β Scribed by Richard F. Patterson
No coin nor oath required. For personal study only.
In 1900 Pringsheim presented the following notion of convergence for double sequences: a double sequence [x] converges to L provided that given Ο΅ > 0 there exists K β N such that |x k,l -L| < Ο΅ whenever k, l > K . Using this definition Hamilton, in 1926 and, respectively, presented the following definition for the regularity of four dimensional matrices. A four dimensional matrix A is regular if it maps every bounded convergent sequence into a convergent sequence with the same limit. These notions shall be used to present simple conditions to ensure that T m,n = β m,n k,l=0,0 a m,n,k,l x k,l and Ο m,n = β β,β k,l=0,0 a m,n,k,l x k,l are included by convergence.
π SIMILAR VOLUMES
Let Ο be a rapidly decreasing one-dimensional wavelet. We show that the wavelet expansion of any L p function converges pointwise almost everywhere under the wavelet projection, hard sampling, and soft sampling summation methods, for 1 < p < β. In fact, the partial sums are uniformly dominated by th
## Abstract Inclusion, convexity and Tauberian theorems are proved for certain generalized NΓΆrlund methods and power series methods applied to double sequences. Families of summability methods are developed which form a hierarchy of methods and can be used to connect matrix and power series methods