ), for which the norm k x k X = (cid:18)Z ∞ k x ( λ ) k X ( λ ) dm ( λ ) (cid:19) / is finite. Definition 1.
We will say that a vector f from H belongs to the Paley-Wienerspace P W ω ( D ) if the support of the Spectral transform F D f belong to [0 , ω ]. Fora vector f ∈ P W ω ( D ) the notation ω f will be used for a positive number such that[0 , ω f ] is the smallest interval which contains the support of the Spectral transform F D f .Using the spectral resolution of identity P λ we define the unitary group of oper-ators by the formula e itD f = Z ∞ e itτ dP τ f, f ∈ H , t ∈ R . Let us introduce the operator(2.2) R ωD f = ωπ X k ∈ Z ( − k − ( k − / e i ( πω ( k − / ) D f, f ∈ H , ω > . Since (cid:13)(cid:13) e it L f (cid:13)(cid:13) = k f k and(2.3) ωπ X k ∈ Z k − / = ω, BESOV VECTORS AND PALEY-WIENER VECTORS IN HILBERT SPACES the series in (2.2) is convergent and it shows that R ωD is a bounded operator in H with the norm ω :(2.4) k R ωD f k ≤ ω k f k , f ∈ H . The next theorem contains generalizations of several results from the classicalharmonic analysis (in particular the Paley-Wiener theorem) and it follows essen-tially from our results in [13], [14], [19].
Theorem 2.1.
The following statements hold: (1) the set S ω> P W ω ( D ) is dense in H ; (2) the space P W ω ( D ) is a linear closed subspace in H ; (3) a function f ∈ H belongs to P W ω ( D ) if and only if it belongs to the set D ∞ = ∞ \ k =1 D k ( D ) , and for all s ∈ R + the following Bernstein inequality takes place (2.5) k D s f k ≤ ω s k f k ;(4) a vector f ∈ H belongs to the space P W ω f ( D ) , < ω f < ∞ , if and only if f belongs to the set D ∞ , the limit lim k →∞ k D k f k /k exists and (2.6) lim k →∞ k D k f k /k = ω f . (5) a vector f ∈ H belongs to P W ω ( D ) if and only if f ∈ D ∞ and the upperbound (2.7) sup k ∈ N (cid:0) ω − k k D k f k (cid:1) < ∞ is finite, (6) a vector f ∈ H belongs to P W ω ( D ) if and only if f ∈ D ∞ and (2.8) lim k →∞ k D k f k /k = ω < ∞ . In this case ω = ω f . (7) a vector f ∈ H belongs to P W ω ( D ) if and only if it belongs to the to theset D ∞ and the following Riesz interpolation formula holds (2.9) ( iD ) n f = ( R ωD ) n f, n ∈ N ;(8) f ∈ P W ω ( D ) if and only if for every g ∈ H the scalar-valued function ofthe real variable (cid:10) e itD f, g (cid:11) , t ∈ R , is bounded on the real line and has anextension to the complex plane as an entire function of the exponential type ω ; (9) f ∈ P W ω ( D ) if and only if the abstract-valued function e itD f is boundedon the real line and has an extension to the complex plane as an entirefunction of the exponential type ω ; (10) f ∈ P W ω ( D ) if and only if the solution u ( t ) , t ∈ R of the Cauchy problemfor the corresponding abstract Schrodinger equation i ∂u ( t ) ∂t = Du ( t ) , u (0) = f, i = √− , ESOV VECTORS AND PALEY-WIENER VECTORS IN HILBERT SPACES 5 has analytic extension u ( z ) to the complex plane C as an entire functionand satisfies the estimate k u ( z ) k H ≤ e ω |ℑ z | k f k H . Direct and Inverse Approximation Theorems
