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<p>In the space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(l_\infty (\mathbb Z^3,\mathbb C)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>l</mi> <mi>∞</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="double-struck">Z</mi> <mn>3</mn> </msup> <mo>,</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we consider the difference operator <Equation ID="Equ10"> <EquationSource Format="TEX">\(\begin{aligned} (\mathcal {A}x)_{n}=\sum _{k\in \mathbb {Z}^3}a_{k} x_{n-k}, \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow> <mi>k</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>3</mn> </msup> </mrow> </munder> <msub> <mi>a</mi> <mi>k</mi> </msub> <msub> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>which is invariant under the action of the group generated by rotations around the coordinate axes by the angle <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\pi /2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. The equality of the coefficients <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(a_k\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(k\in \mathbb Z^3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <msup> <mi mathvariant="double-struck">Z</mi> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, corresponding to the same orbit is established. A representation of the operator based on this property is proposed.</p>

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REPRESENTATION OF DIFFERENCE OPERATORS ON \(\mathbb {Z}^3\) THAT ARE INVARIANT UNDER ROTATIONS

  • 期刊论文

  • 发表日期：

    2026年2月1日

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  • E. G. Alipatov

摘要

In the space \(l_\infty (\mathbb Z^3,\mathbb C)\) l ∞ ( Z 3 , C ) , we consider the difference operator \(\begin{aligned} (\mathcal {A}x)_{n}=\sum _{k\in \mathbb {Z}^3}a_{k} x_{n-k}, \end{aligned}\) ( A x ) n = ∑ k ∈ Z 3 a k x n - k , which is invariant under the action of the group generated by rotations around the coordinate axes by the angle \(\pi /2\) π / 2 . The equality of the coefficients \(a_k\) a k , \(k\in \mathbb Z^3\) k ∈ Z 3 , corresponding to the same orbit is established. A representation of the operator based on this property is proposed.