A Note on BIBO Stability

The statements on the BIBO stability of continuous-time convolution systems found in engineering textbooks are often either too vague (because of lack of hypotheses) or mathematically incorrect. What is more troubling is that they usually exclude the identity operator. The purpose of this note is to clarify the issue while presenting some fixes. In particular, we show that a linear shift-invariant system is BIBO-stable in the $L_\infty$-sense if and only if its impulse response is included in the space of bounded Radon measures, which is a superset of $L_1(\mathbb{R})$ (Lebesgue's space of absolutely integrable functions). As we restrict the scope of this characterization to the convolution operators whose impulse response is a measurable function, we recover the classical statement

Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion

We propose a robust autofocus method for reconstructing digital Fresnel holograms. The numerical reconstruction involves simulating the propagation of a complex wave front to the appropriate distance. Since the latter value is difficult to determine manually, it is desirable to rely on an automatic procedure for finding the optimal distance to achieve high-quality reconstructions. Our algorithm maximizes a sharpness metric related to the sparsity of the signal’s expansion in distance-dependent waveletlike Fresnelet bases. We show results from simulations and experimental situations that confirm its applicability

Market Transparency and Call Markets

This paper reports the results of 16 experimental asset markets that explore the effects of trade transparency on the price formation process and its results using a more realistic design than related studies. The open orderbook does not improve informational efficiency and does not result in higher liquidity (lower transaction costs). An increase in information intensity leads to both higher trading volume and higher volatility in both orderbook treatments. The comparison shows that they only differ in price volatility which is higher with an open orderbook. The market results mentioned above are confirmed by analyses on the individual level. --Market Microstructure,Experimental Asset Markets,Orderbook Transparency,Individual Behavior in Call Markets

On the Continuity of Characteristic Functionals and Sparse Stochastic Modeling

The characteristic functional is the infinite-dimensional generalization of the Fourier transform for measures on function spaces. It characterizes the statistical law of the associated stochastic process in the same way as a characteristic function specifies the probability distribution of its corresponding random variable. Our goal in this work is to lay the foundations of the innovation model, a (possibly) non-Gaussian probabilistic model for sparse signals. This is achieved by using the characteristic functional to specify sparse stochastic processes that are defined as linear transformations of general continuous-domain white noises (also called innovation processes). We prove the existence of a broad class of sparse processes by using the Minlos-Bochner theorem. This requires a careful study of the regularity properties, especially the boundedness in Lp-spaces, of the characteristic functional of the innovations. We are especially interested in the functionals that are only defined for p<1 since they appear to be associated with the sparser kind of processes. Finally, we apply our main theorem of existence to two specific subclasses of processes with specific invariance properties.Comment: 24 page

Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes

The fractional Laplacian $(-\triangle)^{\gamma/2}$ commutes with the primary coordination transformations in the Euclidean space \RR^d: dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For $0<\gamma<d$, its inverse is the classical Riesz potential $I_\gamma$ which is dilation-invariant and translation-invariant. In this work, we investigate the functional properties (continuity, decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular, we extend the definition of the classical Riesz potential $I_\gamma$ to any non-integer number $\gamma$ larger than $d$ and show that it is the unique left-inverse of the fractional Laplacian $(-\triangle)^{\gamma/2}$ which is dilation-invariant and translation-invariant. We observe that, for any $1\le p\le \infty$ and $\gamma\ge d(1-1/p)$, there exists a Schwartz function $f$ such that $I_\gamma f$ is not $p$-integrable. We then introduce the new unique left-inverse $I_{\gamma, p}$ of the fractional Laplacian $(-\triangle)^{\gamma/2}$ with the property that $I_{\gamma, p}$ is dilation-invariant (but not translation-invariant) and that $I_{\gamma, p}f$ is $p$-integrable for any Schwartz function $f$. We finally apply that linear operator $I_{\gamma, p}$ with $p=1$ to solve the stochastic partial differential equation $(-\triangle)^{\gamma/2} \Phi=w$ with white Poisson noise as its driving term $w$.Comment: Advances in Computational Mathematics, accepte