Now we are going to use the notion of the best approximation (1.2) to introduceApproximation spaces E α ,q ( D ) , < α < r, r ∈ N , ≤ q ≤ ∞ , as spaces for whichthe following norm is finite(3.1) k f k E α ,q ( D ) = k f k + (cid:18)Z ∞ ( s α E ( f, s )) q dss (cid:19) /q , where 0 < α < r, ≤ q < ∞ , or 0 ≤ α ≤ r, q = ∞ . It is easy to verify that thisnorm is equivalent to the following ”discrete” norm(3.2) k f k + X j ∈ N (cid:0) a jα E ( f, a j ) (cid:1) q /q , a > , The Plancherel Theorem for F D also gives the following inequality(3.3) E ( f, ω ) ≤ ω − k (cid:18)Z ∞ ω kF D ( D k f )( λ ) k X ( λ ) dm ( λ ) (cid:19) / ≤ ω − k k D k f k . In the classical Approximation theory the Direct and Inverse Theorems giveequivalence of the Approximation and Besov spaces. Our goal is to extend theseresults to a more general setting.For any f ∈ H we introduce a difference operator of order m ∈ N as(3.4) ∆ mτ f = ( − m +1 m X j =0 ( − j − C jm e jτ ( iD ) f, τ ∈ R . and the modulus of continuity is defined as(3.5) Ω m ( f, s ) = sup | τ |≤ s k ∆ mτ f k The following Theorem is a generalization of the classical Direct ApproximationTheorem by entire functions of exponential type [10].
Theorem 3.1.
There exists a constant
C > such that for all ω > and all f (3.6) E ( f, ω ) ≤ Cω k Ω m − k (cid:0) D k f, /ω (cid:1) , ≤ k ≤ m. In particular the following embeddings hold true (3.7) B α ,q ( D ) ⊂ E αq ( D ) , ≤ q ≤ ∞ . Proof. If h ∈ L ( R ) is an entire function of exponential type ω then for any f ∈ H the vector g = Z ∞−∞ h ( t ) e itD f dt belongs to P W ω ( D ) . Indeed, for every real τ we have e iτD g = Z ∞−∞ h ( t ) e i ( t + τ ) D f dt = Z ∞−∞ h ( t − τ ) e itD f dt. BESOV VECTORS AND PALEY-WIENER VECTORS IN HILBERT SPACES
Using this formula we can extend the abstract function e iτD g to the complex planeas e izD g = Z ∞−∞ h ( t − z ) e itD f dt. Since by assumption h ∈ L ( R ) is an entire function of exponential type ω we have k e izD g k ≤ k f k Z ∞−∞ | h ( t − z ) | dt ≤ k f k e ω | z | Z ∞−∞ | h ( t ) | dt. It shows that for every functional g ∗ ∈ H the function (cid:10) e izD g, g ∗ (cid:11) is an entirefunction and (cid:12)(cid:12)(cid:10) e izD g, g ∗ (cid:11)(cid:12)(cid:12) ≤ k g ∗ kk f k e ω | z | Z ∞−∞ | h ( t ) | dt. In other words (cid:10) e izD g, g ∗ (cid:11) is an entire function of the exponential type ω which isbounded on the real line and application of the classical Bernstein theorem givesthe following inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ddt (cid:19) k (cid:10) e itD g, g ∗ (cid:11)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ω k sup t ∈ R (cid:12)(cid:12)(cid:10) e itD g, g ∗ (cid:11)(cid:12)(cid:12) . Since (cid:18) ddt (cid:19) k (cid:10) e itD g, g ∗ (cid:11) = (cid:10) e itD ( iD ) k g, g ∗ (cid:11) we obtain for t = 0 (cid:12)(cid:12)(cid:10) D k g, g ∗ (cid:11)(cid:12)(cid:12) ≤ ω k k g ∗ kk f k Z ∞−∞ | h ( τ ) | dτ. Choosing g ∗ such that k g ∗ k = 1 and (cid:10) D k g, g ∗ (cid:11) = k D k g k we obtain the followinginequality k D