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The conversion of analog signals into digital signals and vice versa, performed by sampling and interpolation, respectively, is an essential operation in signal processing. When digital computers are used to compute the analog signals, it is important to effectively control the approximation error. In this paper we analyze the computability, i.e., the effective approximation of bandlimited signals in the Bernstein spaces $\mathcal {B}_{\pi }^p$ , $1 \leq p , and of the corresponding discrete-time signals that are obtained by sampling. We show that for $1 , computability of the discrete-time signal implies computability of the continuous-time signal. For $p=1$ this correspondence no longer holds. Further, we give a necessary and sufficient condition for computability and show that the Shannon sampling series provides a canonical approximation algorithm for $p>1$ . We discuss BIBO stable LTI systems and the time-domain concentration behavior of bandlimited signals as applications.

Downsampling and the computation of the bandlimited interpolation of discrete-time signals are two important concepts in signal processing. In this paper we analyze the downsampling operation regarding its impact on the existence and computability of the bounded bandlimited interpolation. We assume that the discrete-time signal is obtained by downsampling the samples of a bounded bandlimited signal that vanishes at infinity, and we study two problems. First, we investigate the existence of the bounded bandlimited interpolation for such discrete-time signals from a signal theoretic perspective and show that there exist signals for which the bounded bandlimited interpolation does not exist. Second, we analyze the algorithmic generation of the bounded bandlimited interpolation, using the concept of Turing computability. Turing computability models what is theoretically implementable on a digital computer. Interestingly, it turns out that even if the bounded bandlimited interpolation exists analytically, it is not always computable, which implies that there exists no algorithm on a digital computer that can always compute it. Computability is important in order that the approximation error be controlled. If a signal is not computable, we cannot ascertain whether the computed signal is sufficiently close to the true signal, i.e., we cannot verify every approximation accuracy.

Convolution sum system representations are commonly used in signal processing. It is known that the convolution sum, treated as the limit of its partial sums, can be divergent for certain continuous signals and stable linear time-invariant (LTI) systems, even when the convergence of the partial sums is treated in a distributional setting. In this paper, we ask a far more general question: is it at all possible to define a generalized convolution sum with natural properties that works for all absolutely integrable continuous signals that vanish at infinity and all stable LTI systems? We prove that the answer is “no.” Further, for certain subspaces, we give a sufficient and necessary condition for uniform convergence. Finally, we discuss the implications of our results on the effectiveness of window functions in the convolution sum.

We study the time-domain concentration of bandlimited signals form a computational point of view. To this end we employ the concept of Turing computability that exactly describes what can be theoretically computed on a digital machine. A previous definition of computability for bandlimited signals is based on the idea of effective approximation with finite Shannon sampling series. In this paper we provide a different definition that uses the time-domain concentration of the signals. For computable bandlimited signals with finite Lp-norm, we prove that both definitions are equivalent. We further show that local computability together with the computability of the Lp-norm imply the computability of the signal itself. This provides a simple test for computability.

In this paper, we analytically analyze the tone reservation method for the reduction of the peak to average power ratio (PAPR) in code division multiple access systems that employ the Walsh functions. We find the best possible reduction of the PAPR and give one optimal information set that achieves this reduction. Interestingly, when using more than one information carrier, the smallest possible extension constant is independent of the size of the information set and has the value $\sqrt{2}$ . We further show that the minimal extension constant can also be achieved with finite compensation sets, and illustrate the findings with a numerical example. For certain special cases we are also able to provide results for the system of complex exponentials, which is employed in orthogonal frequency division multiplexing.

In this paper, we study the validity of the usual convolution sum sampling representation of linear time-invariant (LTI) systems. We consider continuous input signals with finite energy that are absolutely integrable and vanish at infinity. Even for these benign signals, the convolution sum does not always converge. There exist LTI systems and signals such that the convolution sum diverges even in a distributional sense. This result shows that the practice of multiplying a signal with a Dirac comb and convolving subsequently with the impulse response of the LTI system is not valid for this signal space. We further fully characterize the LTI systems for which we have convergence for all signals in the space, and establish a connection between the pointwise, uniform, and distributional convergence. In particular, we show that the convolution sum converges in a distributional sense if and only it converges in a classical pointwise sense. Hence, for this signal space, nothing can be gained by treating the convergence in a distributional sense.

