Multichannel Boxcar Deconvolution with Growing Number of Channels

We consider the problem of estimating the unknown response function in the multichannel deconvolution model with a boxcar-like kernel which is of particular interest in signal processing. It is known that, when the number of channels is finite, the precision of reconstruction of the response function increases as the number of channels $M$ grow (even when the total number of observations $n$ for all channels $M$ remains constant) and this requires that the parameter of the channels form a Badly Approximable $M$-tuple. Recent advances in data collection and recording techniques made it of urgent interest to study the case when the number of channels $M=M_n$ grow with the total number of observations $n$. However, in real-life situations, the number of channels $M = M_n$ usually refers to the number of physical devices and, consequently, may grow to infinity only at a slow rate as $n \rightarrow \infty$. When $M=M_n$ grows slowly as $n$ increases, we develop a procedure for the construction of a Badly Approximable $M$-tuple on a specified interval, of a non-asymptotic length, together with a lower bound associated with this $M$-tuple, which explicitly shows its dependence on $M$ as $M$ is growing. This result is further used for the evaluation of the $L^2$-risk of the suggested adaptive wavelet thresholding estimator of the unknown response function and, furthermore, for the choice of the optimal number of channels $M$ which minimizes the $L^2$-risk.

In signal processing, this model is referred to as a multichannel deconvolution model, where M is the number of channels and N is the number of observations per channel, so that n = M N is the total number of observations.We assume that the measurements t i in each channel are equispaced but the observer can choose the number of channels M and the points u l , l = 1, . . ., M , in (1.1) prior to the experiment as a part of experimental design.In order to be able to access convergence rates depending on the number of channels M and the choice of points u l , l = 1, 2, . . ., M , we shall further assume that the total number of observations n is fixed and very large (i.e., n → ∞).The objective is to choose M and u l , l = 1, 2, . . ., M , which ensure the construction of an estimator of the response function f with the highest possible convergence rates in terms of n.
Note that standard deconvolution (i.e., when a = b) with the boxcar kernel (i.e., when γ(u) = 1/u, for some fixed u > 0) is a common model in many areas of signal and image processing which include, for instance, LIDAR remote sensing and reconstruction of blurred images.LIDAR is a lazer device which emits pulses, reflections of which are gathered by a telescope aligned with the lazer, see, e.g., Park et al. (1997) and Harsdorf & Reuter (2000).The return signal is used to determine distance and the position of the reflecting material.However, if the system response function of the LIDAR is longer than the time resolution interval, then the measured LIDAR signal is blurred and the effective accuracy of the LIDAR decreases.This loss of precision can be corrected by deconvolution.In practice, measured LIDAR signals are corrupted by additional noise which renders direct deconvolution impossible.If M ≥ 2, then we talk about a multichannel deconvolution model with blurring functions g l (t) = g(u l , t).
Although standard deconvolution models are traditionally solved using the Fourier transform or the Fourier series, if the corresponding blurring function g(•) is a boxcar-like kernel, implementation of the standard Fourier series based technique is impossible.This happens when the Fourier transform of g(•) has real zeros, e.g., when g(•) is the boxcar kernel g(x) = (2u) −1 I(|x| ≤ u), for some fixed u > 0. When M = 1, Johnstone et al. (2004) and Johnstone & Raimondo (2004) managed to circumvent this obstacle by considering a boxcar kernel g(•) with irrational scale.Their method is based on the fact that the Fourier coefficients of the boxcar kernel do not vanish at frequencies (πku) when u is a Badly Approximable (BA) number.An irrational number u is BA if the terms a n = a n (u) of its continued fraction expansion [a 0 ; a 1 , a 2 . ..],where a 0 is an integer and a 1 , a 2 , . . . is an infinite sequence of positive integers, are bounded, i.e., sup n a n (u) < ∞.This notion is related to the fact that a BA number cannot be approximated well by a rational number which leads to the fact that f can be recovered reasonably well.Since standard deconvolution is a particular example of linear statistical ill-posed inverse problems in the sense of Hadamard, i.e., the inversion does not depend continuously on the observed data, Johnstone & Raimondo (2004) used number theory to prove that the degree of ill-posedness in boxcar deconvolution is ν = 3/2.Roughly speaking, the degree of ill-posedness specifies how much the error in the right-hand side of the equation is amplified in the solution.For example, if f belongs to a space with a smoothness index s > 0 and the degree of ill-posedness is ν > 0, then, the quadratic risk of the best possible estimator of the response function f is of the order O n − 2s 2s+2ν+1 .De Canditiis & Pensky (2004Pensky ( , 2006)), following mathematical ideas of Casey & Walnut (1994), extended the results of Johnstone et al. (2004) and Johnstone & Raimondo (2004), and showed that if M is finite, M ≥ 2, one of the u l 's is a BA number, and u 1 , u 2 , . . ., u M is a BA M -tuple, then the degree of ill-posedness is ν = 1+ 1/(2M ).The notion of a BA M -tuple refers to a collection of M irrational numbers which are difficult to approximate simultaneosly by fractions with the same denominator.It will be discussed in depth in Section 2. Therefore, in the case of M channels, the estimation problem requires a construction of a BA M -tuple which has been accomplished by the number theory community (it is described in, e.g., Schmidt (1969Schmidt ( , 1980))).