k g k ≤ ω k k f k Z ∞−∞ | h ( τ ) | dτ which implies that g belongs to P W ω ( D ).Let(3.8) h ( t ) = a (cid:18) sin( t/n ) t (cid:19) n where n ≥ m + 3 is an even integer and a = (cid:18)Z ∞−∞ (cid:18) sin( t/n ) t (cid:19) n dt (cid:19) − . With such choice of a and n the function h will have the following properties:(1) h is an even nonnegative entire function of exponential type one;(2) h belongs to L ( R ) and its L ( R )-norm is 1;(3) the integral(3.9) Z ∞−∞ h ( t ) | t | m dt is finite. ESOV VECTORS AND PALEY-WIENER VECTORS IN HILBERT SPACES 7
Consider the following vector(3.10) Q ω,mh ( f ) = Z ∞−∞ h ( t ) n ( − m − ∆ mt/ω f + f o dt, where(3.11) ( − m +1 ∆ ms f = ( − m +1 m X j =0 ( − j − C jm e js ( iD ) f = m X j =1 b j e js ( iD ) f − f, and(3.12) b + b + ... + b m = 1 . The formulas (3.10) and (3.11) imply the following formula Q ω,mh ( f ) = Z ∞−∞ h ( t ) m X j =1 b j e j tω ( iD ) f dt = Z ∞−∞ Φ( t ) e t ( iD ) f dt. where Φ( t ) = m X j =1 b j (cid:18) ωj (cid:19) h (cid:18) t ωj (cid:19) . Since the function h ( t ) is of the exponential type one every function h ( tω/j ) is ofthe type ω/j . It also shows that the function Φ( t ) is of the exponential type ω aswell.Now we estimate the error of approximation of f by Q ω,mh ( f ). If the modulus ofcontinuity is defined as(3.13) Ω m ( f, s ) = sup | τ |≤ s k ∆ mτ f k then since by (3.10) f − Q ω,mh ( f ) = Z ∞−∞ h ( t )∆ mt/ω f dt we obtain E ( f, ω ) ≤ k f − Q ω,mh ( f ) k ≤ Z ∞−∞ h ( t ) (cid:13)(cid:13)(cid:13) ∆ mt/ω f (cid:13)(cid:13)(cid:13) dt ≤ Z ∞−∞ h ( t )Ω m ( f, t/ω ) dt. Now we are going to use the following inequalities(3.14) Ω m ( f, s ) ≤ s k Ω m − k ( D k f, s )(3.15) Ω m ( f, as ) ≤ (1 + a ) m Ω m ( f, s ) , a ∈ R + . The first one follows from the identity(3.16) ∆ kt f = (cid:0) e itD − I (cid:1) k f = Z t ... Z t e i ( τ + ...τ k ) D D k f dτ ...dτ k , where I is the identity operator and k ∈ N . The second one follows from theproperty Ω ( f, s + s ) ≤ Ω ( f, s ) + Ω ( f, s )which is easy to verify. We can continue our estimation of E ( f, ω ). E ( f, ω ) ≤ Z ∞−∞ h ( t )Ω m ( f, t/ω ) dt ≤ Ω m − k (cid:0) D k f, /ω (cid:1) ω k Z ∞−∞ h ( t ) | t | k (1 + | t | ) m − k dt ≤ BESOV VECTORS AND PALEY-WIENER VECTORS IN HILBERT SPACES C hm,k ω k Ω m − k (cid:0) D k f, /ω (cid:1) , where the integral C hm,k = Z ∞−∞ h ( t ) | t | k (1 + | t | ) m − k dt is finite by the choice of h . The inequality (3.6) is proved and it implies the secondpart of the Theorem. (cid:3) In fact we proved a little bit more. Namely for the same choice of the function h the following holds. Corollary 3.1.
For any ≤ k ≤ m, k, m ∈ N , here exists a constant C hm,k suchthat for all < ω < ∞ and all f ∈ H the following inequality holds (3.17) E ( f, ω ) ≤ kQ ω,mh ( f ) − f k ≤ C hm,k ω k Ω m − k (cid:0) D k f, /ω (cid:1) , where C hm,k = Z ∞−∞ h ( t ) | t | k (1 + | t | ) m dt, ≤ k ≤ m, and the operator Q ω,mh : H →
P W ω ( D ) is defined in (3.10). Next, we are going to obtain the Inverse Approximation Theorem in the case q = ∞ . Lemma 1.