In this paper, we study the tone reservation method for reducing the peak-to-average power ratio (PAPR) in general orthonormal transmission systems. We prove that strong solvability, where the peak value of the transmit signal has to be bounded by a constant times the energy of the information symbols, is equivalent to weak solvability, where the peak value of the transmit signal has to be only bounded. Further, we show that in the case where the PAPR problem is not weakly solvable, almost all information sequences lead to an unbounded transmit signal.

In this paper we study the spaceability of divergence sets of sequences of bounded linear operators on Banach spaces. For Banach spaces with the s-property, we can give a sufficient condition that guarantees the unbounded divergence on a set that contains an infinite dimensional closed subspace after the zero element has been added. This generalizes the classical Banach–Steinhaus theorem which implies that the divergence set is a residual set. We further prove that many important spaces, e.g., \(\ell ^p\), \(1\le p < \infty \), C[0, 1], \(L^p\), \(1< p

In this paper the structure of the set of functions for which the peak value of the Shannon sampling series is strongly divergent is analyzed. Strong divergence is closely linked to the non-existence of adaptive reconstruction methods. Signals in the Paley–Wiener space PW π 1 of bandlimited functions with absolutely integrable Fourier transform are considered, and it is shown that the set of strong divergence is spaceable and dense-lineable, i.e., that there exist an infinite dimensional closed subspace and an infinite dimensional dense subspace, such that we have strong divergence of the peak value of the Shannon sampling series for all functions from these sets, except the zero function. Further, it is proved that this result is not restricted to the Shannon sampling series, but rather holds for an entire class of reconstruction processes.

The approximation of linear time-invariant systems by sampling series is studied for bandlimited input functions in the Paley–Wiener space PW π 1 , i.e., bandlimited signals with absolutely integrable Fourier transform. It has been known that there exist functions and systems such that the approximation process diverges. In this paper we identify a signal set and a system set with divergence, i.e., a finite time blowup of the Shannon sampling expression. We analyze the structure of these sets and prove that they are jointly spaceable, i.e., each of them contains an infinite-dimensional closed subspace such that for any function and system pair from these subspaces, except for the zero elements, we have divergence.

In this paper we study the Fourier transform and the possibility to determine the binary expansion of the values of the Fourier transform in the Zermelo–Fraenkel set theory with the axiom of choice included (ZFC). We construct a computable absolutely integrable bandlimited function with continuous Fourier transform such that ZFC (if arithmetically sound) cannot determine a single binary digit of the binary expansion of the Fourier transform at zero. This result implies that ZFC cannot determine for every precision goal a rational number that approximates the Fourier transform at zero. Further, we discuss connections to Turing computability.

In this paper we study downsampling for bandlimited signals. Downsampling in the discrete-time domain corresponds to a removal of samples. For any downsampled signal that was created from a bandlimited signal with finite energy, we can always compute a bandlimited continuous-time signal such that the samples of this signal, taken at Nyquist rate, are equal to the downsampled discrete-time signal. However, as we show, this is no longer true for the space of bounded bandlimited signals that vanish at infinity. We explicitly construct a signal in this space, which after downsampling does not have a bounded bandlimited interpolation. This shows that downsampling in this signal space is an operation that can lead out of the set of discrete-time signals for which we have a one-to-one correspondence with continuous-time signals.

In this paper we study the divergence behavior of linear approximation processes in general Banach spaces. We are interested in the structure of the set of vectors creating divergence. The Banach–Steinhaus theory gives some information about this set, however, it cannot be used to answer the question whether this set contains subspaces with linear structure. We give necessary and sufficient conditions for the lineability and the spaceability of the set of vectors creating divergence.

In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space \(\mathcal {PW}_{\pi }^2\), i.e., the set of bandlimited functions with finite \(L^2\)-norm, by convolution sums. It is possible to use either, the convolution sum where the time variable is in the argument of the bandlimited impulse response, or the convolution sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the \(L^2\)-norm in our case, is important. While it is well-known that both convolution sums converge uniformly on the whole real axis, the \(L^2\)-norm of the second convolution sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.

In this paper we analyze the tone reservation method to reduce the peak to average power ratio (PAPR) in orthogonal frequency division multiplexing (OFDM) systems. We consider the case of a reduced compensation set, where only the positive carrier frequencies are used, and fully characterize the information sets for which the PAPR problem is solvable. It turns out that the reduction of the compensation set does not affect the solvability of the PAPR problem, however, the optimal constant are worse in general.

