Basic relations valid for the Bernstein space B 2 σ and their extensions to functions from larger spaces in terms of their distances from B 2 σ

Transcription

1 Basic relations valid for the Bernstein space B 2 σ and their extensions to functions from larger spaces in terms of their distances from B 2 σ Part 3: Distance functional approach of Part 2 applied to fundamental theorems of Part 1 P. L. Butzer, G. Schmeisser, R. L. Stens New Trends and Directions in Harmonic Analysis, Fractional Operator Theory, and Image Analysis Summer School, Inzell, September 17 21, 2012

2 Outline 1 Error estimates with rates Approximate sampling theorem Reproducing kernel formula General Parseval decomposition formula Approximate sampling theorem for derivatives 2 Boas-type formulae for higher derivatives Boas-type formulae for bandlimited functions Boas-type formulae for non-bandlimited functions 3 Bernstein inequalities for higher order derivatives 4 Hilbert transforms 5 Applications

3 Approximate sampling theorem (AST) recall Lecture 1 Theorem (Weiss 1963, Brown 1967, Butzer-Splettstößer 1977) Let f F 2 Sπ/σ 2 with σ > 0. Then f (t) = k Z ( kπ f σ ) ( σ ) sinc π t k + (Rσ WKS f )(t) (t R). We have (R WKS σ f )(t) 2 π v σ f (v) dv = 2 π dist 1(f, B 2 σ) = o(1).

4 Corollary If f F 2, then one has for any r N, σ > 0 and t R the derivative free error estimate (Rσ WKS f )(t) 2 π dist 1(f, Bσ) 2 ( c ω r f, v 1, L 2 (R) ) dv. v If in addition f Lip r (α, L 2 (R)) for 1/2 < α r, then uniformly in t R. (R WKS σ f )(t) = O ( σ α+1/2) (σ ) σ

5 Corollary Let f W r,2 (R) C(R) for some r N. Then for t R, (R WKS σ f )(t) c σ r+1/2 f (r) L 2 (R) (σ > 0). If moreover f (r) Lip 1 (α, L 2 (R)), 0 < α 1, then (R WKS σ f )(t) = O ( σ r α+1/2) (σ ). Or, if f (r) M 2,1, then Corollary (R WKS σ f )(t) = O ( σ r ) (σ ). If f H 2 (S d ), then for t R, (Rσ WKS f )(t) c d e dσ f H 2 (S d ) (σ > 0).

6 Function spaces { F 2 := f : R C : f L 2 (R) C(R), f } L 1 (R), S p h {f := : R C : ( f (hk) ) } k Z lp (Z). Lipschitz spaces: ω r (f ; δ; L 2 (R)) := sup h δ r ( ) r ( 1) r j f (u + jh) j j=0 L 2 (R), Lip r (α; L 2 (R)) := { f L 2 (R) : ω r (f ; δ; L 2 (R)) = O(δ α ), δ 0+ }. Sobolev spaces: W r,p (R) := { f : R C : f AC r 1 loc (R), f (k) L p (R), 0 k r }.

7 Function spaces, continued Hardy spaces: { H p (S d ) := f : f analytic on { z C : Iz < d }, f H p (S d ) := { sup 0<y<d R f (t iy) p + f (t + iy) p Wiener amalgam space modulation space: M 2,1 := { f : R C : 1 h n Z { n+1 n 2 1/p } dt} <. ( v ) } 2 1/2 f dv <, h > 0}, h { M 2,1 := f M 2,1 : series converges uniformly in h } on bounded subintervals of (0, ).