If there exist r > α − n > , α > , r, n ∈ N , such that the quantity (3.18) b α ∞ ,n,r ( f ) = sup s> (cid:0) s n − α Ω r ( D n f, s ) (cid:1) is finite, then there exists a constant A = A ( n, r ) for which (3.19) sup s> s α E ( f, s ) ≤ A ( n, r ) b α ∞ ,n,r ( f ) . Proof.
Assume that (3.18) holds, thenΩ r ( D n f, s ) ≤ b α ∞ ,n,r ( f ) s α − n and (3.17) implies E ( f, s ) ≤ C hn + r,n s − n b α ∞ ,n,r ( f ) s n − α =(3.20) A ( n, r ) b α ∞ ,n,r ( f ) s − α . Lemma is proved. (cid:3)
Lemma 2.
If for an f ∈ H and for an α > the following upper bound is finite (3.21) sup s> s α E ( f, s ) = T ( f, α ) < ∞ , then for every r > α − n > , α > , r, n ∈ N , there exists a constant C ( α, n, r ) suchthat the next inequality holds (3.22) b α ∞ ,n,r ( f ) ≤ C ( α, n, r ) ( k f k + T ( f, α )) . ESOV VECTORS AND PALEY-WIENER VECTORS IN HILBERT SPACES 9
Proof.
The assumption implies that for a given f ∈ H and a sequence of numbers a j , a > , j = 0 , , , ... one can find a sequence g j ∈ P W a j ( D ) such that k f − g j k ≤ T ( f, α ) a − jα , a > . Then for(3.23) f = g , f j = g j − g j − ∈ P W a j ( D ) , the series(3.24) f = f + f + f + .... converges in H . Moreover, we have the following estimates k f k = k g k ≤ k g − f k + k f k ≤ k f k + T ( f, α ) , k f j k ≤ k f − g j k + k f − g j − k ≤ T ( f, α ) a − jα + T ( f, α ) a − ( j − α = T ( f, α )(1+ a α ) a − jα , which imply the following inequality(3.25) k f j k ≤ C ( a, α ) a − jα ( k f k + T ( f, α )) , j ∈ N . Since f j ∈ P W a j ( D ) we have for any n ∈ N (3.26) k D n f j k ≤ a jn k f j k , a > , we obtain k D n f j k ≤ C ( a, α ) a − j ( α − n ) ( k f k + T ( f, α ))which shows that the series X j ∈ N D n f j converges in H and because the operator D n is closed the sum f of this seriesbelongs to the domain of D n and D n f = X j ∈ N D n f j . Next, let F j = D n f j then we have that D n f = P j F j , where F j ∈ P W a j ( D ) andaccording to (3.25) and (3.26)(3.27) k F j k = k D n f j k ≤ a jn k f j k ≤ C ( a, α ) a − j ( α − n ) ( k f k + T ( f, α )) . Pick a positive t and a natural N such that(3.28) a − N ≤ t < a − N +1 , a > , then we obviously have the following formula for any natural r (3.29) ∆ rt D n f = N − X j =0 ∆ rt F j + ∞ X j = N ∆ rt F j , where ∆ rt is defined in (3.4). Note, that the Bernstein inequality and the formula(3.16) imply that if f ∈ P W ω ( D ), then(3.30) k ∆ rt f k ≤ ( tω ) r k f k . Since (3.27) and (3.28) hold we obtain for j ≤ N − k ∆ rt F j k ≤ ( a j t ) r k F j k ≤ C ( a, α )( k f k + T ( f, α )) a j ( n + r − α ) − ( N − r ) , a > . These inequalities imply (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N − X j =0 ∆ rt F j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C ( a, α )( k f k + T ( f, α )) a − r ( N − N − X j =0 a ( n + r − α ) j = C ( a, α )( k f k + T ( f, α )) a − r ( N − − a ( n + r − α ) N − a ( n + r − α ) ≤ (3.31) C ( a, α, n, r )( k f k + T ( f, α )) t α − n . By applying the following inequality k ∆ rt F j k ≤ r