We give simple necessary and sufficient conditions on Bessel sequences { f i } and {g i } and operators L 1 , L 2 on a Hilbert space H so that {L 1 f i + L 2 g i } is a frame for H. This allows us to construct a large number of new Hilbert space frames from existing frames.

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A b s t r a c t . We prove a Schur test for mixed-norm spaces L p,q , 1 < p, q < ∞. Also we prove another version of the Schur test for discrete weighted mixed-norm spaces p,q w , 1 < p, q < ∞, and w is a weight. We show that if w1, and w2 are two weight functions on the index sets J × I and K × L respectively, and A = (a ji,kl ) j∈J,i∈I,k∈K,l∈L is an infinite matrix, then under certain conditions, A is a bounded operator from p,q w 1 , 1 < p, q < ∞ to p,q w 2 . This will be a key result in proving boundedness of important operators in our work in time-frequency analysis.

Educational institutions in Saudi Arabia extended e-learning until the third semester of the academic calendar to prevent the spread of COVID-19 infection and to achieve 70% inoculation for the Saudi population. This study assesses the impact of extended e-learning and other associated stressors on the emotional health of university students in Saudi Arabia. An online cross-sectional survey collected data between the months of January–March 2021. The emotional signs of stress were measured by using a subset of items from the COVID-19 Adolescent Symptom and Psychological Experience Questionnaire (CASPE). Data about demographic variables, educational characteristics and academic performance were also collected. A regression analysis was performed to determine predictors of emotional health. A total of 434 university students including females (63%) and males (37%) provided responses. One-third of students (33%) indicated that the COVID-19 pandemic and its resulting changes including online distance studies greatly influenced their daily lives in a negative way. The regression analysis demonstrated that female students and students with average academic performance had increased vulnerability to experience emotional signs of stress (p < 0.05). The factors ‘Not going to university’ and ‘Not having a routine life’ were significant predictors of stress responses (p < 0.01) and (p < 0.001) respectively. E-learning during the COVID-19 pandemic made it possible for students to complete their studies as per academic calendar; simultaneously, it increased the vulnerability to experience stress, particularly for female students and students with average academic performance. These findings imply that academic advising and counseling services should be more readily available during digital studies to support at risk students.