NEW BOUNDS FOR TRUNCATION-TYPE ERRORS ON REGULAR SAMPLING EXPANSIONS

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1 NEW BOUNDS FOR TRUNCATION-TYPE ERRORS ON REGULAR SAMPLING EXPANSIONS Nikolaos D. Atreas Department of Mathematics, Aristotle University of Thessaloniki, 54006, Greece, Abstract We give some new estimates for the Truncation Error of sampling series of functions on regular sampling subspaces of L R. These estimates lower the well known Jagerman s bound on Shannon s sampling expansions. Mathematics Subject Classification: 4A7, 4A80, 65G99. Keywords: Sampling Expansion, Truncation Error, Regular Function.. Introduction Let U be a closed subspace of L R i.e. U is the space of all Lebesgue square integrable functions defined on R with the property that any f U has the following sampling expansion: f. = fns. n, x R, where the convergence of is in the L R sense and where S. is defined to be the sampling function of U, i.e. Sn = δ 0,n, δ 0,n is the Kronecker s delta. Clearly the collection {S. n, n Z} is a basis of U. For a survey of sampling theorems we propose [?], [?] or [?]. It is known see [?], [?] and [?] that the existence of a sampling function implies that U has a reproducing kernel Kx,. U, such that for every f U we have fx = f, Kx,., where.,. is the inner product of L R. Notice also that the L R-convergence of implies uniform convergence in the intervals where the kernel Kx, x is bounded. Moreover, for any σ > 0 we define U σ to be those closed sampling subspaces of L R, such that for any f U σ we can write: f. = fn/σsσ. n, where Sσ. is the sampling function of U σ. There is a variety of examples derived from : Example The sampling expansion of Shannon is given by the formula: Research supported by EU Project IST IMCOMP

2 fx = sin[σπx n] fn/σ σπx n, x R, 3 for f belonging in the Paley-Wiener space of πσ-bandlimited functions i.e. f satisfies fγ = 0 for γ > πσ, where f is the Fourier Transform of f. Example [?] We define a family U σ of subspaces of L R such that the multiresolution sampling formula is valid for any f U σ. Each subspace U σ is the closure of the linear span of an orthonormal set {σ / ϕσ. n, n Z}, where ϕ L R satisfies the following conditions: i ϕx cons. Bx x a, where a and Bx is a bounded l-periodic function with B0 = 0, ii the series ϕne inγ converges absolutely to a function which has no real zero on [ π, π]. In this case see [?] the sampling function Sx arises from ϕ via the relation: Ŝγ = ϕγ, γ R. ϕne inγ We define now the Truncation Error of sampling formulas: Definition Given a closed subspace U σ of L R and f U σ, h = σ and N < N N, N Z, the function: R N,N fx = fx N n=n fnhsh x n, x N h, N h 4 is called the Truncation Error of the sampling expansion. An extensive survey about Truncation Errors can be found in [?]. For simplicity se shall denote R N,N fx by R N fx. For the case of Shannon s sampling expansion, Jagerman in [?] estimated the following: R N fx sinh πx π [ ] K N Nh x / + L N Nh + x /, 5 where h = σ and where K N = h n>n fnh / and LN = h n< N fnh /. We proved in [?] a Jagermann-type error: R N fx cons. Bh x a / h a [ K N Nh x a / + ] L N Nh + x a /, 6 for f belonging in the multiresolution sampling spaces of Example. Notice that we assumed in [?] that Sx is an α-regular sampling function, that is: Sx cons. Bx x a, x, where a and Bx is a bounded l-periodic function with B0 = 0. Recall that U is said to be a regular sampling subspace of L R if it possesses a regular sampling function. In this work we lower the Jagerman-type errors 5 and 6. In Section we give certain new