Large peak to average power ratios (PAPRs) are problematic for communication systems. One possible approach to control the PAPR is the tone reservation method. We analyze the tone reservation method for general complete orthonormal systems, and consider two solvability concepts: strong solvability and weak solvability. Strong solvability requires a rather strong control of the peak value of the transmit signal by the energy of the information signal, and thus might be to restrictive for practical applications. Therefore, the concept of weak solvability was introduced, which only requires the boundedness of the transmit signal. In this paper we prove that weak solvability and strong solvability are equivalent for arbitrary complete orthonormal systems.

In this paper we analyze the divergence behavior of linear approximation processes in general Banach spaces. If the approximation process is given by a sequence of bounded linear operators, often the Banach–Steinhaus theorem is employed to perform a divergence analysis. However, the Banach–Steinhaus theorem gives only divergence in terms of the limit superior, and no strong divergence in terms of the limit. Strong divergence is closely related to the question of whether adaptive approaches, where the approximation process is adapted to the signal, can be successful. Here we study the structure of the set of signals for which the approximation process is strongly divergent, i.e., does not have a convergent subsequence, and show that, under mild conditions, this set is at most a meager set. Further, for the Shannon sampling series, which is a special case of the general theory, we prove that the set of signals in $$PW_\pi ^1$$ with strong divergence is lineable. That is, there exists an infinite dimensional subspace such that we have strong divergence for all signals, except the zero signal, from this subspace.

The approximation of stable linear time-invariant systems is a central task in many applications. Therefore, it is important to know if a given approximation process is stable and converges for all signals from the signal space or if it is unstable and diverges for certain signals. Further, in the case of divergence, it is interesting to know whether the set of signals with divergent approximation process has some additional property or inherent structure, like containing subsets that are linear, shift invariant, or closed. In this paper we analyze this question. We will discuss connections to Carleson's theorem, and show that the problem of dense lineability of certain signal sets is equivalent to the Riemann hypothesis.

The Banach-Steinhaus theorem, one of the fundamental results in functional analysis, completely characterizes the convergence of linear approximation processes. If the condition of boundedness is violated, then the principle of uniform boundedness implies the unbounded divergence of the approximation process on a residual set. In this paper we give a sufficient condition for Banach spaces that guarantees the unbounded divergence not only for a residual set but rather for a set that contains an infinite dimensional closed subspace after the zero element has been added. We further show that many important signal space, e.g., Paley-Wiener and Bernstein spaces, possess this property, and demonstrate consequences for the convergence behavior of sampling series and system approximation processes.

In this paper we analyze the peak-to-average power ratio (PAPR) reduction by tone reservation for orthogonal frequency division multiplexing (OFDM) schemes. In addition to the strong solvability of the PAPR reduction problem, where the PAPR has to be bounded by some constant, we consider a weaker form of solvability, where only the boundedness of the peak value of the signal is required. We show that for OFDM both forms of solvability are equivalent. Further, we show that in the case where the PAPR problem is not solvable, the set of input signals that lead to an unbounded OFDM signal is a residual set. As a consequence, if the upper density of the carriers, used for information transmission, is positive, the set of input signals that lead to a bounded OFDM signal is a meager set.

In this paper we analyze the convergence behavior of a sampling based system approximation process, where the time variable is in the argument of the signal and not in the argument of the bandlimited impulse response. We consider the Paley-Wiener space PW π 2 of bandlimited signals with finite energy and stable linear time-invariant (LTI) systems, and show that there are signals and systems such that the approximation process diverges in the L2-norm, i.e., the norm of the signal space. We prove that the sets of signals and systems creating divergence are jointly spaceable, i.e., there exists an infinite dimensional closed subspace of PW π 2 and an infinite dimensional closed subspace of the space of all stable LTI systems, such that the approximation process diverges for any non-zero pair of signal and system from these subspaces.

It is known that there exist signals in Paley-Wiener space PW π 1 of bandlimited signals with absolutely integrable Fourier transform, for which the peak value of the Shannon sampling series diverges unboundedly. In this paper we analyze the structure of the set of signals which lead to strong divergence. Strong divergence is closely linked to the existence of adaptive methods. We prove that there exists an infinite dimensional closed subspace of PW1 π 1, all signals of which, except the zero signal, lead to strong divergence of the peak value of the Shannon sampling series.