Recent advances in data collection and recording techniques made it of urgent interest to study the case when the number of channels M = M n grow with the total number of observations n.It turns out that when the number of channels M = M n grows fast as the total number of observations n increases, one does not need to make a special choice of the points u l , l = 1, 2 . . ., M , and it is sufficient to take them to be equidistant.Indeed, Pensky & Sapatinas (2010) considered the discrete multichannel deconvolution model (1.1) as observations on the continuous functional deconvolution model where z(u, t) is assumed to be a two-dimensional Gaussian white noise, i.e., a generalized twodimensional Gaussian field with covariance function  (1.4) assumes that one can measure y(u, t) at any u ∈ U and t ∈ T and that n −1/2 marks the precision of these observations.Nevertheless, this does not happen in real-life situations where one observes y(u, t) only at the points u l ∈ U, t i = (i − 1)/N for l = 1, 2, . . ., M and i = 1, 2, . . ., N .Pensky & Sapatinas (2010) showed that the degree of illposedness in the continuous deconvolution model (1.4) is ν = 1 and that it can be attained in the discrete deconvolution model (1.1) if M = M n ≥ c 0 n 1/3 for some constant c 0 > 0, independent of n.Indeed, in this case, one does not need to employ BA numbers or BA M -tuples: it is sufficient to observe the discrete deconvolution model (1.1) at equidistant points u l = a + l(b − a)/M , l = 1, 2, . . ., M .This set up provides the "best possible" minimax convergence rates (under the L 2 -risk and over a wide range of Besov balls) in the model.
However, in real-life situations, the number of channels M usually refers to the number of physical devices and, consequently, cannot be very big.Therefore, M = M n ≥ c 0 n 1/3 may be impossible although it is natural to assume that M = M n may grow to infinity at a slower rate as n → ∞.Unfortunately, the theoretical results obtained by De Canditiis & Pensky (2006) cannot be blindly applied to accommodate the case when M = M n → ∞ as n → ∞.This is due to the fact that, to the best of our knowledge, so far no one have studied the construction of a BA M -tuple of a growing length on a specified interval, of a non-asymptotic length, of the real line.Therefore, this generalization requires non-trivial results in number theory.
Our aim is to investigate the situation when M = M n grows slowly with n and to derive necessary new results in number theory in order to devise a technique which allows to approach minimax convergence rates (under the L 2 -risk and over a wide range of Besov balls) in the continuous model (1.4) with a factor which grows slower than any power of n.This situation seems to be of a particular interest nowadays since data recording equipment is getting cheaper and cheaper while overall volumes of data is growing very fast.
When M = M n grows slowly as n increases, we develop a procedure for the construction of a BA M -tuple on a specified interval, of a non-asymptotic length, together with a lower bound associated with this M -tuple, which explicitly shows its dependence on M as M is growing.This result is further used for evaluation of the L 2 -risk of the suggested adaptive wavelet thresholding estimator of the unknown response function and, furthermore, for the choice of the optimal number of channels M which minimizes the L 2 -risk.
The theoretical results that we have obtained provide a cross-area between number theory, statistics and signal processing.We hope to alert the number theory community to a new problem of constructing a BA M -tuple on a specified interval, of a non-asymptotic length, of the real line, as M is growing.On the other hand, we believe that our findings will also be of interest to researchers in statistics and signal processing.
