Summer School on Applied Analysis 2011 TU Chemnitz, September 26-30, 2011 Polynomial approximation via de la Vall´ ee Poussin means Woula Themistoclakis CNR - National Research Council of Italy Institute for Computational Applications “Mauro Picone”, Naples, Italy. Woula Themistoclakis, TU Chemnitz, September 26-30, 2011 1
◮ Lecture 1: De la Vall´ ee Poussin means • Approximation properties • Basic facts on polynomial approximation • Application to prove boundedness of some CSIO in Lipschitz type spaces ◮ Lecture 2: Discrete de la Vall´ ee Poussin means • Approximation properties • Comparison with Lagrange interpolation • Interpolating de la Vall´ ee Poussin polynomials ◮ Lecture 3: Applications of de la Vall´ ee Poussin type interpolation • Part 1: A numerical method for solving the generalized airfoil equation • Part 2: Construction of interpolating polynomial wavelets on [-1,1] Woula Themistoclakis, TU Chemnitz, September 26-30, 2011 2
APPROXIMATION THEOREM: [ (1885) Karl Weierstrass ] For any f ∈ C [ − 1 , 1] and each ǫ > 0 , there exists an algebraic polynomial Q such that � f − Q � ∞ < ǫ � ( f − Q ) u � p < ǫ, 1 ≤ p ≤ ∞ Weighted extensions: where u ( x ) := v α,β ( x ) = (1 − x ) α (1 + x ) β ∈ L p [ − 1 , 1] is a Jacobi weight and if 1 ≤ p < ∞ , we assume f ∈ L p u with L p u := { f : � fu � p < ∞} while in the case p = ∞ , we suppose f ∈ C 0 u , with C 0 u := { f ∈ C ( − 1 , 1) : lim | x |→ 1 f ( x ) u ( x ) = 0 } , if α, β > 0 , C 0 u := { f ∈ C ( − 1 , 1] : lim x →− 1 f ( x ) u ( x ) = 0 } , if α = 0 < β C 0 u := { f ∈ C [ − 1 , 1) : lim x → +1 f ( x ) u ( x ) = 0 } , if α > 0 = β Woula Themistoclakis, TU Chemnitz, September 26-30, 2011 3
n � S n ( w, f, x ) := c k ( w, f ) p k ( w, x ) Fourier partial sums : k =0 where w ( x ) := v α,β ( x ) = (1 − x ) α (1 + x ) β , α, β > − 1 , is a Jacobi weight, { p k ( w, x ) } k denotes the corresponding system of orthonormal Jacobi � 1 polynomials and c k ( w, f ) := − 1 p k ( w, y ) f ( y ) w ( y ) dy . ◮ Invariance: S n ( w, P ) = P, P ∈ P n := { P : deg( P ) ≤ n } . ◮ Boundedness in L p u : Under some conditions on u, w , we have 1 < p < ∞ = ⇒ sup n � S n ( w ) � L p u < ∞ , u → L p ◮ Critical cases: p = 1 , ∞ = ⇒ sup n � S n ( w ) � L p u = + ∞ , ∀ u, w u → L p Woula Themistoclakis, TU Chemnitz, September 26-30, 2011 4
m � 1 V m n ( w, f, x ) := S k ( w, f, x ) De la Vall´ ee Poussin means : m − n + 1 k = n ◮ Quasi-projection: V m n ( w, P ) = P whenever P ∈ P n , but n < m . � 1 V m H m In integral form: n ( w, f, x ) := n ( w, x, y ) f ( y ) w ( y ) dy − 1 m � 1 H m n ( w, x, y ) := K r ( w, x, y ) de la Vall´ ee Poussin kernel m − n +1 r = n where r � K r ( w, x, y ) := p j ( w, x ) p j ( w, y ) Darboux kernel j =0 Woula Themistoclakis, TU Chemnitz, September 26-30, 2011 5