The peak value of a signal is a characteristic that has to be controlled in many applications. In this paper we analyze the peak value of the Hilbert transform for the space $\mathcal{B}_\pi ^\infty $ of bounded bandlimited signals. It is known that for this space the Hilbert transform cannot be calculated by the common principal value integral, because there are signals for which it diverges everywhere. Although the classical definition fails for $\mathcal{B}_\pi ^\infty $ , there is a more general definition of the Hilbert transform, which is based on the abstract H 1-BMO(?) duality. It was recently shown in [1] that, in addition to this abstract definition, there exists an explicit formula for calculating the Hilbert transform. Based on this formula we study properties of the Hilbert transform for the space $\mathcal{B}_\pi ^\infty $ of bounded bandlimited signals. We analyze its asymptotic growth behavior, and thereby solve the peak value problem of the Hilbert transform for this space. Further, we obtain results for the growth behavior of the Hilbert transform for the subspace $\mathcal{B}_{\pi ,0}^\infty $ of bounded bandlimited signals that vanish at infinity. By studying the properties of the Hilbert transform, we continue the work [2].

In this paper we analyze the structure of bandlimited BMO-functions. Using a recently found equation for the calculation of the Hilbert transform of bounded bandlimited functions, we derive a decomposition result for bandlimited BMO-functions, which is similar to the well-known Fefferman–Stein decomposition. Based on this decomposition we characterize the range of the Hilbert transform. Moreover, we present interesting applications of this result. We characterize the peak value behavior of bandlimited BMO-functions, show that the derivative of bandlimited BMO-functions is bounded, and prove a sampling theorem for bandlimited BMO-functions.

One interesting question is how the good local approximation behavior of the Shannon sampling series for the Paley–Wiener space $\mathcal {PW}_{\pi}^{1}$ is affected if the samples are disturbed by the non-linear threshold operator. This operator, which is important in many applications, sets all samples whose absolute value is smaller than some threshold to zero. In this paper we analyze a generalization of this problem, in which not the Shannon sampling series is disturbed by the threshold operator but a more general system approximation process, were a stable linear time-invariant system is involved. We completely characterize the stable linear time-invariant systems that, for some functions in $\mathcal {PW}_{\pi}^{1}$ , lead to a diverging approximation process as the threshold is decreased to zero. Further, we show that if there exists one such function then the set of functions for which divergence occurs is in fact a residual set. We study the pointwise behavior as well as the behavior of the L ∞-norm of the approximation process. It is known that oversampling does not lead to stable approximation processes in the presence of thresholding. An interesting open problem is the characterization of the systems that can be stably approximated with oversampling.

The approximation of linear time-invariant (LTI) systems by sampling series is a central task of signal processing. For the Paley-Wiener space PW1 π of bandlimited signals with absolutely integrable Fourier transform, it is known that there exist signals and stable LTI systems such that the canonical approximation process diverges. In this paper we analyze the structure of the sets of signals and systems creating divergence and show that both sets are jointly spaceable, i.e., contain subsets such that every linear combination of signals and systems from these subsets, which is not the zero element, leads to divergence. In signal processing applications the linear structure of the involved signal spaces is essential. Here, we show that the same linear structure also holds for the sets of signals and systems creating divergence.

Knowing whether a reconstruction process, for example the Shannon sampling series, is strongly divergent in terms of the lim or only weakly divergent in terms of the lim sup is important, because strong divergence is linked to the non-existence of adaptive reconstruction processes. For non-adaptive reconstruction processes the existence is answered by the Banach-Steinhaus theory. However, the analysis of adaptive reconstruction processes is more difficult and not covered by the former theory. In this paper we consider the Paley-Wiener space PW π 1 of bandlimited signals with absolutely integrable Fourier transform and analyze the structure of the set of signals for which the peak value of the Shannon sampling series is strongly divergent. We show that this set is lineable, i.e., that there exists an infinite dimensional subspace, all signals of which, except the zero signal, lead to strong divergence. Consequently, for all signals from this subspace, adaptivity in the number of samples that are used in the Shannon sampling series does not create a convergent reconstruction process.

It is well-known that the Shannon sampling series is locally uniformly convergent for all signals in the Paley-Wiener space PW"@p^1. An interesting question is how this good local approximation behavior is affected if the samples are disturbed by the non-linear threshold operator. This operator, which sets to zero all samples with absolute value smaller than some threshold, arises in the modeling of many applications, e.g., in wireless sensor networks. Moreover, it constitutes an essential part of a large class of quantizers, and consequently is important for all digital signal processing applications that involve conversion between analog and digital domains. In this paper, the approximation behavior of the Shannon sampling series that only uses the samples with absolute value larger than or equal to some threshold is analyzed. It is shown that there exists a signal in PW"@p^1 such that the local approximation error increases unboundedly as the threshold tends to zero. Moreover, for a fixed threshold, the local approximation error can grow arbitrarily large on the set of signals whose norm is bounded by one. With this, we generalize results of Butzer et al. that were given in the paper ''On quantization, truncation and jitter errors in the sampling theorem and its generalizations,'' Signal Processing (2) 1980 [1]. We conclude the paper with a discussion about the differences in the reconstruction behavior between the sampling series which is truncated in the domain of the sampled signal, i.e., time-domain truncation, and the sampling series which is truncated in the range of the sampled signal.