The rest of the paper is organized as follows.Section 2 provides some number theory background which is required for understanding the material presented in subsequent sections.Section 3 briefly reviews the adaptive wavelet thresholding estimator introduced in Pensky & Sapatinas (2009).Section 4 explains the relationship between the L 2 -risk of the estimator obtained in Section 3 and the theory of Diophantine approximation, thus, motivating the derivation of the new results in number theory obtained in Section 5.In particular, the objective of Section 5 is the construction of a BA M -tuple on a specified interval when M = M n → ∞ as n → 0 and the development of related asymptotic bounds which are necessary in order to choose an optimal value of M = M n in this case.Section 6 provides the asymptotic upper bounds for the L 2 -risk of the adaptive wavelet thresholding estimator constructed in Section 3 when M = M n is a slowly growing function of n.We conclude in Section 7 with a brief discussion while Section 8 contains the proofs of the theoretical results obtained in earlier sections.

Background Results in Number Theory
The theory of Diophantine approximation is an important branch of number theory (see, e.g., Edixhoven & Evertse (1993), Lang (1966), Masser et al. (2003) and Schmidt (1980Schmidt ( , 1991))).One important topic of the above theory is the simultaneous approximation of linear forms, which was pursued as early as mid-19th century by Dirichlet and later studied by a number of profound researchers in the field.In particular, it is known that for any real numbers β 1 , β 2 , . . ., β M there exist integer numbers q and p 1 , p 2 , . . ., p M such that max i=1,2,...,M The above result was proved by Minkowski and has been expanded in the recent years to cover systems of linear forms (see, e.g., Schmidt (1980), p. 36, pp.40-41).We note that in the case where M = 1, the constant C(M ) = M/(M + 1) in (2.1) reduces to 1/2 whereas, by Hurwitz's theorem, the best possible value is 1/ √ 5 (see, e.g., Schmidt (1980), Theorem 2F, p. 6).For M = 2, C(M ) takes the value 2/3; the best possible value is unknown although if C 0 (2) denotes the infimum of admissible values of C(M ) for M = 2, then it is known that 2/7 ≤ C 0 (2) ≤ 0.615 (see, e.g., Schmidt (1980), p. 41).Furthermore, the corresponding best constant in the case of systems of linear forms is positive, meaning that it cannot be replaced by arbitrary small constants (see, e.g., Schmidt (1980), Section 4, pp.41-47) .
We, however, are interested in the opposite result.Namely, the real numbers β 1 , β 2 , . . ., β M form a BA M -tuple if for any integer numbers q > 0 and p 1 , p 2 , . . ., p M one has max i=1,2,...,M for some constant B(M ) > 0, dependent on M (and β 1 , β 2 , . . ., β M ) but independent of q and p 1 , p 2 , . . ., p M (see, e.g., Schmidt (1980), p. 42).It is well-known that the set of all BA M -tuples has Lebesgue measure zero, but nevertheless this set is quite large, namely there are uncountably many BA M -tuples (see Cassels (1955), Davenport (1962)) and the Hausdorff dimension of the set of all BA M -tuples is equal to M (see Schmidt (1969)).In the case where M = 1, the number β = β 1 which satisfies (2.2) is referred to in the Diophantine approximation literature as a BA number (see, e.g., Schmidt (1980), p. 22); in view of Hurwitz's theorem, the constant B(1) in this case must satisfy 0 < B(1) < 1/ √ 5 (see, e.g., Schmidt (1980), pp.41-42).Furthermore, a characterization result exists, namely a real number, that is not an integer, is BA if and only if its continued fraction coefficients are bounded.The latter is often used as a definition of a BA number, however, there is no analogous characterization for M > 1 (see, e.g., Schmidt (1980), Theorem 5F, p. 22).The above definitions of BA numbers and BA M -tuples have been also extended to cover BA systems of linear forms (see, e.g., Schmidt (1980), pp.41) and their existence was proved by Perron, providing also an algorithm for constructing BA linear forms (see, e.g., Schmidt (1980), Theorem 4B, p. 43).Furthermore, it has been established the existence of uncountably many BA systems of linear forms (see Schmidt (1969)).
In what follows, we are interested in the case of BA M -tuples.Although, as indicated above, an algorithm is available for constructing BA M -tuples on the real line, these do not necessarily belong to any specified interval of the real line.Furthermore, if M is strictly fixed (independent of q), one can treat B in (2.2) as a positive constant; this, however, becomes impossible if the value of M is growing.Using the technique described in Schmidt (1980), Section 4, pp.43-45, we show that one can construct a BA M -tuple β 1 , β 2 , . . ., β M of real numbers so that it lies in any specified interval (a, b), a < b, of non-asymptotic length, of the real line, and derive a lower bound for B(M ) in (2.2) as M → ∞.This result is proved in Section 5.

