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The Wave Equation in Lp-Spaces [PDF]
Semigroup Forum, 2005 In the first part of the paper, Gaussian estimates are used to study $L^p$-summability of the solution of the wave equation in $L^p(\Omega)$ associated with a general operator in divergence form with bounded coefficients. Secondly, we prove that if $\Omega$ is a cube in $\RR^N$, then the Laplacian with Dirichlet or Neumann boundary conditions generates
Valentin Keyantuo, Mahamadi Warma
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International Mathematics Research Notices, 2010
A representation theorem is established for continuous valuations on L p -spaces whose underlying measure is non-atomic. As a consequence, a complete classification is obtained of continuous translation invariant valuations on and of continuous rotation invariant valuations on L p (S n−1 ).2000 AMS subject classification: 46E30 (52B45)
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A representation theorem is established for continuous valuations on L p -spaces whose underlying measure is non-atomic. As a consequence, a complete classification is obtained of continuous translation invariant valuations on and of continuous rotation invariant valuations on L p (S n−1 ).2000 AMS subject classification: 46E30 (52B45)
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1998
The L p -spaces of functions the pth power of whose absolute value is integrable are introduced. Minkowski’s and Holder’s inequality are shown, and the L p -spaces are seen to be Banach spaces. Also, the approximation of Lp-functions by smooth ones is discussed. These mollifications are also used to show the existence of partitions of unity.
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The L p -spaces of functions the pth power of whose absolute value is integrable are introduced. Minkowski’s and Holder’s inequality are shown, and the L p -spaces are seen to be Banach spaces. Also, the approximation of Lp-functions by smooth ones is discussed. These mollifications are also used to show the existence of partitions of unity.
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1998
For (X, ℜ, μ) a positive measure space, it has already been noted that μ - a.e. equality is an equivalence relation, and the relation ≤ μ-a.e. a preorder, on.This section studies the structure of the equivalence classes into which μ-a,e. equality partitions.Since the set X/X( ℜ) is always u-null (2.7.7 a)), only the function values on the set X(ℜ) have
Corneliu Constantinescu +3 more
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For (X, ℜ, μ) a positive measure space, it has already been noted that μ - a.e. equality is an equivalence relation, and the relation ≤ μ-a.e. a preorder, on.This section studies the structure of the equivalence classes into which μ-a,e. equality partitions.Since the set X/X( ℜ) is always u-null (2.7.7 a)), only the function values on the set X(ℜ) have
Corneliu Constantinescu +3 more
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2002
Let { X,A,μ} be a measure space and let E be a measurable subset of X. A measurable function \(f:E \to {{\mathbb{R}}^{*}}\) is said to be in L P (E) for P≥1 if is integrable on E, i.e., if $$ \left\| f \right\|_p \mathop = \limits^{def} \left( {\smallint _E \left| f \right|^p d\mu } \right)^{1/p} < \infty $$ (1.1 ...
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Let { X,A,μ} be a measure space and let E be a measurable subset of X. A measurable function \(f:E \to {{\mathbb{R}}^{*}}\) is said to be in L P (E) for P≥1 if is integrable on E, i.e., if $$ \left\| f \right\|_p \mathop = \limits^{def} \left( {\smallint _E \left| f \right|^p d\mu } \right)^{1/p} < \infty $$ (1.1 ...
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, 1996 5.1 In this section we prove several fundamental inequalities. For example, if p, q, and r with 1/p + 1/q = 1/r belong to [0, +∞], and if f, g are two functions on Ω such that N p (f) and N q (g) are finite, then N r (fg) ≤ N p (f)N q (g) (Proposition 5.1.2).
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Fractals, 2012
The methodology of fractal sets generates new procedures for the analysis of functions whose graphs have a complex geometric structure. In the present paper, a method for the definition of fractal functions is described. The new mappings are perturbed versions of classical bases as Legendre polynomials, etc.
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The methodology of fractal sets generates new procedures for the analysis of functions whose graphs have a complex geometric structure. In the present paper, a method for the definition of fractal functions is described. The new mappings are perturbed versions of classical bases as Legendre polynomials, etc.
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On the LP space of observables
Fuzzy Sets and Systems, 1999The author studies the subspace \(L^p\) of observables in fuzzy set considerations. The main result states that the subspace \(L^p\) considered with a suitable pseudometric is complete.
Beloslav Riečan, Beloslav Riečan
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2018
The notion of convex function is more stringent than that of continuous function. It is the source of many interesting inequalities in real analysis.
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The notion of convex function is more stringent than that of continuous function. It is the source of many interesting inequalities in real analysis.
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1989
Before discussing an important special class of measurable functions we prove a fundamental inequality, known as Holder’s inequality. There are two variants, one for sums and one for integrals. The original variant for integrals of continuous functions or Riemann integrable functions was extended to measurable functions without additional difficulties.
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Before discussing an important special class of measurable functions we prove a fundamental inequality, known as Holder’s inequality. There are two variants, one for sums and one for integrals. The original variant for integrals of continuous functions or Riemann integrable functions was extended to measurable functions without additional difficulties.
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