NEW BOUNDS FOR TRUNCATION-TYPE ERRORS ON REGULAR SAMPLING EXPANSIONS Nikolaos D. Atreas Department of Mathematics, Aristotle University of Thessaloniki, 54006, Greece, e-mail:natreas@auth.gr 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-000-606 IMCOMP
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
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
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
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
= 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
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, 575-588, 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, 988. 7
[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, 363-390, 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., 993. 8