An adaptive wavelet thresholding estimator
Let ϕ * (•) and ψ * (•) be the Meyer scaling and mother wavelet functions, respectively, in the real line (see, e.g., Meyer (1992) or Mallat (1999)).As usual, are, respectively, the dilated and translated Meyer scaling and wavelet (orthonormal) basis functions at resolution level j and scale position k/2 j .Similarly to Section 2.3 in Johnstone et al.
Using the periodized Meyer wavelet basis described above, and for any j 0 ≥ 0, any (periodic) f (•) ∈ L 2 (T ) can be expanded as Furthermore, by Plancherel's formula, the scaling coefficients, a j 0 k = f, ϕ j 0 k , and the wavelet coefficients, b jk = f, ψ jk , of f (•) can be represented as where C j 0 = {m : ϕ mj 0 k = 0} and, for any j ≥ j 0 , C j = {m : ψ mjk = 0}.Note that both C j 0 and C j , j ≥ j 0 , are subsets of where a j 0 k and b jk are the natural estimates of a j 0 k and b jk , respectively (see (3.1) and (3.2)), given by with f m obtained by here g m (u l ) and y m (u l ), l = 1, 2, . . ., M , are the discrete Fourier coefficients of y(u, •) and g(u, •), respectively, obtained by applying the discrete Fourier transform to the equation (1.3).Note that, in this case, The choices of the resolution levels j 0 and J and the thresholds λ j will be described in Section 6 when we examine an expression for the L 2 -risk of the estimator (3.4) over a collection of Besov balls, leading to an adaptive estimator (i.e., its construction is independent of the Besov ball parameters that are usually unknown in practice).
Among the various characterizations of Besov spaces for periodic functions defined on L p (T ) in terms of wavelet bases, we recall that for an r-regular (0 < r ≤ ∞) multiresolution analysis with 0 < s < r and for a Besov ball B s p,q (A) of radius A > 0 with 1 ≤ p, q ≤ ∞, one has that, with with respective sum(s) replaced by maximum if p = ∞ or q = ∞ (see, e.g., Johnstone et al. (2004), Section 2.4).(Note that, for the Meyer wavelet basis, considered considered above, r = ∞.) The parameter s measures the number of derivatives, where the existence of derivatives is required in an L p -sense, while the parameter q provides a further finer gradation.The Besov spaces include, in particular, the well-known Sobolev and Hölder spaces of smooth functions but in addition less traditional spaces, like the space of functions of bounded variation.The latter functions are of statistical interest because they allow for better models of spatial inhomogeneity (see, e.g., Meyer (1992)).
The precision of the estimator (3.4) is measured by the (maximal) L 2 -risk given by We are interested in the asymptotic rate of convergence of the estimator fn , i.e., we are interested in the following asymptotical upper bounds where {γ n } ∞ n=1 is a positive sequence converging to 0 as n → ∞ and C > 0 is a generic constant, independent of n, which may take different values at different places.

Relation to the theory of Diophantine approximation
By direct evaluations (see also Lemma 5 and its proof in the Appendix), one can show that Since in the case of Meyer wavelets, |ψ mjk | ≤ 2 −j/2 and |C j | ≍ 2 j (see, e.g., Johnstone et al (2004), p. 565), we derive where 2) and (3.6), one has Here, u(m) ≍ v(m) mean that there exist constants C 1 > 0 and Therefore, the risk of the estimator fn (t) defined in (3.4) is determined by the rate of growth of ∆ 1 (j) as j → ∞ which, in turn, depends on the rate at which τ 1 (m) goes to zero as m → ∞.
It is easy to see that for some choices of M and u (e.g., M = 1, u 1 = u = 1), one has min m τ 1 (m; u, M ) = 0 for every m which leads to an infinite variances of the estimated coefficients b jk and, consequently, to an infinite L 2 -risk.Hence, the choice of M and the selection of points u is of an uttermost importance.In particular, we want to choose points (u 1 , u 2 , . . ., u M ) such that M l=1 sin 2 (2πmu l ) is as large as possible for m ∈ C j and large j.
