Polynomial approximation via de la Vall ee Poussin means { - PowerPoint PPT Presentation

Summer School on Applied Analysis 2011 TU Chemnitz, September 26-30, 2011 Polynomial approximation via de la Vall´ ee Poussin means { • Generalized airfoil equation (Part 1) Lecture 3: • Polynomial wavelets (Part 2) Woula Themistoclakis CNR - National Research Council of Italy Institute for Computational Applications “Mauro Picone”, Naples, Italy. Woula Themistoclakis - Chemnitz, September 26-30, 2011 1

GENERALIZED AIRFOIL EQUATION √ 1 − y √ 1 − y ∫ 1 ∫ 1 − 1 f ( y ) 1 + ydy + ν log | x − y | f ( y ) 1 + ydy = g ( x ) , | x | < 1 π y − x π − 1 − 1 where the first integral is in the Cauchy principal value sense, ν is a complex number, g is a known function and f is the sought solution. Df ( x )+ νKf ( x ) = g ( x ) , | x | < 1 ← Operator form ∫ 1 ◮ Cauchy singular integral operator: Df ( x ) = − 1 f ( y ) 1 2 , − 1 y − xv 2 ( y ) dy π − 1 ∫ 1 Kf ( x ) = 1 1 2 , − 1 ◮ Perturbation operator: log | x − y | f ( y ) v 2 ( y ) dy π − 1 Woula Themistoclakis - Chemnitz, September 26-30, 2011 2

For u ( x ) = (1 − x ) γ (1 + x ) δ with γ, δ ≥ 0 , we consider ◮ Weighted spaces of locally continuous functions: { } lim x → 1 ( fu )( x ) = 0 if γ > 0 and C 0 f ∈ C 0 u := loc : lim x →− 1 ( fu )( x ) = 0 if δ > 0 equipped with the norm ∥ f ∥ C 0 u := ∥ fu ∥ ∞ . Z r ( u ) := { f ∈ C 0 ◮ H¨ older–Zygmund subspaces: u : ∥ f ∥ Z r ( u ) < ∞} equipped with the norm ω k ϕ ( f, t ) u, ∞ ( k + 1) r E k ( f ) u, ∞ ∥ f ∥ Z r ( u ) := ∥ fu ∥ ∞ + sup ∼ ∥ fu ∥ ∞ + sup t r t> 0 k> 0 Woula Themistoclakis - Chemnitz, September 26-30, 2011 3

∫ 1 ◮ Mapping properties of Df ( x ) = − 1 f ( y ) 1 2 , − 1 y − x v 2 ( y ) . π − 1 2 , 0 ) → Z r ( v 0 , 1 1 TH. 1: For all r > 0 , the map D : Z r ( v 2 ) is linear, bounded, ∫ 1 Df ( x ) = 1 f ( y ) y − xv − 1 2 , 1 with bounded inverse given by � 2 ( y ) dy. Moreover π − 1 ω k ω k ϕ ( Df, t ) ϕ ( f, t ) v 0 , 1 1 2 , 0 , ∞ 2 , ∞ v sup ∼ sup , k > r > 0 t r t r t> 0 t> 0 Note: More generally, in the first lecture we studied ∫ 1 D α, − α f ( x ) := cos παf ( x ) v α, − α ( x ) − sin πα f ( y ) y − xv α, − α ( y ) dy, π − 1 establishing TH.1 for the map D α, − α : Z r ( v α, 0 ) → Z r ( v 0 ,α ) . Woula Themistoclakis - Chemnitz, September 26-30, 2011 4

∫ 1 ◮ Mapping properties of Kf ( x ) = 1 1 2 , − 1 log | x − y | f ( y ) v 2 ( y ) dy π − 1 1 2 , 0 ) → Z r +1 is bounded and TH. 2: For all r > 0 , the map K : Z r ( v ω k ϕ ( f, t ) ω k +1 ( Kf, t ) ∞ 1 2 , 0 , ∞ 1 ϕ v 2 , 0 ∥ ∞ , ∥ Kf ∥ ∞ ≤ C ∥ fv sup ≤ C sup t r +1 t r t> 0 t> 0 1 2 , 0 ) . hold for all k > r , C > 0 being independent of f ∈ Z r ( v Note that: • The identity ( Kf ) ′ = Df and TH.1 can be used in order to prove the second inequality of TH.2. • Since Z r +1 is compactly embedded into Z s for all s < r + 1 , by TH.2 1 2 , 0 ) → Z r is compact. we also get that the map K : Z r ( v Woula Themistoclakis - Chemnitz, September 26-30, 2011 5

