Temukan di Perpustakaan

We consider the problem of finding a (non-negative) measure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">B</mi><mo>(</mo><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup><mo>)</mo></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∫</mo><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup></msub><msup><mi mathvariant="bold">z</mi><mi mathvariant="bold">k</mi></msup><mi>d</mi><mi>μ</mi><mrow><mo>(</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>s</mi><mi mathvariant="bold">k</mi></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi mathvariant="bold">k</mi><mo>∈</mo><mi mathvariant="script">K</mi></mrow></semantics></math></inline-formula>. Here, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">K</mi></semantics></math></inline-formula> is an arbitrary finite subset of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">Z</mi><mo>+</mo><mi>n</mi></msubsup></semantics></math></inline-formula>, which contains <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>s</mi><mi mathvariant="bold">k</mi></msub></semantics></math></inline-formula> are prescribed complex numbers (we use the usual notations for multi-indices). There are two possible interpretations of this problem. Firstly, one may consider this problem as an extension of the truncated multidimensional moment problem on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, where the support of the measure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> is allowed to lie in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup></semantics></math></inline-formula>. Secondly, the moment problem is a particular case of the truncated moment problem in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup></semantics></math></inline-formula>, with special truncations. We give simple conditions for the solvability of the above moment problem. As a corollary, we have an integral representation with a non-negative measure for linear functionals on some linear subspaces of polynomials.