Moreover, for any choice of M and any selection of points u, one has τ 1 (m; u, M ) ≤ K 1 m −2 for some constant K 1 > 0 independent of m, the choice of M and the selection points u, so that, for any j and selection of M and u, for some constant K 2 > 0, independent of j.It turns out that if M = M n increases at least as fast as n 1/3 , then, by sampling u l , l = 1, 2, . . ., M , uniformly on U , i.e., by selecting u l = a+(b−a)l/M , l = 1, 2, . . ., M , one can attain ∆ 1 (j) ≤ K 3 2 2j for some constant K 3 > 0, independent of j, so that the upper and the lower bounds in this case coincide up to a constant independent of n (see Pensky & Sapatinas (2010)).Unfortunately, the above results do not hold for finite values of M or when M = M n is a slowly growing function of n.Indeed, in the case of small values of M , both τ 1 (m; u, M ) and ∆ 1 (j) have completely different dynamics from large M .Indeed, if M = 1, Johnstone & Raimondo (2004) and Johnstone et al. (2004) showed that in the case of γ(u) = 1/u, u 1 = u * = a = b, one has ∆ 1 (j) ≥ K 4 2 3j for any choice of u * and some constant K 4 > 0, independent of j.Johnstone et al. (2004) also demonstrated that if u * is selected to be a BA number, then the lower bound for ∆ 1 (j) is attainable, i.e., ∆ 1 (j) ≤ K 5 2 3j for some constant K 5 > 0, independent of j.Hence, in this case, ∆ 1 (j) ≍ 2 3j .These results were extended by De Canditiis & Pensky (2006) who studied the multichannel deconvolution model with a boxcar kernel and showed that the convergence rates obtained by Johnstone et al. (2004) for M = 1 can be improved by sampling at several different points.In particular, they demonstrated that if M is finite, M ≥ 2, one of the u 1 , u 2 , . . ., u M is a BA number, and u is a BA M -tuple defined in (2.2), then for some positive C(M ).In particular, when M is growing with n, the value of C(M ) depends on n and, hence, affects the convergence rates of the estimator fn (•) as n → ∞.
The relation between the convergence rates of the estimator fn (•), given by (3.4), of f (•) in the model (1.3) and the theory of Diophantine approximation becomes obvious when one notes that in (4.2), for any m ∈ Z \ {0} and any u l , l = 1, 2, . . ., M , one has, combining the periodic behavior of the sine function together with a first order (linear) approximation, where a = inf{|a − k|, k ∈ Z} denotes the distance from a real number a to the nearest integer number.Hence, (4.2) becomes so that the convergence rates of the estimator fn (•) depend on the lower bound, in terms of m, of the expression (4.6).
The value of C(M ) in (4.4) is related to the value of B(M ) in (2.2).To the best of our knowledge, there has not been developed a procedure for construction of a BA M -tuple on a specified interval, of non-asymptotic length, of the real line, and there are no asymptotic lower bounds, in terms of M , on B(M ) in (2.2) when the value of M is growing.For this reason, in order to find upper bounds of estimator (3.4) and choose an optimal relation between the sample size n and the number of channels M when M = M n is a slowly growing function of n, we need to obtain new original results in Diophantine approximations.In particular, the objective of the next section is to construct a BA M -tuple on the non-asymptotic interval U , of the real line, and to obtain a lower bound on B(M ) in terms of M for this BA M-tuple when M grows slowly with n.

Construction of a BA M-tuple on a specified interval
Below, we construct a BA M -tuple β = (β 1 , β 2 , . . ., β M ) of real numbers on a specified interval (a, b), of a non-asymptotic length, of the real line, and derive the lower bound on B(M ) in formula (2.2).For this construction, we use the technique described in Schmidt (1980), Section 4, pp.43-45.In particular, we shall provide an algorithm for construction of an M -tuple β 1 , β 2 , . . ., β M of real numbers such that, as M → ∞, 1. it lies in any specified interval (a, b), a < b, of nonasymptotic length, of the real line, and 2. it satisfies max i=1,2,...,M for any integer numbers q > 0 and p 1 , p 2 , . . ., p M , and for some constant B 0 > 0, independent of M , q and p 1 , p 2 , . . ., p M , so that B(M Assume that M is large enough, fix a positive integer Q and consider a monic polynomial (i.e., a polynomial with a unit leading coefficient) of the degree M .Let ξ 1 , ξ 2 , . . ., ξ M be the roots of a polynomial (5.2).Recall that ξ is called an algebraic integer number if it is a root of some monic polynomial with coefficients being integer numbers.Algebraic integers are called conjugate if they are roots of the same monic polynomial with integer coefficients.