Theorem [ M.R.Capobianco, T. ] For all Jacobi weights w = v α,β , for any pair of sufficiently large integers n ∼ m ∼ m − n (i.e. n < m < C 1 n , and C 2 m < m − n < m with C 1 , C 2 > 0 independent of n, m ) and for each � � c c x, y ∈ − 1 + m 2 , 1 − ( c > 0 fixed), we have m 2 4 , − β 4 , − β n ( w, x, y ) | ≤ Cmv − α 2 − 1 2 − 1 4 ( x ) v − α 2 − 1 2 − 1 | H m 4 ( y ) , (1) If in addition x � = y , then we have 4 , − β 4 , − β Cv − α 2 − 1 2 − 1 4 ( x ) v − α 2 − 1 2 − 1 4 ( y ) | H m (2) n ( w, x, y ) | ≤ E ± ( x, y ) , m | x − y | where, in the case | x − y | ≥ a > 0 , E ± ( x, y ) ≤ C , and generally E ± ( x, y ) := ( √ 1 ± x + √ 1 ± y ) 2 √ 1 ± x + √ 1 ± y + 1 − y 2 . √ 1 − x 2 � | x − y | m In all the previous estimates, C is a positive constant independent of n, x, y . Woula Themistoclakis, TU Chemnitz, September 26-30, 2011 6
Theorem [ M.R.Capobianco, T. ] Let 1 ≤ p ≤ ∞ and consider the map u , where n ∼ m ∼ m − n and w = v α,β , u = v γ,δ satisfy V m n ( w ) : L p u → L p the inequalities α 2 − 1 < γ + 1 α 2 + 5 0 < γ + 1 p < 4 , and p < α + 1 , 4 β 2 − 1 < δ + 1 β 2 + 5 0 < δ + 1 p < 4 , and p < β + 1 , 4 � � � � � γ − δ − α − β � � � ≤ 1 . Then for all f ∈ L p Moreover assume u , we have 2 � ( V m n ( w, f ) u � p ≤ C � fu � p , C � = C ( n, m, f ) . (3) OPEN PROBLEM: State necessary and sufficient conditions for (3) Woula Themistoclakis, TU Chemnitz, September 26-30, 2011 7
Theorem [ G.Mastroianni, T. ] Let 1 ≤ p ≤ ∞ , and w = v α,β , u = v γ,δ be such that the following bounds α 2 + 1 < γ + 1 α 2 + 5 0 < γ + 1 4 − ν p < 4 − ν, and p < α + 1 , β 2 + 1 < δ + 1 β 2 + 5 0 < δ + 1 4 − ν p < 4 − ν, and p < β + 1 , are satisfied for some ν ∈ [0 , 1 / 2] . Then for all positive integers n ∼ m ∼ m − n and any f ∈ L p u , we have � ( V m n ( w, f ) u � p ≤ C � fu � p , C � = C ( n, m, f ) . � p = ∞ and u = 1 ( if α, β < 1 Limiting cases also possible: 2 ) p = 1 and u = w Woula Themistoclakis, TU Chemnitz, September 26-30, 2011 8
Comparison with Fourier sums � 1 S n ( w, f, x ) = − 1 K n ( w, x, y ) f ( y ) w ( y ) dy, � 1 V m − 1 H m n ( w, f, x ) = n ( w, x, y ) f ( y ) w ( y ) dy, m > n � S n ( w, P ) = P, ∀ P ∈ P n (projection on P n ) ◮ Invariance: V m n ( w, P ) = P, ∀ P ∈ P n (quasi-projection on P m ) ◮ Boundedness in C 0 u and L 1 u : Let n ∼ m ∼ m − n . Then � sup n � S n ( w ) � C 0 u = sup n � S n ( w ) � L 1 u = ∞ for any u, w u → C 0 u → L 1 sup n � V m u = sup n � V m n ( w ) � C 0 n ( w ) � L 1 u < ∞ for suitable u, w u → C 0 u → L 1 Woula Themistoclakis, TU Chemnitz, September 26-30, 2011 9