8 Reproducing kernel formula (RKF) Theorem (extended, Butzer et al. 2011) Let f F 2 Sσ/π 2, σ > 0. Then f (t) = σ π Furthermore, R (Rσ RKF f (t) 1 2π ( σ ) f (u) sinc π (t u) du + (Rσ RKF f )(t) (t R), (Rσ RKF f )(t) := 1 2π v >σ v >σ Three corollaries above are also valid here. f (v)e itv dv. f (v) dv = 1 dist 1 (f, Bσ) 2 = o(1). 2π

9 General Parseval decomposition formula (GDPF) Theorem (Butzer Gessinger 1995/97) Let f F 2 S 1 π/σ and g F 2. Then for σ > 0 R R σ (f, g) = f (u)g(u) du = π σ R k Z (Rσ WKS f )(u)g(u) du π 1 2 σ ( π ) ( π ) f σ k g σ k + R σ (f, g), k Z ( kπ f σ k Z ) v σ R σ (f, g) π 1 ( kπ ) Rσ WKS f L 2 g L 2+ f 2 σ σ ĝ(v) e ikπv/σ dv. v >σ ĝ(v) dv.

10 Theorem Let f F 2 S 1 π/σ and g F 2. Then for σ > 0 R f (u)g(u) du = π σ k Z ( π ) ( π ) f σ k g σ k + R σ (f, g), Rσ (f, g) R WKS π 1 ( kπ ) σ f L 2 g L 2+ f 2 σ σ If f, g W 1,2 (R) C(R), then k Z v >σ ĝ(v) dv. R σ (f, g) C { dist 2 (f, B 2 σ σ) + dist 2 (g, Bσ) 2 + π } σ dist 2(f, Bσ) 2 dist 2 (g, Bσ) 2.

11 Recall Lecture 2 If f W 1,2 (R) C(R) with v f (v) L 1 (R), then for each r N, { dist 2 (f, Bσ) 2 [ ( c ωr f, v 1, L 2 (R) )] } 1/2 2 dv. If in addition f Lip r (α, L 2 (R)), 0 < α r, then σ dist 2 (f, B 2 σ) = O ( σ 1 α) (σ ).

12 Corollary If f, g W r,2 (R) C(R), r 2, and f (r), g (r) Lip 1 (α, L 2 (R)), 0 < α 1, then R σ (f, g) = O ( σ r α) (σ ). If instead f (r), g (r) M 2,1, then R σ (f, g) = O ( σ r 1/2) (σ ). Corollary If f, g H 2 (S d ), then R σ (f, g) c exp( σd) f H 2 (S d ) g H 2 (S d ) (σ > 0).

13 Approximate sampling theorem for derivatives Theorem For f F 2 S 2 π/σ, σ > 0, with v s f (v) L 1 (R) L 2 (R) for some s N 0. Then f (s) (t) = k Z ( kπ f σ (Rs,σ WKS f )(t) := 1 2π ) sinc (s) σ ( t kπ π σ v >σ ) + (R WKS s,σ f )(t) (t R), f (v) [(iv) r e ivt [(iv) r e ivt ] ] dv. [(iv) s e ivt ] is the 2π-periodic extens. of (iv) s e ivt from ( π, π) to R. Furthermore, 2 (Rs,σ WKS f )(t) v s f 2 (v) dv = π π dist 1(f (s), Bσ) 2 = o(1) v σ uniformly for t R for σ.

14 Corollary If f F 2, s N, and v s f (v) L 1 (R), then for any r N, (R WKS s,σ f )(t) 2 π dist 1(f (s), B 2 σ) c σ ( ω r f (s), v 1, L 2 (R) ) dv. v If in addition f (s) Lip r (α, L 2 (R)) for 1/2 < α r, then (R WKS s,σ f )(t) = O ( σ α+1/2) (σ ).

15 Corollary Let s N, f W r,2 (R) C(R) for some r s + 1. Then for t R, (Rs,σ WKS f )(t) c σ r+s+1/2 f (r) L 2 (R) (σ > 0). If moreover f (r) Lip 1 (α, L 2 (R)), 0 < α 1, then Corollary (R WKS s,σ f )(t) = O ( σ r α+s+1/2) (σ ). If f H 2 (S d ), then for t R, (Rs,σ WKS f )(t) c d σ s e dσ f H 2 (S d ) (σ > 0).