Then, the following statement is valid.
Lemma 2 provides a BA M -tuple which, however, does not necessarily belong to the specified interval (a, b), of a non-asymptotic length, of the real line.Assume, without loss of generality, that both a and b are rational numbers, otherwise, replace (a, b) by (a * , b * ) ∈ (a, b), where a * and b * are rational numbers.Let a = p a /q 0 and b = p b /q 0 for some integer numbers p a , p b and q 0 , and let z be an integer number such that z − 1 < 30 exp(3M ln M ) ≤ z.Define where α = (α 1 , α 2 , . . ., α M ) is the BA M -tuple constructed in Lemma 2.
The following theorem confirms that β = (β 1 , β 2 , . . ., β M ), as constructed above, forms a BA M -tuple on the specified (a, b), of a non-asymptotic length, of the real line.
6 Asymptotical upper bounds for the L 2 -risk of the adaptive wavelet thresholding estimator In Section 5, we constructed a BA M -tuple and derived a lower bound on B(M ) in (2.2), as M → ∞.We can now choose the resolution levels j 0 and J, the thresholds λ j in (3.4) and the optimal relation between the total number of observations n and the number of channels M = M n and derive asymptotical upper bounds for the L 2 -risk of the estimator fn (•) given by (3.4) over a collection of Besov balls.
In order to formulate and prove Theorem 2 we first need to obtain some preliminary results.Recall that a denotes the distance from a real number a to the nearest integer number.For this purpose, we recall the equidistribution lemma (see Lemma 3), proved in Johnstone & Raimondo (2004), which we state here for completeness, and formulate a new lemma (i.e., Lemma 4) which is based on application of Lemma 3 to the BA M -tuple.
Lemma 3 (Lemma 1 in Johnstone & Raimondo ( 2004)) Let p/q and p ′ /q ′ be successive convergents in the continued fraction expansion of a real number a. Let N be a positive integer number with N + q < q ′ .Let h be a non-increasing function.Then Lemma 4 Let β 1 , β 2 , . . ., β M be a BA M -tuple constructed in Theorem 1 and let β 1 be a BA number.Let r 0 be an arbitrary fixed positive real number.Denote where Ω j is defined as If M is large enough, then, as j → ∞, ℵ k (j, M ) = O j 2 j(1+(2k−1)/M ) e 6(2k−1)M ln M , k = 1, 2, 3, 4. (6.3) We also need the following two lemmas which evaluate the precision of estimation of a j 0 k and b jk .
Lemma 5 Let β = (β 1 , β 2 , . . ., β M ) be a BA M -tuple constructed on the interval (2a, 2b) according to Theorem 1 and let one of β 1 , β 2 , . . ., β M be a BA number.Let the equation (1.3) be evaluated at the the point u with components u l = β l /2, l = 1, 2, . . ., M .Then, for all j ≥ j 0 , as n → ∞, Lemma 6 Let u 1 , u 2 , . . ., u M be as in Lemma 5.If η > 0 is a constant large enough, then, for all j ≥ j 0 , as n → ∞, We are now ready to formulate Theorem 2. Let the resolution levels j 0 and J and the thresholds λ j be such that for some constant η > 0, where Note that since the construction of j 0 , J and λ j is independent of the Besov ball parameters, s, p, q and A, the suggested wavelet thresholding estimator fn (•) given by (3.4) is adaptive with respect to these parameters.
The following statement provides the asymptotical upper bounds for the L 2 -risk, over a collection of Besov balls.

Discussion
We considered the estimation problem of the unknown response function in the multichannel boxcar deconvolution model with a boxcar-like kernel when the number of channels grows as the total number of observations increases.This situation seems to be of a particular interest nowadays since data recording equipment is getting cheaper and cheaper while overall volumes of data is growing very fast.Our aim was to investigate the situation when the number of channels M = M n grows slowly with the number of observations n.
For this purpose, we obtained new original results in the field of Diophantine approximation in order to devise a technique which allows the reconstruction of the unknown response function with a precision that differs from the best possible convergence rates (which can be attained in the corresponding continuous functional deconvolution model (1.4)) by a factor which grows slower than any power of n.
