In this work, we prove the existence of a spectral function for singular q-Sturm-Liouville operator. Further, we establish a Parseval equality and expansion formula in eigenfunctions by terms of the spectral function.

In this study, we establish a Parseval equality and an expansion formula for a Sturm– Liouville operator on semi-unbounded time scales. AMS subject classification: 34L10.

• Pre-Hilbert spaces: definition • Cauchy-Schwarz-Bunyakowski inequality • Example: spaces ` • Triangle inequality, associated metric, continuity issues • Hilbert spaces, completions, infinite sums • Minimum principle • Orthogonal projections to closed subspaces • Orthogonal complements W⊥ • Riesz-Fischer theorem on linear functionals • Orthonormal sets, separability • Parseval equality, Bessel...

Contrary to the simple structure of tensor product quaternionic Hilbert space, octonionic situation becomes more involved. It turns out that an space can be decomposed as orthogonal direct sum two subspaces, each them isomorphic a irreducible with real space. As application, we find for given basis, Parseval equality holds if and only basis is weak associative. Fortunately, there always exists ...

Decomposition in a basis: (b j) j∈J basis of L 2 ([0, 1]). ∀f ∈ L 2 ([0, 1], f = j∈J f, b j b j with f, b j = 1 0 f (t)b j (t)dt. The Parseval-Bessel norm energy conservation equality yields f 2 = j∈J |f, b j | 2 1.2 Linear and non linear approximation For any I ∈ P(J), one defines f |I by f |I = j∈I f, b j b j. The question is now how to select " a good subset " of a given size M. A first poss...

In this paper we propose a general method to estimate periodic unknown input signals of finitedimensional linear time-varying systems. We present an infinite-dimensional observer that reconstructs the coefficients of the Fourier decomposition of such systems. Although the overall system is infinite dimensional, convergence of the observer can be proven using a standard Lyapunov approach along w...