| Summary: | We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo>{</mo> <msub> <mi mathvariant="script">F</mi> <mi>k</mi> </msub> <mo>}</mo> </mrow> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </semantics> </math> </inline-formula> where each <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">F</mi> <mi>k</mi> </msub> </semantics> </math> </inline-formula> maps <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">H</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> <mo>→</mo> <mi mathvariant="script">H</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and arises from an iterated function system. Employing the recently-developed theory of non-stationary versions of fixed points and the concept of forward and backward trajectories, we present new classes of fractal functions exhibiting different local and global behavior and extend fractal interpolation to this new, more flexible setting.
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