Abstract. We introduce and study Hermite expansions and various integral transformations on the spaces S'( m p) and of tempered ultradistributions of Beurling and Roumieu type. In particular, we investigate the Wigner distribution and the Fourier, Bargmann and Laplace transforms.

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We give a simple proof of the Kernel theorem for the space of tempered ultradistributions of Beurling -Komatsu type, using the characterization of Fourier-Hermite coefficients of the elements of the space. We prove in details that the test space of tempered ultradistributions of Beurling -Komatsu type can be identified with the space of sequences of ultrapolynomal falloff and its dual space with the space of sequences of ultrapolynomial growth. As a consequence of the Kernel theorem we have that the Weyl transform can be extended on a space of tempered ultradistributions of Beurling -Komatsu type.

A probabilistic crop forecast based on ensembles of crop model output (CMO) estimates offers a myriad of possible realizations and probabilistic forecasts of green water components (precipitation and evapotranspiration), crop yields and green water footprints (GWFs) on monthly or seasonal scales. The present paper presents part of the results of an ongoing study related to the application of ensemble forecasting concepts for agricultural production. The methodology used to produce the ensemble CMO using the ensemble seasonal weather forecasts as the crop model input meteorological data without the perturbation of initial soil or crop conditions is presented and tested for accuracy, as are its results. The selected case study is for winter wheat growth in Austria and Serbia during the 2006–2014 period modelled with the SIRIUS crop model. The historical seasonal forecasts for a 6-month period (1 March-31 August) were collected for the period 2006–2014 and were assimilated from the European Centre for Medium-range Weather Forecast and the Meteorological Archival and Retrieval System. The seasonal ensemble forecasting results obtained for winter wheat phenology dynamics, yield and GWF showed a narrow range of estimates. These results indicate that the use of seasonal weather forecasting in agriculture and its applications for probabilistic crop forecasting can optimize field operations (e.g., soil cultivation, plant protection, fertilizing, irrigation) and takes advantage of the predictions of crop development and yield a few weeks or months in advance.