K. K. Pandey, S. Verma, and P. Viswanathan
Additive Decomposition of Vector-Valued Continuous Functions with Some Dimensional Considerations of the Summands
Dynamic Systems and Applications 32 (2023) 351-368
https://doi.org/10.46719/dsa2023.32.19
ABSTRACT.
Additive decomposition of a continuous real-valued function on the unit interval in the framework of fractal dimensions of the graphs of the summands has received significant attention recently. This is intimately connected with a rather old problem concerning the fractal
dimensions of the graph of a generic continuous function. The primary objective of the current note is to revisit the aforementioned results on the decomposition of a continuous real-valued function and provide certain aspects of suitable vector-valued analogues. We show, for instance, that a continuous function f : [0, 1] → Rn with a suitable choice of β ∈ [1, n + 1] can be decomposed as the sum of two continuous functions such that their graphs have the Hausdorff dimension β. Similar results regarding decompositions of a continuous vector-valued function in the light of packing dimension and box dimension of the graphs of the summands are indicated. Along the way, some elementary properties of the set-valued maps that arise in connection with the additive decomposition are also provided.
AMS (MOS) Subject Classification. 28A78, 26A30, 46E15.
Key Words and Phrases. Prevalent, Generic, Hausdorff dimension, Decomposition, Multivalued map.