◮ Boundedness in L p u : Let n ∼ m ∼ m − n and 1 < p < ∞ . Then � sup n � S n ( w ) � L p u < ∞ for suitable (more restricted) u u → L p sup n � V m n ( w ) � L p u < ∞ for suitable u u → L p In particular if w = v α,β and u = v γ,δ satisfy the technical requirements 0 < γ + 1 p < α + 1 and 0 < δ + 1 p < β + 1 , then we have α 2 + 1 < γ + 1 α 2 + 3 p < 4 , 4 sup n � S n ( w ) � L p u < ∞ ⇔ u → L p β 2 + 1 < δ + 1 β 2 + 3 p < 4 , 4 α 2 + 1 4 − ν < γ + 1 p < α 2 + 5 4 − ν, 0 ≤ ν ≤ 1 n � V m sup n ( w ) � L p u < ∞ ⇐ u → L p β 2 + 1 4 − ν < δ + 1 p < β 2 + 5 2 4 − ν, Woula Themistoclakis, TU Chemnitz, September 26-30, 2011 10
ERROR OF BEST POLYNOMIAL APPROXIMATION IN L p u E n ( f ) u,p := deg ( P n ) ≤ n � ( f − P n ) u � p inf 1 ≤ p ≤ ∞ , ◮ Invariance on P n implies f − V m n ( w, f ) = ( f − P n ) − V m n ( w, f − P n ) , i.e. � [ f − V m n ( w, f )] u � p ≤ (1 + � V m n ( w ) � L p u ) E n ( f ) u,p u → L p ◮ Boundedness in L p u is equivalent to: E m ( f ) u,p ≤ � [ f − V m n ( w, f )] u � p ≤ CE n ( f ) u,p i.e. V m n ( w, f ) is a near best polynomial approximating f ∈ L p u . Woula Themistoclakis, TU Chemnitz, September 26-30, 2011 11
Ditzian–Totik moduli of smoothness: Non weighted case: ω r � ∆ r ϕ ( f, t ) p := sup hϕ f � L p [ − 1 , 1] 0 <h ≤ t Ω r � (∆ r ϕ ( f, t ) u,p := sup hϕ f ) u � L p [ − 1+4 r 2 h 2 , 1 − 4 r 2 h 2 ] ← ( main–part) 0 <h ≤ t ω r Ω r ϕ ( f, t ) u,p := ϕ ( f, t ) u,p + deg ( P ) <r � ( f − P ) u � L p [ − 1 , − 1+4 r 2 t 2 ] inf + deg ( P ) <r � ( f − P ) u � L p [1 − 4 r 2 t 2 , 1] inf √ 1 − x 2 and ∆ r where we set ϕ ( x ) := hϕ f is the central r th difference of f of variable step size hϕ ( x ) , i.e. � � � � x + h x − h ∆ r hϕ f = ∆∆ r − 1 ∆ hϕ f ( x ) := f 2 ϕ ( x ) − f 2 ϕ ( x ) , hϕ Woula Themistoclakis, TU Chemnitz, September 26-30, 2011 12
Some basic properties: (worth for Ω r ϕ ( f, t ) u,p too) lim t → 0 ω r ∀ f ∈ L p ( f ∈ C 0 ( i ) ϕ ( f, t ) u,p = 0 , u when p = ∞ ) u ⇒ ω r ϕ ( f, t 1 ) u,p ≤ ω r ( ii ) t 1 ≤ t 2 = ϕ ( f, t 2 ) u,p ω r ϕ ( f, t ) u,p ≤ Cω r − 1 ( iii ) ( f, t ) u,p ϕ ω r ϕ ( f, λt ) u,p ≤ Cλ r ω r ( iv ) ϕ ( f, t ) u,p , ( λ > 1) ω r ϕ ( f, t ) u,p ≤ C tω r − 1 ( f ′ , t ) uϕ,p , ( f ∈ AC loc : � f ′ uϕ � p < ∞ ) ( v ) ϕ ω r K r ϕ ( f, t r ) u,p ϕ ( f, t ) u,p ∼ Equivalent K –functionals: where: ˜ Ω r K r ϕ ( f, t r ) u,p ϕ ( f, t ) u,p ∼ {� ( f − g ) u � p + t � g ( r ) ϕ r u � p } K r ϕ ( f, t ) u,p := inf g ( r − 1) ∈ AC loc ˜ {� ( f − g ) u � L p ( I r,h ) + h � g ( r ) ϕ r u � L p ( I r,h ) } K r ϕ ( f, t ) u,p := sup inf g ( r − 1) ∈ AC loc 0 <h ≤ t I r,h := [ − 1 + 4 r 2 h 2 , 1 − 4 r 2 h 2 ] Woula Themistoclakis, TU Chemnitz, September 26-30, 2011 13
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