In this paper we analyze the Hilbert transform and existence of the analytical signal for the space B ? ? of bandlimited signals that are bounded on the real axis. Originally, the theory was developed for signals in L 2(?) and then extended to larger signal spaces. While it is well known that the common integral representation of the Hilbert transform may diverge for some signals in B ? ? and that the Hilbert transform is not a bounded operator on B ? ? , it is nevertheless possible to define the Hilbert transform for the space B ? ? . We use a definition that is based on the H 1-BMO(?) duality. This abstract definition, which can be used for general bounded signals, gives no constructive procedure to compute the Hilbert transform. However, for the practically important special case of bounded bandlimited signals, we can provide such an explicit procedure by giving a closed-form expression for the Hilbert transform. Further, it is shown that the Hilbert transform of a signal in B ? ? is still bandlimited but not necessarily bounded. With these results we continue the work of [1, 2].

Arbitrary bounded bandlimited signals that are real on the real axis cannot be reconstructed from their zero crossings in general, because they can have complex zeros. However, a simple invertible transformation, which consists of taking the difference of such a signal with a properly chosen sine function, creates a function that has only simple and real zeros. It can be shown that the original signal can be reconstructed from these zeros, or, equivalently, from the sine wave crossings of the signal. In this paper we study the reconstruction of bandlimited signals from their sine wave crossings by sampling series, which is one possible method of reconstruction. The reconstruction functions and the sampling points that are used in the reconstruction process are determined by the signal which shall be reconstructed and therefore are, in a certain sense, adapted to the signal. We address the convergence behavior of the reconstruction process without oversampling. Although we have local uniform convergence, we conjectured in a previous paper that the adaptivity is not sufficient for global uniform convergence. By developing a rigorous convergence theory for different signal spaces, and by showing that the Paley-Wiener space PW"@p^1 of signals that are the inverse Fourier transform of some absolutely integrable function is the border case, we prove that this conjecture is true. The solution of this problem has far reaching consequences for the approximation of stable linear time-invariant systems by sampling series, because it shows that the adaptivity of the system approximation process is not sufficient for its pointwise convergence. The theory of sine-type functions is an important tool for our analysis, and we give several sufficient conditions for the sine-wave crossings of a signal to be the zero sequence of a sine-type function.

In this paper, we analyze sampling approximations of stable linear time-invariant systems for input signals in the Paley-Wiener space PWπ1. The goal is to approximate a transform Tf by using only the samples of the signal f, which do not necessarily have to be equidistant. We completely characterize the stable linear time-invariant (LTI) systems T and sampling patterns for which the approximation process converges to Tf for all signals in PWπ1. Furthermore, we prove that for every complete interpolating sequence there exist a stable LTI system and a signal in PWπ1 for which the approximation error increases unboundedly as the number of samples that are used for the approximation is increased. This shows that a more flexible choice of the sampling points instead of equidistant sampling, where such a divergence behavior is known to exist, is not sufficient to overcome the divergence. Thus, sampling based signal processing has fundamental limits. The calculations further indicate that oversampling cannot resolve the convergence problems.

In this paper, the convergence behavior of the symmetric and the nonsymmetric Shannon sampling series is analyzed for bandlimited continuous-time wide-sense stationary stochastic processes that have absolutely continuous spectral measure. It is shown that the nonsymmetric sampling series converges in the mean-square sense uniformly on compact subsets of the real axis if and only if the power spectral density of the process fulfills a certain integrability condition. Moreover, if this condition is not fulfilled, then the pointwise mean-square approximation error of the nonsymmetric sampling series and the supremum of the mean-square approximation error over the real axis of the symmetric sampling series both diverge. This shows that there is a significant difference between the convergence behavior of the symmetric and the nonsymmetric sampling series.