16 Boas formula for the first derivative In 1937 Boas established a differentiation formula which may be presented as follows: Let f B σ, where σ > 0. Then, for h = π/σ, we have f (t) = 1 h k Z ( 1) k+1 ( ( π(k 1 f t + h k 1 )). 2 )2 2

17 Boas-type formulae for higher derivatives (Schmeisser 2009) The Boas-type formulae to be established will be deduced as applications of the Whittaker-Kotel nikov-shannon sampling theorem for higher order derivatives. Theorem (Boas-type formulae) Let f B σ f (2s 1) (t) = 1 h 2s 1 for some σ > 0, and s N. Then k= ( ( ( 1) k+1 A s,k f t + h k 1 )) 2 (t R), (2s 1)! A s,k := π(k 1 2 )2s s 1 j=0 ( 1) j [ ( π k 1 2j (2j)! 2)] (k Z), the series being absolutely and uniformly convergent. A similar expansion holds for even order derivatives.

18 Proof Assume σ = π. Setting t = 1/2 in derivative sampling theorem yields f (2s 1)( 1 ) ( = f (k) sinc (2s 1) 1 ) 2 2 k. k= The sinc-terms can be evaluated by Leibnitz rule, namely, ( { sinc (2s 1) 1 ) 2 k = ( 1)k+1 (2s 1)! s 1 ( 1) j[ π(k 1 2 )] } 2j π(k 1. 2 )2s (2j)! j=0 }{{} =( 1) k+1 A s,k f (2s 1)( 1 ) = 2 k= f (k)( 1) k+1 A s,k. For arbitrary σ > 0, t R apply this to u f (hu + t h).

19 Theorem (Boas-type formulae, extended to non-bandlimited funct.) Let s N, f F 2 and let v 2s 1 f (v) L 1 (R). Then f (2s 1) exists and for h > 0, σ := π/h there holds f (2s 1) (t) = 1 ( ( h 2s 1 ( 1) k+1 A s,k f t+h k 1 )) +(R 2s 1,σ f )(t), 2 k Z (R2s 1,σ f )(t) 2 v 2s 1 f (v) dv π v σ = 2 π dist 1(f (2s 1), B 2 σ). The corollaries stated above for the remainder of the approximate sampling theorem are also valid for R 2s 1,σ f.

20 Theorem (Bernstein inequality) For f B p σ, 1 p, σ > 0, there holds f (s) L p (R) σ s f L p (R) (s N). Corollary (Bernstein inequality for trigonometric polynomials) If f (t) = t n (t) = n k= n c ke ikt, a trigonometric polynomial of degree n, then t n B n, and t (s) n L (R) n s t n L (R) (s N). Theorem (Extended Bernstein inequality) Let s N, f F 2 and suppose that v s f (v) L 1 (R) L 2 (R) as a function of v. Then, for any σ > 0, we have f (s) L 2 (R) σ s f L 2 (R) + dist 2 (f (s), B 2 σ). The proofs follow from the Boas-type formulae above.

21 The Hilbert transform Hilbert transform or conjugate function of f L 2 (R) C(R), is defined by Cauchy principal value f (t u) f (t) := lim du = PV 1 f (t u) du, δ 0+ u π u u >δ It defines a bounded linear operator from L 2 (R) into itself, and [ f ] (v) = ( i sgn v) f (v) a. e. If v s f (v) L 1 (R), then by Fourier inversion formula, ] (s)(t) 1 [ f = f (v)( i sgn v)(iv) s e ivt dv (t R). 2π Thus [ f ] (s) = f (s) ; i. e. derivation and taking Hilbert transform are commutative operations. Since sinc (v) = 1/ 2π for v π and = 0 otherwise, sinc (t) = 1 cos πt πt ( ) = sin2 πt 2. πt 2

22 Theorem (Boas-type formulae for the Hilbert transform) Let f B 2 σ, σ > 0 and h := π/σ. Then for s N, f (2s 1) (t) = 1 h 2s 1 k= à s,0 := ( 1) s π2s 1, 2s à s,k := { (2s 1)! πk 2s ( 1) k ( 1) k+1 à s,k f (t + hk) s 1 j=0 ( 1) j } ( ) 2j πk (2j)! (t R). (k 0). The proof is based on the sampling theorem for the Hilbert transform, f (2s 1) (0) = k= f (k) sinc (2s 1) ( k).

