On approximation properties of Kantorovich-type sampling operators

Transcription

1 On approximation properties of Kantorovich-type sampling operators Olga Orlova, Gert Tamberg 1 Department of Mathematics, Tallinn University of Technology, ESTONIA Modern Time-Frequency Analysis Strobl 2014 AUSTRIA June 5, This is a joint work with Andi Kivinukk (Tallinn University) This research was supported by the Estonian Science Foundation, grant 9383 and by the Estonian Min. of Educ. and Research, project SF s09. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

2 Sampling theory The aim of this talk is to study approximation properties of a generalized Shannon sampling operators (S w f )(t) := k= f ( k )s(wt k) w with a bandlimited kernel s, which is a Fourier transform of a certain even window function λ C [ 1,1], λ(0) = 1, λ(u) = 0 ( u 1), i.e. s(t) := s(λ; t) := 1 0 λ(u) cos(πtu) du. We give the conditions for the kernel s, which allow us to have the estimate of order of approximation via modulus of smoothness in form: f S r wf M r ω 2r (f ; 1 w ). O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

3 Preliminary results Def The Bernstein class B p σ for σ 0 and 1 p consists of those bounded functions f L p (R) which can be extended to an entire function f (z) (z C)of exponential type σ, i. e., f (z) e σ y f C (z = x + iy C). The class B p σ is a Banach space if one takes the norm of L p (R). Lemma We have B 1 σ B p σ B r σ B σ, 1 p r, B p α B p β, 0 α β < O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

4 Whittaker-Kotelnikov-Shannon sampling series Whittaker-Kotelnikov-Shannon sampling theorem Theorem (Whittaker-Kotelnikov-Shannon) If g B p πw, 1 p <, or g B σ g(t) = k= g( k W for some 0 σ < πw, then ) sinc(wt k) =: (Ssinc W g)(t), (1) the series being uniformly convergent on each compact subset of R. The Bernstein class B p πw forms the set of fixed points of the sampling operators (1). Now it is natural to ask what happens in WKS Theorem if instead of g B p πw we consider uniformly continuous and bounded functions f C(R)? M. Theis [Theis 19] showed that the equality in (1) does not hold for any g C(R). O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

5 Whittaker-Kotelnikov-Shannon sampling series Whittaker-Kotelnikov-Shannon sampling theorem Theorem (Whittaker-Kotelnikov-Shannon) If g B p πw, 1 p <, or g B σ g(t) = k= g( k W for some 0 σ < πw, then ) sinc(wt k) =: (Ssinc W g)(t), (1) the series being uniformly convergent on each compact subset of R. The Bernstein class B p πw forms the set of fixed points of the sampling operators (1). Now it is natural to ask what happens in WKS Theorem if instead of g B p πw we consider uniformly continuous and bounded functions f C(R)? M. Theis [Theis 19] showed that the equality in (1) does not hold for any g C(R). O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

6 Generalized sampling series Generalized Shannon sampling series We get uniform convergence f S w f C 0 (w ) for any f C(R) if, instead of the cardinal sine, we use different kernel functions s L 1 (R) C(R) (sinc L 1 (R)), k Z s(u k) = 1, u R. This approach gives us the generalized sampling series for t R, w > 0 (S w f )(t) := f ( k )s(wt k). (2) w k= Whereas a generalized sampling series with a particular kernel were already considered by M. Theis in 1919, a study of those series for arbitrary kernel functions was initiated at RWTH Aachen University (P. L. Butzer et al) and widely studied there since 1977 and in Perugia since O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

7 Generalized sampling series Generalized Shannon sampling series We get uniform convergence f S w f C 0 (w ) for any f C(R) if, instead of the cardinal sine, we use different kernel functions s L 1 (R) C(R) (sinc L 1 (R)), k Z s(u k) = 1, u R. This approach gives us the generalized sampling series for t R, w > 0 (S w f )(t) := f ( k )s(wt k). (2) w k= Whereas a generalized sampling series with a particular kernel were already considered by M. Theis in 1919, a study of those series for arbitrary kernel functions was initiated at RWTH Aachen University (P. L. Butzer et al) and widely studied there since 1977 and in Perugia since O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

8 Generalized sampling series Take a function f and compute the corresponding sampling series (S 2 f )(t) := k= and the approximation error S 2 f f. f ( ) k s(2t k), (W = 2) 2 O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

9 Generalized sampling series Now we compute the sampling series, taking W = 6, (S 6 f )(t) := k= and the approximation error S 6 f f. f ( ) k s(6t k) 6 O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