In this paper we analyze the distributional convergence behavior of time-domain convolution type system representations on the Paley-Wiener space PWπ1. Two convolution integrals as well as the discrete counterpart, the convolution sum, are treated. It is shown that there exist stable linear time-invariant (LTI) systems for which the convolution integral representation does not exist because the integral is divergent, even if the convergence is interpreted in a distributional sense. Furthermore, we completely characterize all stable LTI systems for which a convolution representation is possible by giving a necessary and sufficient condition for convergence. The classical and the distributional convergence behavior are compared, and differences between the convergence of the convolution integral and the convolution sum are discussed. Finally, the results are illustrated by numerical examples.

The convergence behavior of a convolution representation of stable linear time-invariant (LTI) systems operating on the Zakai class of bandlimited signals is analyzed. It is shown that there are signals in the Zakai class for which the convolution integral diverges if the system is the Hilbert transform or the ideal low-pass filter with bandwidth less than or equal to the signal bandwidth. Moreover, using a previously obtained result of Habib, it is proved that the class of stable LTI systems that map the Zakai class into itself does not include the Hilbert transform and the ideal low-pass filter with bandwidth less than or equal to the signal bandwidth. Finally, it is shown that the concept of the analytical signal, which is used in communications, is problematic for the signal space Zπ, because the operator for its computation is unbounded and discontinuous.

In this paper non-equidistant sampling series are studied for bounded bandlimited signals. We consider sampling patterns that are made of the zeros of sine-type functions and analyze the local and global convergence behavior of the sampling series. It is shown that the series converge locally uniformly for bounded bandlimited signals that vanish at infinity. Moreover, we discuss the influence of oversampling on the global approximation behavior and the convergence speed of the sampling series.

The process of quantization generates a loss of information, and, thus, the original signal cannot be reconstructed exactly from the quantized samples in general. However, it is desirable to keep the error as small as possible. In this paper, the quantization error is quantified in terms of several distortion measures. All these measures employ the difference between the original signal and the reconstructed signal, which is obtained by bandlimited interpolation of the quantized samples. We assume that the signals are bandlimited and that the samples are taken at Nyquist rate. It is shown that for signals in the Paley-Wiener space PW ? 1, the supremum of the reconstructed signal, and, hence, the quantization error cannot be bounded in the sense that there exists a bounded subset of PW ? 1 on which both quantities can increase unboundedly. This unexpected behavior is due to the nonlinearity of the quantization operator and the slow decay of the sinc function. The nonlinearity is essential for this behavior because every linear operator that fulfills a certain property of the quantization operator would otherwise have to be bounded. Furthermore, it is proven that for a fixed signal the possible quantization error increases as the quantization step size tends to zero. The treatment of the quantization error in this paper is completely deterministic.

In this paper, we analyze the approximation of the outputs of linear time-invariant systems by sampling series that use only the samples of the input signal. The samples are disturbed by the threshold operator, which sets all samples with an absolute value smaller than some threshold to zero. We do the analysis for the space of Paley-Wiener signals with absolutely integrable Fourier transform and show for the Hilbert transform that the peak approximation error can grow arbitrarily large for some signals in this space when the threshold approaches zero. This behavior is counterintuitive because one would expect a better behavior if the threshold was decreased. Since we consider oversampling and all kernels from a certain meaningful set, the results are valid not only for one specific approximation process, but for a whole class of approximation processes. Furthermore, we give a game theoretic interpretation of the problem in the setting of a game against nature and show that nature has a universal strategy to win this game.

The convergence of sampling series with non-equidistant sampling points cannot be guaranteed for the Paley-Wiener space PW"@p^1 if the class of sampling patterns is not restricted. In this paper we consider sampling patterns that are made of the zeros of sine-type functions and analyze the local and global convergence behavior of the sampling series. It is shown that oversampling is necessary for global uniform convergence. If no oversampling is used there exists for every sampling pattern a signal such that the peak value of the approximation error grows arbitrarily large. Furthermore, we use the findings to derive results about the mean-square convergence behavior of the sampling series for bandlimited wide-sense stationary stochastic processes. Finally, a procedure is given to construct functions of sine type and possible sampling patterns.

In practical applications the sole reconstruction of signals by its samples is sometimes not sufficient. Often, some processed version of the signal is of interest and must be approximated by using only the samples of the signal. In this paper, the possible reconstruction kernels are characterized. Then, the convergence behavior of general approximation processes for translation invariant, linear, and bounded operators is analyzed for signals in the Paley-Wiener space $$PW_\pi ^1$$ and these kernels. It is shown that the Hilbert transform is a universal operator in the sense that the peak value of all possible approximation processes diverges unboundedly for some signal in $$PW_\pi ^1$$ , regardless of the oversampling factor and the kernel. Furthermore, for all approximation processes and all points in time, there exists an operator such that the approximation process diverges in this point. The results are compared to the approximation behavior of the Hilbert transform, operating on continuous-time signals. Moreover, a simple criterion based on the exponential function as test signal is developed for answering the question of whether or not an approximation process is convergent for a given operator.

