We consider a problem that arises in the field of frequency domain system identification. If a discrete-time system has an input-output relation Y (z) = G(z)U(z), with transfer function G, then the problem is to find a rational approximation Ĝn for G. The data given are measurements of input and output spectra in the frequency points zk: {U(zk), Y (zk)}k=1 together with some weight. The approxi...

This paper is devoted to analyze, inside the ∞-many possible bases of a Quantum Universal Enveloping Algebra Uq(g), those that can be considered as “more equal then others”, like orthonormal bases in the Euclidean spaces. The only possible element of selection has been found to be a privileged connection with the corresponding bialgebra. A new parameter z ∈ C –independent from the z := log(q) ∈...

In this paper, we first discuss about canonical dual of g<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font...

In this note I am writing out some of the material from Paul Halmos, Hilbert Space Problem Book, on shift operators. The reason I’m doing this is because shift operators are standard objects in operator theory and every analyst should know their properties and their spectra. A reference to Problem n of Halmos is a reference to Problem n in this book. An orthonormal basis for a Hilbert space is ...

We show that every frame for a Hilbert space H can be written as a (multiple of a) sum of three orthonormal bases for H. We next show that this result is best possible by including a result of N.J. Kalton: A frame can be represented as a linear combination of two orthonormal bases if and only if it is a Riesz basis. We further show that every frame can be written as a (multiple of a) sum of two...

This paper introduces a Quantum Correlation Matrix Memory (QCMM) and Enhanced QCMM (EQCMM), which are useful to work with quantum memories. A version of classical GramSchmidt orthogonalisation process in Dirac notation (called Quantum Orthogonalisation Process: QOP) is presented to convert a nonorthonormal quantum basis, i.e., a set of non-orthonormal quantum vectors (called qudits) to an ortho...

We prove that there does not exist an orthonormal basis {bn} for L(R) such that the sequences {μ(bn)}, {μ(c bn)}, and {∆(bn)∆( bn)} are bounded. A higher dimensional version of this result that involves generalized dispersions is also obtained. The main tool is a time-frequency localization inequality for orthonormal sequences in L(R). On the other hand, for d > 1 we construct a basis {bn} for ...

Let G0 be a connected unipotent group over a finite field Fq, and let G = G0 ⊗Fq Fq, equipped with the Frobenius endomorphism Frq : G −→ G. For every character sheaf M on G such that Frq M ∼= M , we prove that M comes from an irreducible perverse sheaf M0 on G0 such that M0 is pure of weight 0 (as an `-adic complex) and for each integer n ≥ 1 the “trace of Frobenius” function tM0⊗FqFqn on G0(Fq...

Square-integrable functions f ∈ L are those of length ‖f‖L2 = 〈f, f〉 L2 < ∞. A subset D ⊂ L is said to be dense, if any f ∈ L can be approximated by a sequence fn ∈ D. This means limn ‖f − fn‖L2 = 0. An orthonormal basis {Ωn} is a set of orthonormal vectors whose finite linear combinations are dense. A linear transformation T is continuous on L if ‖Tf‖L2 ≤ M‖f‖L2 for some constant M < ∞. A cont...

Exercise 1 Suppose that H = C3 and let V be the one-dimensional subspace spanned by |ψy = ?1/3(|0y+ ei/3|1y |2y). Provide an orthonormal basis for VK. Solution: We need to identify a pair of orthonormal states that are also orthogonal to |ψy. An easy first state to take is |ψK 1 y = 1 2 (|0y ei/3|1y). Any other vector of the form a(|0y+ ei/3|1y+ b|2ywill be orthogonal to |ψK 1 y. To also make i...