Boundedness Of Linear Operators And Their Applications

The dissertation focuses on the boundedness and convergence of some linear operators. Also we gave some applications in determination of values on edges, and signal reconstruction. There are five chapters included.Chapter 1 is the introduction part. We introduced some basic concepts, func-tion spaces and relevant operators.In Chapter 2, local convergence was taken into account. We established the pointwise convergence of multiplier operators via a theorem by K. K. Chen about the convergence of orthogonal series. Compared with [24], we weakened the con-dition of the function f to get the summability of Fourier multiplier transform.In Chapter 3, we introduced the boundedness of Bochner-Riesz means in some function spaces and found the spectrum in those spaces. We established some results on the spectral invariance of Bochner-Riesz means. For bounded Bochner-Riesz operators on IP.1≤P<∞. its spectral on Lp is [0.1]. While for those unbounded Bocher-Riesz operators, its spectral is C On the other hand, those results provided a new alternative to establish the boundedness of Bochner-Riesz means on IP,1≤P< ∞.In Chapter 4, the derivative operators of convolution operators were taken into consideration. We used those operators to determine the value of jumps. And the convergence rate was given. It’s useful to the edge detection. Moreover, we improved the results in [77] about computing the jumps by conjugate convolution operators. We removed the condition’ the kernel of the convolution operator is evenIn Chapter 5, the multiresolution analysis in wavelet theory was introduced. For high-dimensional sparse wavelet signals which have sparse representation in the wavelet basis, we found a deterministic multi-level uniform sampling set so that signals could be recovered from their Fourier measurements on the sampling set via Prony’s method. Moreover, the cardinality of our sampling set is about a multiple of signal sparsity, independent on signal dimension.