In this paper, we analyze the approximation behavior of sampling series, where the sample values-taken equidistantly at Nyquist rate-are disturbed either by the nonlinear threshold operator or the nonlinear quantization operator. We perform the analysis for several spaces of bandlimited signals and completely characterize the spaces for which an approximation is possible. Additionally, we study the approximation of outputs of stable linear time-invariant systems by sampling series with disturbed samples for signals in PW pi 1. We show that there exist stable systems that become unstable under thresholding and quantization and that the approximation error is unbounded irrespective of how small the quantization step size is chosen. Further, we give a necessary and sufficient condition for the pointwise and the uniform convergence of the series. Surprisingly, this condition is the well-known condition for bounded-input bounded-output (BIBO) stability. Finally, we discuss the special case of finite-impulse-response (FIR) filters and give an upper bound for the approximation error.

In this paper the reconstruction of bandlimited signals by sampling series is analyzed for the case where the samples of the signal are disturbed by the non-linear threshold operator. This operator sets all samples whose absolute value is smaller than some threshold to zero. It is shown that the reconstruction error can grow arbitrarily large, regardless of how small the threshold is chosen, when no oversampling is used. However, with oversampling, it is possible to upper bound the reconstruction error. Additionally, we consider the approximation of outputs of stable linear time-invariant systems, by sampling series that use only the samples that are larger than the threshold, and show that there exist stable linear time-invariant systems for which the approximation error is unbounded, even if oversampling is applied. Finally, we consider the case of non-equidistant sampling. It is possible to draw conclusions about the approximation behavior of the sampling series with threshold operator by analyzing the convergence behavior of the sampling series with permuted sampling points and without threshold operator. Since the latter does not have a good approximation behavior in general, we conjecture that the sampling series with non-equidistant samples and threshold operator has no good approximation behavior either.

For signals in the Paley-Wiener space $$PW_\pi ^1$$ a reconstruction in the form of a sampling series that is uniformly convergent on compact subsets of the real axis and uniformly bounded on the whole real axis is not possible in general if the signals are sampled equidistantly at Nyquist rate. We prove that, even if the signal is non-uniformly sampled with an average sampling rate equal to the Nyquist rate, a uniformly convergent reconstruction is not possible. Additionally, we provide a detailed convergence analysis and give a sufficient condition for the uniform convergence of the Shannon sampling series without oversampling. However, if oversampling is applied, a uniformly convergent reconstruction is always possible and as far as convergence is concerned no elaborate kernel design is necessary. Moreover, we show that a projection of the reconstruction process onto the range of signal frequencies is not possible without losing the good convergence behavior.

The bandlimited interpolation of signals and the convergence behavior of the Shannon sampling series are discussed in order to show that it is desirable to have a uniformly convergent reconstruction, for as large a space of signals as possible. In this paper general sampling series are analyzed for the frequently utilized Paley-Wiener space PW pi 1, which is the largest space in the scale of Paley-Wiener spaces. The analysis is done not only for the Shannon sampling series, but for a whole class of axiomatically defined reconstruction processes. It is shown that for this very general class, which contains all common sampling series including the Shannon sampling series, a uniformly convergent reconstruction is not possible for the space PW pi 1. Moreover, a universal signal is identified that causes the divergence behavior for all sampling series. Finally, a lower and an upper bound are derived and used to describe the asymptotic behavior of the peak value of the finite sampling series.

In this paper symmetric and non-symmetric Shanon type sampling series that use samples taken at Nyquist rate are analyzed. It is shown that the symmetric series converges uniformly on the whole real axis, i.e., that it is a stable reconstruction process, for the Paley-Wiener space. This result is surprising, because recently it was shown that for a very general class of reconstruction processes a stable reconstruction is not possible for the Paley-Wiener space. However, there are some important differences. The analyzed Shannon type sampling series uses infinitely many samples for signal reconstruction and is not bandlimited. The characteristics stability, bandlimitedness, and the property of perfect reconstruction for a certain subspace are discussed. Furthermore, the corresponding non-symmetric series is analyzed. An explicit example shows that it is not stable.

