Lower bounds for the truncated Hilbert transform

On the P - Norm of the Truncated Hilbert Transform

The p-norm of the Hilbert transform is the same as the p-norm of its truncation to any Lebesgue measurable set with strictly positive measure. This fact follows from two symmetry properties, the joint presence of which is essentially unique to the Hilbert transform. Our result applies, in particular, to the finite Hilbert transform taken over ( — 1,1), and to the one-sided Hilbert transform tak...

Spectral Analysis of the Truncated Hilbert Transform with Overlap

We study a restriction of the Hilbert transform as an operator HT from L (a2, a4) to L(a1, a3) for real numbers a1 < a2 < a3 < a4. The operator HT arises in tomographic reconstruction from limited data, more precisely in the method of differentiated back-projection (DBP). There, the reconstruction requires recovering a family of one-dimensional functions f supported on compact intervals [a2, a4...

Some lower bounds for the $L$-intersection number of graphs

‎For a set of non-negative integers~$L$‎, ‎the $L$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $A_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|A_u cap A_v|in L$‎. ‎The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the ver...

Some Lower Bounds for Estrada Index

Hilbert Transform and Gain/Phase Error Bounds for Rational Functions

It is well known that a function analytic in the right half plane can be constructed from its real part alone, or (modulo an additive constant) from its imaginary part alone via the Hilbert transform. It is also known that a stable minimum phase transfer function can be reconstructed from its gain alone, or (modulo a multiplicative constant) from its phase alone, via the Bode gain/phase relatio...