10 Generalized sampling series At last we take W = 16, getting (S 16 f )(t) := k= and the approximation error S 16 f f. f ( ) k s(16t k) 16 O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

11 Sampling operators in L p case Since in practice signals are however often discontinuous, this talk is also concerned with the convergence of S w f to f in the L p (R)-norm for 1 p <. The sampling series S w f of an arbitrary L p -function f may be divergent. Take f L p (R), f (k/w) = 1 for a fixed w > 0, then (S w f )(t) 1 L p (R). Also operators S w f do not always provide good results in applications, e.g. in Signal Processing. Generalized sampling operators depend on exact values f (k/w), whilst in practice we often have to deal with the so called "jitter errors" i.e. the impossibility to determine the exact values at the nodes. Replacing the exact value f (k/w) with an average of f around k/w might lead to smaller errors and therefore be useful in applications. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

12 Sampling operators in L p case Z. Burinska, K. Runovski and H. J. Schmeisser introduced in [BRS 06] for 0 < p additional shifts λ: (S w;λ f )(t) := k= f ( k w + λ)s(w(t λ) k) and estimated the order of approximation via modulus of smoothness. C. Bardaro, P. L. Butzer, R. Stens and G. Vinti in [BBSV 06] introduced a suitable subspace of L p (R). They estimated the order of approximation for the classical (Whittaker-Kotel nikov-shannon) operator for 1 < p < and for sampling operators with time-limited kernels for 1 p < in [Butzer, Stens 08] and [BBSV 10] via averaged modulus of smoothness. We used the same approach and estimated the order of approximation for sampling operators with band-limited kernels for 1 < p < in [T 13] via the averaged modulus of smoothness and for 1 p < in [Kivinukk,T 14] and [T 14] via the classical modulus of smoothness. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

13 Sampling operators in L p case Z. Burinska, K. Runovski and H. J. Schmeisser introduced in [BRS 06] for 0 < p additional shifts λ: (S w;λ f )(t) := k= f ( k w + λ)s(w(t λ) k) and estimated the order of approximation via modulus of smoothness. C. Bardaro, P. L. Butzer, R. Stens and G. Vinti in [BBSV 06] introduced a suitable subspace of L p (R). They estimated the order of approximation for the classical (Whittaker-Kotel nikov-shannon) operator for 1 < p < and for sampling operators with time-limited kernels for 1 p < in [Butzer, Stens 08] and [BBSV 10] via averaged modulus of smoothness. We used the same approach and estimated the order of approximation for sampling operators with band-limited kernels for 1 < p < in [T 13] via the averaged modulus of smoothness and for 1 p < in [Kivinukk,T 14] and [T 14] via the classical modulus of smoothness. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

14 Sampling operators in L p case Z. Burinska, K. Runovski and H. J. Schmeisser introduced in [BRS 06] for 0 < p additional shifts λ: (S w;λ f )(t) := k= f ( k w + λ)s(w(t λ) k) and estimated the order of approximation via modulus of smoothness. C. Bardaro, P. L. Butzer, R. Stens and G. Vinti in [BBSV 06] introduced a suitable subspace of L p (R). They estimated the order of approximation for the classical (Whittaker-Kotel nikov-shannon) operator for 1 < p < and for sampling operators with time-limited kernels for 1 p < in [Butzer, Stens 08] and [BBSV 10] via averaged modulus of smoothness. We used the same approach and estimated the order of approximation for sampling operators with band-limited kernels for 1 < p < in [T 13] via the averaged modulus of smoothness and for 1 p < in [Kivinukk,T 14] and [T 14] via the classical modulus of smoothness. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

15 Sampling operators in L p case C. Bardaro, P. L. Butzer, R. Stens and G. Vinti in [BBSV 07] introduced Kantorovich-type sampling operators (k+1)/w (Sw K f )(t) = w f (u) du s(wt k) k= and proved the convergence for 1 p < case. We estimated the order of approximation for Kantorovich-type sampling operators (S I w,nf )(t) = k= nw k/w (2nk+1)/2nw (2nk 1)/2nw f (u) du s(wt k) with band-limited kernels for 1 p in [Orlova,T 14] via the classical modulus of smoothness. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

16 Sampling operators in L p case C. Bardaro, P. L. Butzer, R. Stens and G. Vinti in [BBSV 07] introduced Kantorovich-type sampling operators (k+1)/w (Sw K f )(t) = w f (u) du s(wt k) k= and proved the convergence for 1 p < case. We estimated the order of approximation for Kantorovich-type sampling operators (S I w,nf )(t) = k= nw k/w (2nk+1)/2nw (2nk 1)/2nw f (u) du s(wt k) with band-limited kernels for 1 p in [Orlova,T 14] via the classical modulus of smoothness. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