Specifically, in Section 6, we derived asymptotical upper bounds for the L 2 -risk of the adaptive wavelet thresholding estimator (3.4) of f (•) ∈ L 2 (T ) in the model (1.3).In comparison, it follows from Pensky & Sapatinas (2010) that the choice of a uniform sampling strategy (i.e., for s > 1/ min(p, 2), where s ′ = s + 1/2 − 1/p and Moreover, its has been shown that the above convergence rates with ρ = 0 are the fastest possible ones (see Pensky & Sapatinas (2010)); hence, up to the logarithmic factor (ln n) ̺ , f B n (•) attains the best possible convergence rates.By comparing the convergence rates (6.7) in Theorem 2 to the fastest possible convergence rates (without the extra logarithmic factor (ln n) ρ appearing in (7.1)), one concludes that they differ by the extra factor α n defined in (6.8).
How fast does α n → ∞ as n → ∞?It can be easily seen that α n grows slower than any power of n but faster than any power of ln n, i.e., for any a 1 , a 2 > 0, one has Hence, although choosing M = M n → ∞ at a rate given by (6.6) improves the convergence rates in comparison with the finite values of M (see Pensky & De Canditiis (2006), Theorem 2), these rates are quite a bit worse than in the case when M = M n grows at a faster rate as n → ∞.Since, as we have explained in Section 4, this fast growth of M = M n with n cannot be achieved in a number of practical situations, one has to resign to M n growing slowly with n, in particular, The interesting question, however, is whether the convergence rates (6.7) can be improved.
To uncover an answer to this question, one needs to either come up with another procedure for constructing a BA M -tuple which belongs to a specified interval, of a non-asymptotic length, of the real line and delivers a higher value of B(M ) in (2.2), as M → ∞, or to show that no matter what the value of u = (u 1 , u 2 , . . ., u M ) is, there exist integer numbers q and p 1 , p 2 , . . ., p M such that, as M → ∞, max i=1,2,...,M for some positive constant B 1 independent of M , q and p 1 , p 2 , . . ., p M .At the moment we are unable to provide answers to either of the above questions; we challenge, however, the number theory community to work on the issue.Derivation of these results will not only enrich the theory of Diophantine approximation but will also be valuable for the theory of statistical signal processing.
Proof of Lemma 4. Recall first that any real number a, which is not an integer number, may be uniquely determined by its continued fraction expansion , where a 0 is an integer number and a 1 , a 2 , . . .are strictly positive integer numbers.The convergents p k /q k = p k (a)/q k (a), k = 0, 1, . .., of a are those rational numbers, the continued fraction expansions of which terminate at stage k, that is, p 0 /q 0 = a 0 , p 1 /q 1 = a 0 + 1/a 1 , p 2 /q 2 = a 0 + 1/(a 1 + 1/a 2 ), and so on.The denominators in the above expansions grow at least geometrically q n+i ≥ 2 i/2 q n , if i even, and a n < q n /q n−1 ≤ a n + 1, n ≥ 1.A real number a is BA if sup n a n < ∞, i.e., there exists q > 0 such that q n /q n−1 ≤ q, n ≥ (see, e.g., Schmidt (1980), Sections 3-5, pp.7-23).Let p/q and p ′ /q ′ be successive principal convergents in the continued fraction expansion of β 1 .Let N be a positive integer number with N + q < q ′ .Then, application of Lemma 3 with h since q ′ ≤ qq by (8.7).Now, note that by (5.8) Combination of (5.8), (8.8) and (8.9) implies that Now, observe that the set of indices l in Ω j is symmetric about zero, and so are the components of the sum.Hence, we can consider only the positive part of Ω j which, with some abuse of notation, we keep calling it Ω j .Let q i be the denominators of the convergents of β 1 , and let l be the smallest number such that q l ≥ 2 j .The geometric grows of denominators (8.6) implies that 2 j+r 0 < 2 r 0 q l ≤ q l+2r 0 so that Ω j ⊆ [q l−1 , q l+2r 0 ).If we denote D s = N ∩ [q l+s−1 , q l+s ), s = 0, 1, . . ., 2r 0 , then Since, by (8.6), q i+1 ≤ qq i , there are at most q disjoint blocks of length q l+s−1 that cover D s .Applying (8.10) to each of those blocks, we derive = O e 6(2k−1) M ln M (q l+s−1 ) 1+(2k−1)/M ln q l+s−1 , k = 1, 2, 3, 4.