Coronavirus disease (Covid-19) has been the main agenda of the whole world since it came in sight in December, 2019. It has already caused thousands of causalities and infected several millions worldwide. Any technological tool that can be provided to healthcare practitioners to save time, effort, and possibly lives has crucial importance. The main tools practitioners currently use to diagnose Covid-19 are Reverse transcriptionpolymerase chain reaction (RT-PCR) and Computed Tomography (CT), which require significant time, resources and acknowledged experts. X-ray imaging is a common and easily accessible tool that has great potential for Covid-19 diagnosis. In this study, we propose a novel approach for Covid-19 recognition from chest Xray images. Despite the importance of the problem recent studies in this domain produced not so satisfactory results due to the limited datasets available for training. Recall that Deep Learning techniques can generally provide state-of-the-art performance in many classification tasks when trained properly over large datasets, such data scarcity can be a crucial obstacle when using them for Covid-19 detection. Alternative approaches such as representation-based classification (collaborative or sparse representation) might provide satisfactory performance with limited size datasets, but they generally fall short in performance or speed compared to Machine Learning methods. To address this deficiency, Convolution Support Estimation Network (CSEN) has recently been proposed as a bridge between model-based and Deep Learning approaches by providing a non-iterative real-time mapping from query sample to ideally sparse representation coefficient' support, which is critical information for class decision in representation based techniques.Main premises of this study can be summarized as follows: (i) a benchmark X-ray dataset, namely QaTa-Cov19, containing over 6200 X-ray images is created. Up to date, this is the largest dataset covering 462 X-ray images from Covid-19 patients along with three other classes; bacterial pneumonia, viral pneumonia, and normal. (ii) In such a scarce and imbalanced dataset the proposed CSEN based classification scheme equipped with feature extraction from a state-of-the-art deep neural network solution for X-ray images, CheXNet, achieves over 98% sensitivity and over 95% specificity for Covid-19 recognition directly from raw X-ray images without any pre-or post-processing. (iii) Having such an elegant Covid-19 assistive diagnosis performance, this study further provides solid evidence that Covid-19 induces a unique pattern in X-rays that can be discriminated with a high accuracy.