17 Singular integrals Shannon Sampling Series Singular integrals In order to proceed with Kantorovich-type sampling operators we need to define the singular integral. If χ L 1 (R) is such that χ(u)du = 1, then the convolution integral of f with χ ρ (u) := ρχ(ρu), namely (I χ ρ f )(x) := (f χ ρ )(x) = f (u)ρχ(ρ(x u))du (x R) (3) is called the singular (convolution) integral with kernel χ. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

18 Kantorovich-type sampling operators Kantorovich-type sampling operators As a final step we replace the exact value f (k/w) in (2) with an average of f around k/w, using the singular integral Inwf χ (n N) with k/w as an argument. Thus for f L p (R) (1 p ) the Kantorovich-type sampling operators are given by (t R; w > 0; n N) (Sw,nf K )(t) := f (u)nwχ(nw( k w u))du s(wt k) (4) k= with kernels s, χ L 1 (R), χ(u)du = 1, k Z s(u k) = 1 (u R). Remark. Bardaro, Butzer, Stens and Vinti considered the case χ = χ [0,1], n = 1 only. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

19 Bandlimited kernels Shannon Sampling Series Bandlimited kernels In this talk we consider an even band-limited kernel s, i.e. s B 1 π, defined by an even window function λ C [ 1,1], λ(0) = 1, λ(u) = 0 ( u 1) by the equality s(t) := s(λ; t) := 1 0 λ(u) cos(πtu) du. (5) In fact, this kernel is the Fourier transform of λ L 1 (R), π s(t) = 2 λ (πt). (6) Lemma (Butzer, Splettstößer, Stens 88) If s B 1 π, then for f C(R) S s wf f C I s wf f C. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

20 Many kernels can be defined by Bandlimited kernels s(t) := 1 0 λ(u) cos(πtu) du, e.g. 1) λ(u) = 1 defines the sinc function; 2) λ(u) = 1 u defines the Fejér kernel s F (t) = 1 2 sinc 2 t 2 (cf. [13]); 3) λ j (u) := cos π(j + 1/2)u, j = 0, 1, 2,... defines the Rogosinski-type kernel (see [7]) in the form r j (t) := 1 2 ( sinc(t + j ) + sinc(t j 1 2 ) ) 4) λ H (u) := cos 2 πu 2 = 1 2 (1 + cos πu) defines the Hann kernel (see [8]) (7) s H (t) := 1 sinc t 2 1 t 2 ; (8) O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

21 Bandlimited kernels 5) the general cosine window m λ C,a (u) := a k cos kπu (9) k=0 defines the Blackman-Harris kernel (see [9]) s C,a (t) := 1 2 m k=0 ( ) a k sinc(t k) + sinc(t + k) (10) provided (here and following x is the largest integer less than or equal to x R) m 2 m+1 2 a 2k = a 2k 1 = 1 2. (11) k=0 k=1 We get Hann kernel (8) if we take m = 1 in (10). O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

22 Bandlimited kernels 6) powers of the Hann window (see [6], formula(25a)) ( λ H,m (u) := cos m πu ) (12) 2 = 1 m ( ) m 2 m cos ((k m ) k 2 )πu, (13) k=0 give a general Hann kernel in the form s H,m (t) = 2 m Γ(1 + m) Γ(1 + m 2 t)γ(1 + m 2 + t). (14) Comparing the window function λ H,m in (13) and the general cosine window λ C,a in (9) we see that the general Hann kernel in case of m = 2n (n N) is a special case of the Blackman-Harris kernel. Indeed, s H,2n = s C,a, where the parameter vector a R n+1 has components a0 = 1 ( 2n ) 2 2n n and a k = 1 ( 2n 2 n k) for k = 1, 2,..., n. 2n 1 O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

23 Bandlimited kernels 7) the general Rogosinski-type window m m λ R,a (u) := a j λ j (u) = a j cos π(j + 1/2)u (15) j=0 defines the general Rogosinski-type kernel r a (t) := 1 m ( a j sinc(t 2j + 1 ) + sinc(t + 2j + 1 ) ) provided j=0 j=0 (16) m a j = 1. (17) j=0 Comparing the window function λ H,m in (13) and the general Rogosinski-type window λ R,a in (15) we see that the general Hann kernel in case of m = 2n + 1 (n N 0 ) is a special case of the general Rogosinski-type kernel. Indeed, s H,2n+1 = s R,a, where the parameter vector a R n+1 has components aj = 1 ( 2n+1 ) 2 2n n j for j = 0, 1,..., n. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

