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The inner periodic structure of a function. (English) Zbl 0973.11087

Summary: Let \(L\subset \mathbb R^n\) be a point-lattice of dimension \(r,1\leq r\leq n\), let \(f:\mathbb R^n\to \mathbb R^1\) be a bounded real valued function vanishing outside of a bounded set and put supp\((f):=\{x\in \mathbb R^n:f(x)\neq 0\}\). In the paper periodic properties of \(f\) with respect to the lattice \(L\) are investigated. For this aim two new concepts are introduced: a special decomposition of the set supp\((f)\) defined by \(L\) and periodic extendability of \(f\) to the whole space, respectively. Connections among these two concepts as well as several characterizations of them are proved. The characterizations are of two types. The first uses the set of \(u\in L\) contained in the algebraic difference of supp\((f)\) with itself and a resticted “almost everywhere” form of this set. The second type characterizations are exact conditions of equalities in inequalities among the \(L_1\)-norm of \(f\), sums of some other integrals and some special “Fourier-type” series, respectively, defined by \(L\) and \(f\).

MSC:

11P21 Lattice points in specified regions
26D20 Other analytical inequalities
42B99 Harmonic analysis in several variables
11H06 Lattices and convex bodies (number-theoretic aspects)
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