k F j k to terms with j ≥ N we can continue our estimation as (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = N ∆ rt F j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ r C ( a, α )( k f k + T ( f, α )) ∞ X j = N a − ( α − n ) j = C ( a, α )2 r ( k f k + T ( f, α )) a − N ( α − n ) (1 − a ( n − α ) ) − ≤ (3.32) C ( a, α, n, r )( k f k + T ( f, α )) t α − n . It gives the following inequality k ∆ rt D n f k ≤ C ( a, α, n, r ) t α − n ( k f k + T ( f, α )) , from which one hasΩ r ( D n f, s ) ≤ C ( a, α, n, r )( k f k + T ( f, α )) s α − n , s > , and b α ∞ ,n,r ( f ) ≤ C ( a, α, n, r )( k f k + T ( f, α )) . The Lemma is proved. (cid:3)
Our main result concerning spaces B α , ∞ ( D ) , α > , is the following. Theorem 3.2.
The norm of the space B α , ∞ D ) , α > , is equivalent to the followingnorms (3.33) k f k + sup s> ( s α E ( f, s )) , (3.34) k f k + sup s> ( s α R ( f, s ))) , (3.35) k f k + sup k ∈ N (cid:0) a kα E ( f, a k ) (cid:1) , a > , (3.36) k f k + sup k ∈ N (cid:0) a kα R ( f, a k )) (cid:1) , a > . Moreover, a vector f ∈ H belongs to B α , ∞ ( D ) , α > , if and only if there existsa sequence of vectors f k = f k ( f ) ∈ P W a k ( D ) , a > , such that the series P f k converges to f in H and (3.37) c k f k B α , ∞ ( D ) ≤ sup k ∈ N (cid:0) a kα k f k k (cid:1) ≤ c k f k B α , ∞ ( D ) , a > , for certain c = c ( D, α ) , c = c ( D, α ) which are independent of f ∈ B α , ∞ ( D ) . ESOV VECTORS AND PALEY-WIENER VECTORS IN HILBERT SPACES 11
Proof.
That the norm of B α , ∞ ( D ) , α > , is equivalent to any of the norms (3.33)-(3.36) follows from the last two Lemmas and (1.4).Next, if the norm (3.33) is finite then it was shown in the proof of the last Lemmathat there exists a sequence of vectors f k = f k ( f ) ∈ P W a k ( D ) , a > , such that theseries P f k converges to f in H . Moreover, the inequality (3.25) shows existence ofconstant c which is independent of f ∈ B α , ∞ ( D ) for which the following inequalityholds sup k ∈ N (cid:0) a kα k f k k (cid:1) ≤ c k f k B α , ∞ ( D ) , a > , Conversely, let us assume that there exists a sequence of vectors f k = f k ( f ) ∈ P W a k ( D ) , a > , such that the series P f k converges to f in H andsup k ∈ N (cid:0) a kα k f k k (cid:1) < ∞ . We have E ( f, a N ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f − N − X k =0 f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = ∞ X k = N k f k k ≤ sup k ∈ N (cid:0) a kα k f k k (cid:1) ∞ X k = N a − αj ≤ C sup k ∈ N (cid:0) a kα k f k k (cid:1) a − Nα , or sup N a Nα E ( f, a N ) ≤ C sup k ∈ N (cid:0) a kα k f k k (cid:1) . Since we also have k f k ≤ X k k f k k ≤ sup k ∈ N (cid:0) a kα k f k k (cid:1) X k a − αk , a > , the Theorem is proved. (cid:3) The Theorems 1.1 and 1.2 from Introduction are extensions of the Theorem 3.2to all indices 1 ≤ q ≤ ∞ . Their proofs go essentially along the same lines as theproof of the last Theorem and are omitted. References
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