◮ Solvability of ( D + νK ) f = g : By the previous theorems we can apply the Fredholm alternative theorem to the regularized equation ( I + ν � DK ) f = � Dg , obtaining the following Corollary: Assume ker { D + νK } = { 0 } . Then for any g ∈ Z r ( v 0 , 1 2 ) the 1 2 , 0 ) . generalized airfoil equation has a unique and stable solution f ∈ Z r ( v Note: ( D.Berthold, W.Hoppe, B.Silbermann ) ker { D + νK } = { 0 } , ∀ ν ∈ R Polynomial projection methods attempt to find a polynomial ◮ approximation of f , namely f n , solving the approximate equation ( D + ν P n K ) f n = P n g , where P n is the polynomial projection defining the method. n →∞ ∥ K − P n K ∥ lim 2 ) = 0 Condition to require: 1 2 , 0 ) → Z r ( v 0 , 1 Z r ( v Woula Themistoclakis - Chemnitz, September 26-30, 2011 6

L n : f → L n ( v − 1 2 , 1 2 , f ) ∈ P n − 1 Lagrange Projections: V n,m ( v − 1 2 , 1 2 , f ) ∈ S n,m ( v − 1 2 , 1 V n,m : f → ˜ ˜ 2 ) de la V.P. Both these projections satisfy the required condition, since we have ≤ C n − 1 log n ∥ K − L n K ∥ 1 2 , 0 ) → Z r ( v 0 , 1 Z r ( v 2 ) ∥ K − ˜ ≤ C n − 1 , V n,m K ∥ m = θn, 0 < θ < 1 1 2 , 0 ) → Z r ( v 0 , 1 Z r ( v 2 ) 1 2 , 0 ) → Z r ( v 0 , 1 TH. 3: If D + νK : Z r ( v 2 ) has bounded inverse, then the 2 , 0 ) → Z r ( v 0 , 1 1 same holds for D + ν P n K : Z r ( v 2 ) , where either P n = L n or P n = ˜ V n,m with m = θn , 0 < θ < 1 . Moreover: sup n ∥ ( D + ν P n K ) − 1 ∥ < ∞ , lim n κ ( D + ν P n K ) = κ ( D + νK ) where κ ( A ) := ∥ A ∥∥ A − 1 ∥ . Woula Themistoclakis - Chemnitz, September 26-30, 2011 7

g ∈ Z r ( v 0 , 1 Airfoil equation: ( D + νK ) f = g, 2 ) P n = L n or P n = ˜ Approximate equation: ( D + ν P n K ) f n = P n g, V n,m ◮ Solvability of the approximate equation: There exists a unique stable There exists a unique stable solution f of the airfoil equation = ⇒ solution f n of the approximate equation ◮ Error estimates depend on P n and can be deduced from f − f n = ( I + ν � D P n K ) − 1 [ � DDf − � D P n Df ] taking into account that { V n,m ( v − 1 2 , 1 if P n = ˜ 2 , 0 ∥ ∞ ≤ C 1 2 ) , m = θn 1 ∥ [ � DF − � D P n F ] v n r ∥ F ∥ Z r ( v 0 , 1 if P n = L n ( v − 1 2 , 1 log n 2 ) 2 ) Woula Themistoclakis - Chemnitz, September 26-30, 2011 8

Theorem 4: The solution f n of the approximate equation corresponding to P n = L n or P n = ˜ V n,m with m = θn , 0 < θ < 1 , satisfies the following error estimates, where C > 0 denotes a constant independent of f and n .   ∥ g ∥  Z r ( v 0 , 1  2 )   ∥ f − f n ∥ 2 , 0 ) ≤ C log n, 0 < s ≤ r  1 n r − s Z s ( v ◮ Lagrange case :   ∥ g ∥  Z r ( v 0 , 1  1 2 )  2 , 0 ∥ ∞ ≤ C  ∥ ( f − f n ) v log n, n r   ∥ g ∥  Z r ( v 0 , 1  2 )   ∥ f − f n ∥ 2 , 0 ) ≤ C , 0 < s ≤ r  1 n r − s Z s ( v ◮ De la V.P. case :   ∥ g ∥  Z r ( v 0 , 1   1 2 ) 2 , 0 ∥ ∞ ≤ C  ∥ ( f − f n ) v n r Woula Themistoclakis - Chemnitz, September 26-30, 2011 9