In this paper the convergence behavior of the Shannon sampling series is analyzed for Hardy spaces. It is well known that the Shannon sampling series is locally uniformly convergent. However, for practical applications the global uniform convergence is important. It is shown that there are functions in the Hardy space such that the Shannon sampling series is not uniformly convergent on the whole real axis. In fact, there exists a function in this space such that the peak value of the Shannon sampling series diverges unboundedly. The proof uses Fefferman’s theorem, which states that the dual space of the Hardy space is the space of functions of bounded mean oscillation.

In this paper we discuss the interplay between discrete-time and continuous-time signals and the question, whether certain properties of the signal in one domain carry over to the other domain. The Shannon sampling series and the more general Valiron interpolation series are the appropriate means to obtain the continuous-time, bandlimited signal out of its samples, i.e., the discrete-time signal. We study the effects of the projection operator, that projects the discrete-time signal on the positive time axis and, closely related, the differences in the convergence behavior of the symmetric and asymmetric Shannon sampling series. It is well known, that the space of discrete-time signals with finite energy and the space of continuous-time, bandlimited signals with finite energy are isomorphic. Thus, discrete-time and continuous-time, bandlimited signals with finite energy can be used interchangeably. This interchangeability is not restricted to finite energy signals. It is valid for a considerably larger class, but, as we show by stating an explicit example, not for the space of bounded signals: Even if the discrete-time signal is bounded, the corresponding bandlimited interpolation can be unbounded. In addition to the fact that the Shannon sampling series diverges for this signal, we prove that for this signal there is no bounded, bandlimited interpolation at all. Furthermore, by using the Banach-Steinhaus theorem we show that in a certain sense ''almost all'' signals have this property. Finally, some implications for practical applications are discussed.

In this paper we analyze the approximation of stable linear time-invariant systems, like the Hilbert transform, by sampling series for bandlimited functions in the Paley-Wiener space $\mathcal{PW}_{\pi}^{1}$. It is known that there exist systems and functions such that the approximation process is weakly divergent, i.e., divergent for certain subsequences. Here we strengthen this result by proving strong divergence, i.e., divergence for all subsequences. Further, in case of divergence, we give the divergence speed. We consider sampling at Nyquist rate as well as oversampling with adaptive choice of the kernel. Finally, connections between strong divergence and the Banach-Steinhaus theorem, which is not powerful enough to prove strong divergence, are discussed., Comment: Preprint accepted for publication in Problems of Information Transmission. The material in this paper was presented in part at the 2015 IEEE International Conference on Acoustics, Speech, and Signal Processing

It is well-known that there exist bandlimited signals for which certain sampling series are divergent. One possible way of circumventing the divergence is to adapt the sampling series to the signals. In this paper we study adaptivity in the number of summands that are used in each approximation step, and whether this kind of adaptive signal processing can improve the convergence behavior of the sampling series. We approach the problem by considering approximation processes in general Banach spaces and show that adaptivity reduces the set of signals with divergence from a residual set to a meager or empty set. Due to the non-linearity of the adaptive approximation process, this study cannot be done by using the Banach-Steinhaus theory. We present examples from sampling based signal processing, where recently strong divergence, which is connected to the effectiveness of adaptive signal processing, has been observed.

Many divergence results for sampling series are in terms of the limit superior and not the limit. This leaves the possibility of a convergent subsequence. If there exists a convergent subsequence, adaptive signal processing techniques can be used. In this paper we study sampling-based signal reconstruction and system approximation processes for the space PW π 1 of bandlimited signals with absolutely integrable Fourier transform. For all analyzed examples, which include the peak value of the Shannon and the conjugated Shannon sampling series, we prove strong divergence, i.e., divergence for all subsequences. Hence, adaptive signal processing techniques do not help in these cases. We further analyze whether an adaptive choice of the reconstruction functions in the oversampling case can improve the behavior.

The approximation of linear time-invariant (LTI) systems by sampling series is an important topic in signal processing. However, the convergence of the approximation series is not guaranteed: there exist stable LTI systems and bandlimited input signals such that the approximation series diverges, regardless of the oversampling factor and the sampling pattern. Recently, it has been shown that this divergence can be overcome by using measurement functionals instead of pointwise sampling. However, the bandwidth of the approximation series needs to be strictly larger than the signal bandwidth. In this paper we derive a two channel system approximation approach based on measurement functionals that converges for all stable LTI systems and all signals in the Paley-Wiener space PW π 1. Thanks to the two channel structure it is possible to achieve an approximation bandwidth that is equal to the signal bandwidth.