23 Theorem (Boas formulae for the Hilbert transform, extended vers.) Let s N, f F 2 and let v 2s 1 f (v) L 1 (R). Then f (2s 1) exists and for h > 0, σ := π/h, f (2s 1) (t) = 1 h 2s 1 ( 1) k+1 Ã s,k f (t + hk) + ( R 2s 1,σ f )(t), k Z where ( R 2s 1,σ f )(t) 1 2π v σ v 2s 1 f (v) dv = 1 2π dist 1 (f (2s 1), B 2 σ).

24 Proof First assume h = 1, i. e., σ = π. Let f 1 (t) := 1 2π v π f (v)e itv dv. Then f f 1 Bπ difference, i. e., and so the bandlimited case applies to this ( R 2s 1,π f )(t) = ( R2s 1,π (f f 1 ) ) (t)+( R 2s 1,π f 1 )(t) = ( R 2s 1,π f 1 )(t), and we find that for t = 0, ( R 2s 1,π f )(0) = f 1 (2s 1) (0) k= ( 1) k+1 Ã s,k f 1 (k).

25 Proof, continued ( R 2s 1,π f )(0) = f 1 (2s 1) (0) f 1 (2s 1) (0) = 1 2π f 1 (k) = 1 2π v π v π k= ( 1) k+1 Ã s,k f 1 (k). f (v)( i sgn v)(iv) 2s 1 dv f (v)e ikv dv (k Z). Inserting the last two equations into the first one and interchanging summation and integration yields ( R 2s 1,π f )(0) = 1 2π v π f (v) [( i sgn v)(iv) 2s 1 ] ( 1) k+1 Ã s,k e ikv dv. k=

26 Proof, continued The foregoing infinite series is a Fourier series of a 2π-periodic function and can be evaluated to be ( 1) k+1 Ã s,k e ikv = ( 1) s+1 v 2s 1 ( v < π). k= ( R 2s 1,π f )(0) = 1 2π v π f (v) [( i sgn v)(iv) 2s 1 ( 1) s+1[ v 2s 1] ] dv 2 v 2s 1 f (v) 2 dv = π v σ π dist 1(f (2s 1), Bσ). 2 [ v 2s 1 ] is the 2π-periodic extension of v 2s 1 from ( π, π) to R. This is the desired inequality for σ = π, h = 1 and t = 0. For the general case apply this estimate to the function u f (t + hu).

27 Applications Consider g(t) := 1/(1 + t 2 ), t R, Fourier transform π/2 exp( v ), Hilbert transform g(t) = t/(1 + t 2 ). Extended sampling theorem for Hilbert transform takes on concrete form for g, 1 t 2 (1 + t 2 ) 2 k= 1 2π σ 2 σ 2 + (kπ) 2 v σ ( ) ( ) π σt k sin π(σt k) + cos(π(σt k)) 1 π(σt k) 2 π 2 v e v dv = (1 + σ)e σ (σ > 0).

28 Truncation error In practise one has to deal with finite sum, no infinite series. Leads to additional truncation error, (T σ,n f )(t) = k N+1 σ 2 σ 2 + (kπ) 2 ( ) ( ) π σt k sin π(σt k) + cos(π(σt k)) 1 π(σt k) 2 This yields for the truncation error (T σ,n f )(t) σ2 (2γ + 1) π 2 (γ 1) 2σ2 (2γ + 1) π 2 (γ 1) k N+1 for N γσ t and some constant γ > 1. N 1 k 3 1 u 3 du = σ2 (2γ + 1) π 2 (γ 1) N 2.