3 estimates on the Truncation Error 4 for regular sampling functions. Indeed, in Proposition we state the main result: where R N,N fx cons. Bh x a / h a K N E N,h,ax + L N E N,h,a x, E N,h,a x = N + /h x a 3 N + h x a + / N + 3/h x a. In Section 3, Proposition 4, we prove that the above new estimate lowers the Jagerman-type bound 6. Especially for the Shannon sampling expansion 3 we get the following reduced Jagerman-type bound: R N fx sinh πx K N E N,h, x + L N E N,h, x see Example 3 below. π. New Truncation Error Bound In this section we give upper bounds for the Truncation Error 4 for functions belonging to regular sampling subspaces U σ. We assume that each subspace U σ has a reproducing kernel K σ.,. which satisfies the condition K σnh, nh < + h = σ. This condition implies that the sequence {fnh} is in l Z. In fact fnh = f, K σ., nh f σ., nh, K σ., nh = K K σ nh, nh f. As we shall see below, in order to estimate Truncation Errors we need upper bounds for the series of functions n>n n y a /, where y < N and a. This is done in Lemma. Lemma Let {β n } be a positive decreasing sequence and let lim n β n = 0. i If {β n } is strictly convex i.e. β n + β n+ > β n+ and if s = n= n+ β n, then: s [β /, β β /]. iiif {β n } is strictly convex and if the sequence {c n = β n β n+ } is also strictly convex, then: s [3β /4 β /4, β 3β /4 + β 3 /4]. Proof A detailed proof of the Lemma is given in [?]. /, Lemma Let S N,a y = n>n n y where a, N Z and y < N, then: a i ii S N,a y < a / E N,a y, S N,a y < N + y a+ E N, y, 3

4 where the function E N,a y is given by E N,a y = N + / y a 3 N + y a + / N + 3/ y a. Proof i Let f y z = z y a+, where y < N and z N +. For any x N +, there exists θ x x /, x: f y x f y x / = f yθ x /. 7 If z [N + /,, then f yz is negative and strictly increasing, so: and because of 7 we get: thus: f yx / < f yθ x < f yx f y x / f y x = f yθ x > f yx = a x y a > 0, x y a < a In 8 we set x = n, n > N, n Z and we have: S N,a y = n y a < a n>n n>n = a a n>n = a a x / y a x y a n / y a n y a n y a n y a µ>n µ+ µ y a.. 8 The sequence {µ y a+ } µ>n satisfies the conditions of Lemma ii, so: S N,a y < a a N + y a 3 4 N + y a + 4 N + 3 y a. ii If n > N > y, then: S N,a y < N + y a+ S N, y. Now we use the bound in Lemma i for S N, y, to get: S N,a y < N + y a+ E N, y. Remark Let h > 0, N Z and h x < N. By Lemma i, for y = h x we have: S N,a h x = n>n / n h x a < h a / a / E N,h,ax, 9 where E N,h,a x = N + /h x a 3 N + h x a + / N + 3/h x a. 0 4

5 Similarly for h x > N we have: S N,a h x = n<n / h x n a < h a / a / E N,h,a x. Proposition Let U σ be a regular sampling subspace of L R. If K m = h n>m fnh / and L k = h n<k fnh /, where f Uσ, h = σ and k, m Z, then for the Truncation Error we have: R N,N fx cons. Bh x a / h a K N E N,h,ax + L N E N,h,a x, where the function E N,h,a x is given in 0. Proof It is clear that R N,N fx = n>n fnhsh x n + n<n fnhsh x n, where x N h, N h. If n > N, then h x n = n h x, so if S x or S x is the first or the second term of the right hand side of the above equality, using the regularity condition for the sampling function S we get: S x cons. Bh x / fnh n>n n h x a n>n = cons. Bh x h / K N S N,ah x, where S N,a x is as in Lemma. Using 9 we obtain the following upper bound: S x cons. Bh x a / h a K N E N,h,ax. / The proof is similar for the term S x. Indeed, if n < N, then h x n = h x n and S x cons. Bh x h / L N S N,a h x cons. Bh x a / h a L N E N,h,a x use. We combine the upper bounds for S x and S x and we get. Proposition Let U σ be a regular sampling subspace of L R and let the sampling function of U σ satisfies the relation: Sh x n = cons. n Bh xgh x nh x n a, where Gx is bounded on R. If C = sup x R fnhgh x n fn + hgh x n + <, then for the Truncation Error we have: R N,N fx cons. Bh x h a C N + h x a + x N h a Proof R N,N fx = cons.bh x fnhgh n x n n h x a + fnhgh n x n h x n a n>n n<n. 5

