The set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> of complex-valued continuous functions on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$[0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a ring for addition and convolution. It has the quotient field <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Q(\mathcal {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, by which J. Mikusiński developed his operational calculus. In this paper, we revisit a derivation and a transforming operator for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Q(\mathcal {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> discussed in his textbook, and define a new transforming operator related to the q-shift operator, which gives structures of a q-difference field and a difference field of Mahler type to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Q(\mathcal {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

Difference-differential fields of continuous functions

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2026年3月10日

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Seiji Nishioka

摘要

The set $\mathcal {C}$ $C$ of complex-valued continuous functions on $[0,\infty )$ $[0, \infty)$ is a ring for addition and convolution. It has the quotient field $Q(\mathcal {C})$ $Q (C)$ , by which J. Mikusiński developed his operational calculus. In this paper, we revisit a derivation and a transforming operator for $Q(\mathcal {C})$ $Q (C)$ discussed in his textbook, and define a new transforming operator related to the q-shift operator, which gives structures of a q-difference field and a difference field of Mahler type to $Q(\mathcal {C})$ $Q (C)$ .