29 Combined error Combining the aliasing error with truncation error we finally obtain 1 t 2 N (1 + t 2 ) 2 clear (1 + σ)e σ + σ2 (2γ + 1) π 2 (γ 1) N 2 k= N (σ > 0; N γσ t ). Similarly, Boas-type theorem for derivative g takes the form 1 t 2 { π (1 + t 2 ) 2 1 2h 1 + t 2 1 } 2 1 h π(2k + 1) [t + (2k + 1)h] 2 1 2π v σ k= π 2 v e v dv = (1 + σ)e σ (σ > 0).

30 For the second order derivative g one obtains 2t 3 6t (1 + t 2 ) ( π 2πk + 2( 1) k) ( 1) k h 2 π(2k 1) [t + (2k + 1)h] 2 1 2π v σ k= π 2 v 2 e v dv = (2 + 2σ + σ 2 )e σ (σ > 0). These are the aliasing errors for the reconstruction of derivatives of the Hilbert transform in terms of Boas-type formulae. In both cases the truncation errors can be handled in a similar fashion as above.

31 Some Boas-type formulae for bandlimited functions and their Hilbert transform f (3) (t) = 1 h 3 f (t) = 1 h k Z f (t) = π2 3h 2 f (t) + 2 h 2 k= ( 1) k+1 6 π( 1 2 k)4 f (t) = π 2h f (t) + 1 h f (t) = 1 h 2 k= ( 1) k+1 ( ( π(k 1 f t + h k 1 )), 2 )2 2 k= k= k 0 f (t + hk) ( 1)k+1 k 2, [1 π2 ( 1 ) ] 2 ( ( 2 2 k f t + h k 1 )), 2 2 π(2k + 1) 2 f ( t + h(2k + 1) ), )] ( 1) k+1 2[ ( 1) k + (π ( k 1 2 π ( k 1 ) 3 f 2 ( ( t +h k 1 )). 2

32 References I P. L. Butzer, P. J. S. G. Ferreira, J. R. Higgins, G. Schmeisser, and R. L. Stens. The sampling theorem, Poisson s summation formula, general Parseval formula, reproducing kernel formula and the Paley-Wiener theorem for bandlimited signals - their interconnections. Applicable Analysis, 90(3-4): , P. L. Butzer, P. J. S. G. Ferreira, J. R. Higgins, G. Schmeisser, and R. L. Stens. The generalized Parseval decomposition formula, the approximate sampling theorem, the approximate reproducing kernel formula, Poisson s summation formula and Riemann s zeta function; their interconnections for non-bandlimited functions. to appear.

33 References II P. L. Butzer, P. J. S. G. Ferreira, G. Schmeisser, and R. L. Stens. The summation formulae of Euler-Maclaurin, Abel-Plana, Poisson, and their interconnections with the approximate sampling formula of signal analysis. Results Math., 59(3-4): , P. L. Butzer and A. Gessinger. The approximate sampling theorem, Poisson s sum formula, a decomposition theorem for Parseval s equation and their interconnections. Ann. Numer. Math., 4(1-4): , P. L. Butzer, G. Schmeisser, and R. L. Stens. Basic relations valid for the Bernstein space B p σ and their extensions to functions from larger spaces in terms of their distances from B p σ. to appear.

34 References III P. L. Butzer, G. Schmeisser, and R. L. Stens. Shannon s sampling theorem for bandlimited signals and their Hilbert transform, Boas-type formulae for higher order derivatives the aliasing error involved by their extensions from bandlimited to non-bandlimited signals. invited for publication in Entropy. P. L. Butzer and W. Splettstösser. A sampling theorem for duration limited functions with error estimates. Inf. Control, 34:55 65, P. L. Butzer, W. Splettstösser, and R. L. Stens. The sampling theorem and linear prediction in signal analysis. Jber.d.Dt.Math.-Verein., 90:1 70, 1988.

35 References IV G. Schmeisser. Numerical differentiation inspired by a formula of R. P. Boas. J. Approximation Theory, 160: , R. L. Stens. A unified approach to sampling theorems for derivatives and Hilbert transforms. Signal Process., 5(2): , 1983.