24 Kernels for singular integrals Kernels for singular integrals In this talk we consider for the singular integrals an even kernel χ L 1 (R) with absolute moment m 0 (χ) := sup u R χ(u k) <, k Z defined by an even window function µ, µ(0) = 1 by the equality χ(t) := χ(µ; t) := 0 µ(u) cos(πtu) du. (18) In fact, this kernel is also a kernel for generalized sampling operators, when we additionally require µ(2k) = 0, k N. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

25 Kernels for singular integrals Many kernels can be defined by χ(t) := χ(µ; t) := 0 µ(u) cos(πtu) du e.g. ( 1) µ(u) = sinc k u ) (k N) defines the B-spline kernel B k 1, in 2 particular case k = 1 we get the indicator function χ [ 1/2,1/2] ; 2) µ a (u) = m 1 ( a k sinc k+1 u ) defines a spline kernel of degree m if k=0 2 ak = 1; 3) µ(u) = e π( u 2) 2 defines a Gauss-Weierstraß kernel χ GW (t) = e πt2. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

26 Operator norms Norms of the Kantorovich-type sampling operators Norms of the Kantorovich-type sampling operators Theorem For Kantorovich-type sampling operator S K w,n = S s w(i χ nw) we have for 1 p an estimate of the operator norm (1/p + 1/q = 1) S K w,n p p ( ) 1 ( ) 1 p q n s 1 m 0 (χ) χ 1 m 0 (s) Remark. We see that in the case of generalized sampling operators S w (i.e. n ) this estimate is not bounded for 1 p <. Remark. Take χ = χ [ 1/2,1/2]. We proved that in this case for L 1 (R) we have S K w,n 1 1 = n s 1 and for C(R) we have S K w,n C C = m 0 (s). O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

27 Nikolskii s inequality Operator norms Nikolskii s inequality Theorem (Nikolskii s inequality) Let 1 p. Then, for every s B p σ, s p sup u R { k= s(u k) p } 1/p (1 + σ) s p. Our kernels s B 1 π. By Nikolskii s inequality we have which gives the estimate s 1 m 0 (s) = S w, ( Sw,n K p p S w n m 0 (χ) ) 1 p 1 q χ 1 O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

28 Order of approximation Theorem If we have for sampling operator S s w and for the singular integral S s wf f p M 1 ω k (f ; 1 w ) p I χ nwf f p M 2 ω l (f ; 1 w ) p, then for Kantorovich-type sampling operator S K w,n = S s w(i χ nw) where r := min{k, l}. S K w,nf f p M 3 ω r (f ; 1 w ) p, O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

29 Order of approximation Corollary For Kantorovich-type sampling operator S K w,n = S s w(i χ nw), where s is Rogosinski-type, Hann, Blackman-Harris or B-spline kernel and χ is a indicator function or Gauss-Weierstraß kernel we have S K w,nf f p Mω 2 (f ; 1 w ) p. Indeed, for Rogosinski-type, Hann, Blackman-Harris and B-spline kernels we have the estimates of order of approximation for corresponding generalized sampling operators via modulus of smoothness order 2. For singular integrals with indicator function and Gauss-Weierstraß kernels we have the estimates of the order of approximation via modulus of smoothness order 2. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

30 Order of approximation The main theorem Theorem Let Kantorovich-type sampling operator S K w,n = S s w(i χ nw) (w > 0, n N) be defined by the kernel s B 1 π with window λ and by the kernel χ L 1 with window µ. If for some r N ( u ) ν n (u) := λ(u)µ = 1 n Then for f L p (R) (1 p ) c j u 2j, j=r c j. (19) j=r S K w,nf f p M r ω 2r (f ; 1 w ) p. (20) The constants M r are independent of f and w. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

31 Idea of the proof Order of approximation The main theorem Lemma (cf. Butzer, Stens 85) If g Bαπw p and s Bβπ 1 (α + β = 2), then S s wg = I s wg. Take g B p πw/n, then Iχ nwg B p πw and for S K w,n = S s w(i χ nw) S K w,nf f p S K w,n p p f g p + I s w(i χ nwg) g p + g f p. I s w(i χ nwg) = s w χ nw g = (s χ n ) w g = ϕ w g = I ϕ wg = S ϕ wg where ϕ B 1 π is defined ϕ(t) := 1 0 ν n (u) cos πut du = 1 0 ( u ) λ(u)µ cos πut du. n O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