6 = S x + S x, where S x or S x is the first or the second term of the above parenthesis. We define β n x := n h x a, where n > N and x N h, N h. We fix x and we observe that the sequence {β n x} is positive, decreasing and strictly convex, thus by Lemma i we have: n λ β λ x β N +x. 3 λ=n + We define Γ n h, x = fnhgh x n n Z and we apply the Abel Summation formula for the term S x to get: S x = cons.bh x k n n>n λ=n + λ+k β λ x[γ n h, x Γ n+ h, x], where k = 0 or k =, if N is odd or even respectively. By 3 we get: S x cons. Bh x Cβ N +x = cons. Bh x h a CN + h x a. Using similar arguments we can prove that S x cons. Bh x h a Cx N h a. Remark The Shannon sampling function S h x = sinh πx/πx satisfies the hypotheses of Proposition for Bx = sinπx, cons. = /π, Gx = and a =. Proposition 3 Let U σ be a regular sampling subspace of L R and let f U σ with the property fnh <, then for the Truncation Error we have: R N,N fx cons.h a Bh x fnh N n>n + h x a + fnh x N n<n h a. Proof It is an immediate consequence of 4 and the regularity of the sampling function. 3. Reducing Jagerman s Bound Let V σ be those closed subspaces of L R which have been defined in example for which the generator function ϕ and the sampling function S coincide. Then we have: Corollary Let V σ be as above, then for any f V σ the Truncation Error satisfies the relation: R N,N fx cons. Bh x a / h a K N E N,h,ax + L N E N,h,a x 4 where the function E N,h,a x is defined in 0 and where K N, L N are given in Proposition. Proof Sh x is the sampling function of V σ. Since ϕx = Sx where ϕ is α-regular, we have: Sh x cons. Bh x h a x a. We apply and we have the result. Remark 3 If N = N and N = N, N then by 4 we deduce: R N fx cons. Bh x a / h a K N E N,h,a x + L N E N,h,a x. 5 6

7 Now we shall compare 5 with the estimate 6 that we have given in [?]. Proposition 4 The Truncation bound 5 is less than the Truncation bound 6. Proof It suffices to show that for any x < Nh there holds E N,h,a ±x < Nh ± x a+/. We observe that: E N,h,a ±x < N + /h ± x a N + h ± x a, so it suffices to show that N + /h ± x a N + h ± x a < Nh ± x a. The above inequality is equivalent to + 0.5γ a + γ a < for γ = h Nh ± x > 0, which is obviously valid for any γ > 0. Example 3 Shannon V σ are the Paley-Wiener spaces of πσ-bandlimited functions f with generator function ϕx = sinπx/πx. Since ϕx = Sx inequality 5 becomes: R N fx sinh πx K N E N,h, x + L N E N,h, x, π and Proposition 4 implies that the above estimate lowers Jagerman s bound. Proposition 5 If the sampling function of U σ is bounded in a neighborhood of zero and has bounded support, then the Truncation Error is zero. Proof We suppose that Sx x a, where a. We use Lemma ii to get: R N fx h a K N /N + h x a E N, h x + L N /N + h + x a E N, h x. a Since h < N + h ± x, the product [h/n + h ± x] a E N, ±h h x is equal to N+h±x multiplied by a bounded function which of course converges to zero when a, thus the Truncation Error is zero. Example 4 Haar The spaces V σ consist of piecewise constant functions with possible jumps at the points n/σ. We have ϕx = Sx = χ [0, x, where χ E x is the characteristic function on the set E, thus by Proposition 5 the Truncation error is zero. Example 5 Lemarie of order Sx = x when x and Sx = 0 elsewhere, so the Truncation Error is zero see [?] for Lemarie s sampling expansion. References [] Atreas N., Karanikas C., Truncation Error on Wavelet Sampling Expansions, J. Comp. Anal. and Applications,,, 000. [] Atreas N., Karanikas C., Gibbs Phenomenon on Sampling Series based on Shannon s and Meyer s Wavelet Analysis, J. Fourier Anal. and Applications 5, 6, , 999. [3] Butzer P., L., Stens R.L. and Splettstober W., The Sampling Theorem and Linear Prediction in Signal Analysis, Jber. D. Dt. Math.-Verein, 90, -70,

8 [4] Jagerman D., Bounds for Truncation Error of the Sampling Expansion, SIAM J. Appl. Math., 4, 4, 74-73, 966. [5] Jerri A., Error Analysis in Applications of Generalizations of the Sampling Theorem, Advanced Topics in Shannon Sampling and Interpolation Theory, Marks II R. J. Ed., Springer-Verlag, New York, 993. [6] Nashed Z., Walter G. G., General sampling theorems for functions in reproducing kernel Hilbert spaces, Math. Control, Signal and Systems, 4, , 99. [7] Walter G.G., Wavelets and Other Orthogonal Systems with Applications, CRC Press, 994. [8] Young R. M., An Introduction to non-harmonic Fourier Series, Academic Press Inc., 980. [9] Zayed A. I., Advances in Shannon s Sampling Theory, CRC Press Inc.,