On an analogue of Hardy's inequality for the Walsh-Fourier

Abstract

Bi-lateral force measurements on the supporting limbs in postural sway while standing still were carried out to evaluate postural stability of craniocerebral injured (CCI) patients. Brain pathology of these patients was characterised by CT scans and MRI, as well as by their cognitive and behavioural disturbances. Normal subjects of the same age group were tested as controls. From the force tracings obtained, three oscillation frequencies were identified, with orders of magnitudes of 6, 1.5 and 0.1 Hz, respectively, of which the middle frequency, i.e. that corresponding to 1.5 Hz, was selected for subsequent processing and analysis, which included the determination of relative sequence of the force vectors on both feet and evaluation of timings and amplitudes of the waveforms. Weight-bearing imbalance was defined in the vertical direction to express the difference between the average forces supported by each of the legs. In the horizontal plane, two parameters were defined: Sway Total Activity (sii), to represent the vector summation of the absolute values of the horizontal force components acting on both legs; and Asymmetry (ASYM), to express the difference in activities between the two legs. The results presented disclose the reactive force patterns acting on each of the legs of CCI individuals, in comparison with normal individuals. Although the'forces were shown to act synchronously on both legs, in most of the patients, they appeared to be asymmetrical in nature, with a typical vectorial pattern for every individual, which generally differed from that of normal subjects. Sway activity was found to be significantly higher in the CCI group as compared with that of the normal controls. These results justify the necessity to study the activity of each of the legs, rather than that corresponding to the resultant forces acting on the whole body, when treating the external forces involved in the stabilisation and regulation of the standing posture of these patients. An attempt to grade the patients according to ascending order of neurological deficits disclosed that their sway disturbances, as revealed by the magnirude of sway total activity and asymmetry, had a tendency to be related to the severity of neurobehavioural disturbances.

Модифицированные Операторы Харди И Харди - Литтлвуда И Их Поведение В Различных Пространствах

“…В § 4 методами теории приближений исследуется поведение модифицированных операторов Харди и Харди-Литтлвуда в пространствах Гёльдера-Бесова. Наконец, в § 5 обобщается тождество типа Титчмарша-Голубова в случае распределений, в том числе функций из L p (R + ), 2 < p < ∞, и дается оценка преобразования Фурье функций из пространств Харди, обобщающая результаты из [8], [9].…”