32 The space Λ p Order of approximation The main theorem Definition (Bardaro, Butzer, Stens, Vinti 2006) Let Σ := (x j ) j Z R be an admissible partition of R, i.e. 0 < inf j sup j <, j := x j x j 1 and let the discrete j Z j Z l p (Σ)-seminorm of a sequence of function values f Σ on Σ of a function f : R C be defined for 1 p < by f l p (Σ) := f (x j ) p j j Z The space Λ p for 1 p < is defined by Λ p := {f M(R); f l p (Σ) < for each admissible sequence Σ}. 1/p. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

33 Order of approximation The main theorem Denote for 1 p < X p (R) := Λ p and X (R) := C(R). B p σ X p (R). Theorem (Kivinukk,T 14) Let sampling operator S r w (w > 0) be defined by the kernel s r B 1 π with λ = λ r and for some r N let λ r (u) := 1 c j u 2j, j=r c j. (21) j=r Then for f X p (R) (1 p ) S r W f f p M r ω 2r (f ; 1 w ) p. (22) The constants M r are independent of f and w. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

34 Order of approximation The main theorem We have for g B p πw/n S K w,ng g p = S ϕ wg g p M r ω 2r (g; 1 w ) p M r ω 2r (f ; 1 w ) p + M r ω 2r (g f ; 1 w ) p M r ω 2r (f ; 1 w ) p + M r 2 2r g f p. Theorem (Jackson-type theorem) Given f C(R) or f L p (R) (1 p < ). Then there exists a g σ B p σ (1 p ) and a constant C k > 0 (depending only on k N) such that f g σ p C k ω k (f, 1 σ ) p. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

35 Examples Order of approximation Examples Theorem Let C I w,a = S s w(i χ nw) be a Kantorovich-type sampling operator whith s = s C,a (a R m+1 ) and χ = χ [ 1/2,1/2], then for f L p (R) (1 p ) Moreover, if there holds C I w,a,nf f p M a,2 ω 2 (f ; 1 w ) p. m a k k 2 1 = (2 2 1)2 2 n 2, k=1 then C I w,a,nf f p M a,4 ω 4 (f ; 1 w ) p. O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

36 Order of approximation Examples The Bibliographic References I Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals. J. Math. Anal. Appl. 316, (2006) Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Prediction by samples from the past with error estimates covering discontinious signals. IEEE Transactions on Information Theory 56, (2010) Burinska, Z., Runovski, K., Schmeisser, H.J.: On the approximation by generalized sampling series in L p -metrics. Sampling Theory in Signal and Image Processing 5, (2006) O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

37 Order of approximation Examples The Bibliographic References II Butzer, P.L., Splettstößer, W., Stens, R.L.: The sampling theorems and linear prediction in signal analysis. Jahresber. Deutsch. Math-Verein 90, 1 70 (1988) Butzer, P.L., Stens, R.L.: Reconstruction of signals in L p (R)-space by generalized sampling series based on linear combinations of B-splines. Integral Transforms Spec. Funct. 19, (2008) Harris, F.J.: On the use of windows for harmonic analysis. Proc. of the IEEE 66, (1978) Kivinukk, A., Tamberg, G.: On sampling series based on some combinations of sinc functions. Proc. of the Estonian Academy of Sciences. Physics Mathematics 51, (2002) O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

38 Order of approximation Examples The Bibliographic References III Kivinukk, A., Tamberg, G.: On sampling operators defined by the Hann window and some of their extensions. Sampling Theory in Signal and Image Processing 2, (2003) Kivinukk, A., Tamberg, G.: On Blackman-Harris windows for Shannon sampling series. Sampling Theory in Signal and Image Processing 6, (2007) Kivinukk, A., Tamberg, G.: Interpolating generalized Shannon sampling operators, their norms and approximation properties. Sampling Theory in Signal and Image Processing 8, (2009) O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

39 Order of approximation Examples The Bibliographic References IV Kivinukk, A., Tamberg, G.: On window methods in generalized Shannon sampling operators. In: New Perspectives on Approximation and Sampling Theory- Festschrift in honor of Paul Butzer s 85th birthday. Birkhauser Verlag (2014 (to appear)). In press Tamberg, G.: Approximation error of generalized shannon sampling operators with bandlimited kernels in terms of an averaged modulus of smoothness. Dolomites Research Notes on Approximation 6, (2013) Theis, M.: Über eine interpolationsformel von de la Vallee-Poussin. Math. Z. 3, (1919) O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

40 Order of approximation Examples Thank you! O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34

41 Order of approximation Examples [4],[13],[1],[2],[11], [5], [12],[10], [3] O. Orlova, G. Tamberg (TUT) Kantorovich-type Sampling Operators Strobl / 34