Outline. Approximate sampling theorem (AST) recall Lecture 1. P. L. Butzer, G. Schmeisser, R. L. Stens

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1 Outline Basic relations valid or the Bernstein space B and their extensions to unctions rom larger spaces in terms o their distances rom B Part 3: Distance unctional approach o Part applied to undamental theorems o Part P. L. Butzer, G. Schmeisser,. L. Stens New Trends and Directions in Harmonic Analysis, Fractional Operator Theory, and Image Analysis Summer School, Inzell, September 7, 0 Error estimates with rates Approximate sampling theorem eproducing kernel ormula General Parseval decomposition ormula Approximate sampling theorem or derivatives Boas-type ormulae or higher derivatives Boas-type ormulae or bandlimited unctions Boas-type ormulae or non-bandlimited unctions 3 Bernstein inequalities or higher order derivatives 4 Hilbert transorms 5 Applications Approximate sampling theorem AST recall Lecture Theorem Weiss 963, Brown 967, Butzer-Splettstößer 977 Let F Sπ/ with > 0. Then t = kπ We have t π sinc π t k + t t. v dv = π dist, B = o. I F, then one has or any r N, > 0 and t the derivative ree error estimate t π dist, B c ω r, v, L dv. v I in addition Lip r α, L or / < α r, then uniormly in t. t = O α+/

2 Let W r, C or some r N. Then or t, t c r+/ r L > 0. I moreover r Lip α, L, 0 < α, then t = O r α+/. Or, i r M,, then t = O r. I H S d, then or t, t c d e d H S d > 0. Function spaces F := : C : L C, } L, S p h := : C : hk } lp Z. Lipschitz spaces: ω r ; δ; L := sup h δ r r r j u + jh j j=0 L, Lip r α; L := L : ω r ; δ; L = Oδ α, δ 0+ }. Sobolev spaces: W r,p := : C : AC r loc, k L p, 0 k r }. Function spaces, continued Hardy spaces: H p S d := : analytic on z C : Iz < d }, H p S d := sup 0<y<d t iy p + t + iy p Wiener amalgam space modulation space: M, := : C : h n Z M, := M, n+ n /p } dt} <. v } / dv <, h > 0}, h : series converges uniormly in h } on bounded subintervals o 0,. eproducing kernel ormula KF Theorem extended, Butzer et al. 0 Let F S/π, > 0. Then t = π Furthermore, KF t π u sinc π t u du + KF t t, KF t := π v > v > Three corollaries above are also valid here. ve itv dv. v dv = dist, B = o. π

3 General Parseval decomposition ormula GDPF Theorem Butzer Gessinger 995/97 Let F S π/ and g F. Then or > 0, g = ugu du = π ugu du π π π k g k +, g, kπ, g π kπ L g L + ĝv e ikπv/ dv. v > ĝv dv. Theorem Let F S π/ and g F. Then or > 0 ugu du = π π π k g k +, g,, g π kπ L g L + I, g W, C, then v > ĝv dv., g C dist, B + dist g, B + π } dist, B dist g, B. ecall Lecture I W, C with v v L, then or each r N, dist, B [ c ωr, v, L ] } / dv. I in addition Lip r α, L, 0 < α r, then dist, B = O α. I, g W r, C, r, and r, g r Lip α, L, 0 < α, then, g = O r α I instead r, g r M,, then I, g H S d, then, g = O r /.., g c exp d H S d g H S d > 0.

4 Approximate sampling theorem or derivatives Theorem For F S π/, > 0, with v s v L L or some s N 0. Then s t = kπ s, t := π sinc s t kπ π v > + s, t t, v [iv r e ivt [iv r e ivt ] ] dv. [iv s e ivt ] is the π-periodic extens. o iv s e ivt rom π, π to. Furthermore, s, t v s v dv = π π dist s, B = o I F, s N, and v s v L, then or any r N, s, t π dist s, B c ω r s, v, L dv. v I in addition s Lip r α, L or / < α r, then s, t = O α+/. uniormly or t or. Let s N, W r, C or some r s +. Then or t, s, t c r+s+/ r L > 0. I moreover r Lip α, L, 0 < α, then s, t = O r α+s+/. Boas ormula or the irst derivative In 937 Boas established a dierentiation ormula which may be presented as ollows: Let B, where > 0. Then, or h = π/, we have t = h k+ πk t + h k. I H S d, then or t, s, t c d s e d H S d > 0.