∫ 1 ∫ 1 Df ( x ) = − 1 f ( y ) Df ( x ) = 1 f ( y ) 1 2 , − 1 y − xv − 1 2 , 1 � y − xv 2 ( y ) dy, 2 ( y ) dy π π − 1 − 1 1 2 , − 1 Theorem 5 The operator D maps the space S n,m ( v 2 ) into the space S n,m ( v − 1 2 , 1 2 ) . This correspondence is bijective and its inverse is D − 1 = � D : S n,m ( v − 1 2 , 1 1 2 , − 1 2 ) → S n,m ( v 2 ) . 2 , − 1 1 2 ) = p k ( v − 1 2 , 1 1 2 , − 1 2 ) = q k ( v − 1 2 , 1 Proof. Dp k ( v 2 ) = ⇒ Dq k ( v 2 ) . ✷ V n,m ( v − 1 2 , 1 V n,m ( v − 1 2 , 1 Df n + ν ˜ 2 , Kf n ) = ˜ Notes on : 2 , g ) 1 2 , − 1 ◮ Its solution f n ∈ S n,m ( v 2 ) . V n,m ( v − 1 2 , 1 V n,m ( v − 1 2 , 1 ˜ 2 , Df n + νKf n ) = ˜ ◮ It is equivalent to: 2 , g ) Woula Themistoclakis - Chemnitz, September 26-30, 2011 10

COMPUTATION OF THE APPROXIMATE SOLUTIONS ∫ 1 w := v − 1 2 , 1 Notations: and < f, g > w := − 1 f ( x ) g ( x ) w ( x ) dx 2 Df n + ν ˜ V n,m ( w, Kf n ) = ˜ ◮ De la Vall´ ee Poussin case: V n,m ( w, g ) We compute f n = ∑ n − 1 k =0 a k q k ( w − 1 ) ∈ S n,m ( w − 1 ) by requiring that < Df n + ν ˜ = < ˜ V n,m ( w, Kf n ) , q h ( w ) > w V n,m ( w, g ) , q h ( w ) > w < q h ( w ) , q h ( w ) > w < q h ( w ) , q h ( w ) > w h = 0 , . . . , n − 1 ◮ Lagrange case: Df n + νL n ( w, f n ) = L n ( w, g ) We compute f n = ∑ n − 1 k =0 b k p k ( w − 1 ) ∈ P n − 1 by requiring that < Df n + νL n ( w, Kf n ) , p h ( w ) > w = < L n ( w, g ) , p h ( w ) > w h = 0 , . . . , n − 1 Woula Themistoclakis - Chemnitz, September 26-30, 2011 11

Linear system by de la V.P. projection method For h = 0 , . . . , n − 1 , set w := v − 1 2 , 1 2 , we have < Df n + ν ˜ = < ˜ V n,m ( w, Kf n ) , q h ( w ) > w V n,m ( w, g ) , q h ( w ) > w < q h ( w ) , q h ( w ) > w < q h ( w ) , q h ( w ) > w which, by f n = ∑ n − 1 k =0 a k q k ( w − 1 ) and Dq k ( w − 1 ) = q k ( w ) , gives [ ] n − 1 ∑ δ h,k + ν< ˜ = < ˜ V n,m ( w, Kq k ( w − 1 )) , q h ( w ) > w V n,m ( w, g ) , q h ( w ) > w a k < q h ( w ) , q h ( w ) > w < q h ( w ) , q h ( w ) > w k =0 [∑ n ] V n,m ( w, f ) = ∑ n − 1 But ˜ j =1 λ n,j p h ( w, x n,j ) f ( x n,j ) q h ( w ) , hence h =0   n − 1 n n ∑ ∑ ∑  δ h,k + ν λ n,j Kq k ( w − 1 )( x n,j ) p h ( w, x n,j )  = a k λ n,j g ( x n,j ) p h ( w, x n,j ) k =0 j =1 j =1 Woula Themistoclakis - Chemnitz, September 26-30, 2011 12