Generalizing algebraically defined norms

We extend the algebraic construction of finite dimensional varying exponent $$L^{p(\cdot )}$$ L p ( · ) space norms, defined in terms of Cauchy polynomials to a more general setting, including varying exponent $$L^{p(\cdot )}$$ L p ( · ) spaces. This boils down to reformulating the Musielak–Orlicz or Nakano space norm in an algebraic fashion where the infimum appearing in the definition of the norm should become a (uniquely attained) minimum. The latter may easily fail, as turns out, and in this connection we examine the Fatou type semicontinuity conditions on the modulars. Norms defined by ODEs are applied in studying such semicontinuity properties of $$L^{p(\cdot )}$$ L p ( · ) space norms with $$p(\cdot )$$ p ( · ) unbounded.

Calculations on a Class of Candidate Solutions for the Distortion Problem on l 2

In 1994 E. Odell and Th. Schlumprecht showed that the Hilbert space (` 2 ) was arbitrarily distortable.However, we do not have a defined sequence of norms that would perform this distortion. G. An-droulakis and the author defined a class of norms equivalent to the Hilbert space norm. These normsare candidate sequences of norms that might distort ` 2 . These norms are recursively defined andcalculations with these norms are quite difficult. Here we present a tool, a Python program, thatcalculates the norms defined by G. Androulakis and the author.

Time and space adaptivity of the wave equation discretized in time by a second-order scheme

The aim of this paper is to obtain a posteriori error bounds of optimal order in time and space for the linear second-order wave equation discretized by the Newmark scheme in time and the finite element method in space. An error estimate is derived in the $L^{\infty }$-in-time/energy-in-space norm. Numerical experiments are reported for several test cases and confirm equivalence of the proposed estimator and the true error.

DIFFERENCE-DIFFERENCIAL GAME OF APPROACH AND EVASION IN HILBERT SPACE, I

In this paper the dynamic guidance-to-the-functional set game for conflict controlled differential system with time delay is studied. Fulfillment of conditions of saddle point relative to the right side of the system is not supposed to be. The proof is based on Hilbert space norm.

Multiresolution analysis for compactly supported interpolating tensor product wavelets

We construct multidimensional interpolating tensor product multiresolution analyses (MRA's) of the function spaces C0(ℝn, K), K = ℝ or K = ℂ, consisting of real or complex valued functions on ℝn vanishing at infinity and the function spaces Cu(ℝn, K) consisting of bounded and uniformly continuous functions on ℝn. We also construct an interpolating dual MRA for both of these spaces. The theory of the tensor products of Banach spaces is used. We generalize the Besov space norm equivalence from the one-dimensional case to our n-dimensional construction.