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## 1 Introduction 

The focus of this paper is on convolution equations arising in models with measurement error. Reviews of measurement error models are in Carroll and Stefanski (2006), Chen, Hong and Nekipelov (2011), Meister(2009). The convolution equations that are examined here also arise in other contexts; various models that go beyond measurement error models are enumerated in Zinde-Walsh (2012). This paper is devoted to the mathematical treatment of such equations. 

Start with the classical measurement error where the variable of interest x ∗ is observed with error, u : 

<img src='content_image/300276.jpg'>

Here the density of x ∗ is the function of interest, denoted g ; the observed z has density w ; suppose that the measurement/contamination error u is independent of x ∗ and has density f. Then the convolution equation 

<img src='content_image/300277.jpg'>

holds. When densities exist, g ∗ f denotes ∫ g ( z − u ) f ( u ) du . In problems that often arise in image processing, epidemiology, medicine the error density f could be assumed known, e.g. the error is Gaussian noise. 

The assumption of known error distribution may not be realistic; if f is 