5 Boas-type ormulae or higher derivatives Schmeisser 009 Proo The Boas-type ormulae to be established will be deduced as applications o the Whittaker-Kotel nikov-shannon sampling theorem or higher order derivatives. Theorem Boas-type ormulae Let B s t = h s or some > 0, and s N. Then s! A s,k := πk s k+ A s,k t + h k s j=0 j [ π k j j! ] k Z, the series being absolutely and uniormly convergent. A similar expansion holds or even order derivatives. t, Assume = π. Setting t = / in derivative sampling theorem yields s = k sinc s k. The sinc-terms can be evaluated by Leibnitz rule, namely, sinc s k = k+ s! s j[ πk ] } j πk. s j! j=0 }} = k+ A s,k s = k k+ A s,k. For arbitrary > 0, t apply this to u hu + t h. Theorem Boas-type ormulae, extended to non-bandlimited unct. Let s N, F and let v s v L. Then s exists and or h > 0, := π/h there holds s t = h s k+ A s,k t+h k + s, t, s, t v s v dv π = π dist s, B. The corollaries stated above or the remainder o the approximate sampling theorem are also valid or s,. Theorem Bernstein inequality For B p, p, > 0, there holds s L p s L p s N. Bernstein inequality or trigonometric polynomials I t = t n t = n k= n c ke ikt, a trigonometric polynomial o degree n, then t n B n, and t s n L n s t n L s N. Theorem Extended Bernstein inequality Let s N, F and suppose that v s v L L as a unction o v. Then, or any > 0, we have s L s L + dist s, B. The proos ollow rom the Boas-type ormulae above.

6 The Hilbert transorm Hilbert transorm or conjugate unction o L C, is deined by Cauchy principal value t u t := lim du = PV t u du, δ 0+ u π u u >δ It deines a bounded linear operator rom L into itsel, and [ ] v = i sgn v v a. e. I v s v L, then by Fourier inversion ormula, ] st [ = v i sgn viv s e ivt dv t. π Thus [ ] s = s ; i. e. derivation and taking Hilbert transorm are commutative operations. Since sinc v = / π or v π and = 0 otherwise, sinc t = cos πt πt = sin πt. πt Theorem Boas-type ormulae or the Hilbert transorm Let B, > 0 and h := π/. Then or s N, s t = h s à s,0 := s πs, s à s,k := s! πk s k k+ à s,k t + hk s j=0 j } j πk j! t. k 0. The proo is based on the sampling theorem or the Hilbert transorm, s 0 = k sinc s k. Proo Theorem Boas ormulae or the Hilbert transorm, extended vers. Let s N, F and let v s v L. Then s exists and or h > 0, := π/h, s t = h s k+ à s,k t + hk + s, t, where s, t π v s v dv = π dist s, B. First assume h =, i. e., = π. Let t := π Then Bπ dierence, i. e., ve itv dv. and so the bandlimited case applies to this s,π t = s,π t+ s,π t = s,π t, and we ind that or t = 0, s,π 0 = s 0 k+ à s,k k.

7 Proo, continued s,π 0 = s 0 s 0 = π k = π k+ Ã s,k k. v i sgn viv s dv ve ikv dv k Z. Inserting the last two equations into the irst one and interchanging summation and integration yields s,π 0 = π v [ i sgn viv s k+ Ã s,k e ikv ] dv. Proo, continued The oregoing ininite series is a Fourier series o a π-periodic unction and can be evaluated to be k+ Ã s,k e ikv = s+ v s v < π. s,π 0 = π v [ i sgn viv s s+[ v s ] ] dv v s v dv = π π dist s, B. [ v s ] is the π-periodic extension o v s rom π, π to. This is the desired inequality or = π, h = and t = 0. For the general case apply this estimate to the unction u t + hu. Applications Consider gt := / + t, t, Fourier transorm π/ exp v, Hilbert transorm gt = t/ + t. Extended sampling theorem or Hilbert transorm takes on concrete orm or g, t + t π + kπ π t k sin πt k + cosπt k πt k π v e v dv = + e > 0. Truncation error In practise one has to deal with inite sum, no ininite series. Leads to additional truncation error, T,N t = k N+ + kπ This yields or the truncation error T,N t γ + π γ π t k sin πt k + cosπt k πt k γ + π γ k N+ N k 3 u 3 du = γ + π γ N. or N γ t and some constant γ >.

8 Combined error Combining the aliasing error with truncation error we inally obtain t N + t clear + e + γ + π γ N k= N > 0; N γ t. Similarly, Boas-type theorem or derivative g takes the orm t π + t h + t } h πk + + [t + k + h] π π v e v dv = + e > 0. For the second order derivative g one obtains t 3 6t + t 3 8 π πk + k k h πk 3 + [t + k + h] π π v e v dv = + + e > 0. These are the aliasing errors or the reconstruction o derivatives o the Hilbert transorm in terms o Boas-type ormulae. In both cases the truncation errors can be handled in a similar ashion as above. Some Boas-type ormulae or bandlimited unctions and their Hilbert transorm 3 t = h 3 t = h t = π 3h t + h k+ 6 π k4 t = π h t + h t = h k+ πk t + h k, k 0 t + hk k+ k, [ π ] k t + h k, πk + t + hk +, ] k+ [ k + π k π k 3 t +h k. eerences I P. L. Butzer, P. J. S. G. Ferreira, J.. Higgins, G. Schmeisser, and. L. Stens. The sampling theorem, Poisson s summation ormula, general Parseval ormula, reproducing kernel ormula and the Paley-Wiener theorem or bandlimited signals - their interconnections. Applicable Analysis, 903-4:43 46, 0. P. L. Butzer, P. J. S. G. Ferreira, J.. Higgins, G. Schmeisser, and. L. Stens. The generalized Parseval decomposition ormula, the approximate sampling theorem, the approximate reproducing kernel ormula, Poisson s summation ormula and iemann s zeta unction; their interconnections or non-bandlimited unctions. to appear.

9 eerences II eerences III P. L. Butzer, P. J. S. G. Ferreira, G. Schmeisser, and. L. Stens. The summation ormulae o Euler-Maclaurin, Abel-Plana, Poisson, and their interconnections with the approximate sampling ormula o signal analysis. esults Math., 593-4: , 0. P. L. Butzer and A. Gessinger. The approximate sampling theorem, Poisson s sum ormula, a decomposition theorem or Parseval s equation and their interconnections. Ann. Numer. Math., 4-4:43 60, 997. P. L. Butzer, G. Schmeisser, and. L. Stens. Basic relations valid or the Bernstein space B p and their extensions to unctions rom larger spaces in terms o their distances rom B p. to appear. P. L. Butzer, G. Schmeisser, and. L. Stens. Shannon s sampling theorem or bandlimited signals and their Hilbert transorm, Boas-type ormulae or higher order derivatives the aliasing error involved by their extensions rom bandlimited to non-bandlimited signals. invited or publication in Entropy. P. L. Butzer and W. Splettstösser. A sampling theorem or duration limited unctions with error estimates. In. Control, 34:55 65, 977. P. L. Butzer, W. Splettstösser, and. L. Stens. The sampling theorem and linear prediction in signal analysis. Jber.d.Dt.Math.-Verein., 90: 70, 988. eerences IV G. Schmeisser. Numerical dierentiation inspired by a ormula o. P. Boas. J. Approximation Theory, 60:0, L. Stens. A uniied approach to sampling theorems or derivatives and Hilbert transorms. Signal Process., 5:39 5, 983.