Hardy variation

2020 Mathematics Subject Classification: Primary: 26B30 Secondary: 26A4526-XX [MSN][ZBL]

A generalization to functions of several variables of the Variation of a function of one variable, proposed by Hardy in [Har] (see also [Ha]). However the modern theory of functions of bounded variation uses a different generalization (see Function of bounded variation and Variation of a function). Therefore the Hardy variation is seldomly used nowadays.

Consider a rectangle $R:= [a_1, b_1]\times \ldots \times [a_n, b_n]\subset \mathbb R^n$ and a function $f:R\to \mathbb R$. We define \[ \Delta_{h_k} (f, x) := f (x_1, \ldots, x_k+ h_k, \ldots, x_n) - f(x_1, \ldots, x_k, \ldots x_n) \] and, recursively, \[ \Delta_{h_1h_2\ldots h_k} (f, x):= \Delta_{h_k} \left(\Delta_{h_1\ldots h_{k-1}} , x\right)\, . \] Consider next the collection $\Pi_k$ of finite ordered families $\pi_k$ of points $t_k^1 < t_k^2< \ldots < t_k^{N_k+1}\in [a_k, b_k]$. For each such $\pi_k$ we denote by $h^i_k$ the difference $t_k^{i+1}- t_k^i$.

Definition We define $\tilde{H}_n (f)$ as the supremum over $(\pi_1, \ldots, \pi_n)\in \Pi_1\times \ldots \times \Pi_n$ of the sums \[ \sum_{i_1=1}^{N_1} \ldots \sum_{i_n=1}^{N_n} \left|\Delta_{h^{i_1}_1\ldots h^{i_n}_n} \left(f, \left(x^{i_1}_1, \ldots x^{i_n}_n\right)\right)\right|\, . \] Finally, let $\alpha, \bar{\alpha}$ be pair of ordered subsets which gives a partition of $\{1, \ldots, x_n\}$. For each such pair and for each \[ (y_1, \ldots, y_s)\in [a_{\alpha_1}, b_{\alpha_1}]\times \ldots \times [a_{\alpha_s}, b_{\alpha_s}] \] we denote by $f^y_\alpha$ the function of $n-s$ variables $z_1, \ldots, z_{n-s}$ given by $f(x_1, \ldots, x_n)$ where $x_{\alpha_i} = y_i$ and $x_{\bar{\alpha}_j}= z_j$. The Hardy variation of $f$ is then given by \[ V_H (f) = \sup_{\alpha}\; \sup_y\; \tilde{H}_{n-s} \left(f^y_\alpha\right)\, . \] If $V_H (f)<\infty$, then one says that the function $f$ has bounded (finite) Hardy variation.

The original definition of Hardy considered the case $n=2$ and the author introduced it to generalize the Jordan criterion on the converge of Fourier series to Fourier double series. In particular in [Har] he proved the following

Theorem Assume a function $f:[0,2\pi]^2\to \mathbb R$ has finite Hardy variation. Then at every point $(x_1, x_2)$ the rectangular partial sums of the Fourier double series of $f$ converges to \[ \frac{1}{4}\left(f(x_1^+, x_2^+) + f (x_1^-, x_2^-) - f(x_1^+, x_2^-)- f(x_1^-, x_2^+)\right) \] (where \[ f (x_1^+, x_2^+) = \lim_{(y_1, y_2)\to 0, y_1>0, y_2>0} f(x_1+y_1, x_2+y_2) \] and the limits $f (x_1^-, x_2^-)$, $f(x_1^+, x_2^-)$, $f(x_1^-, x_2^+)$ are defined analogously).

A function $f$ has finite Hardy variation if and only if it can be written as the difference of two functions $f^+-f^-$ such that $\Delta_{h_1, \ldots, h_n} (f, x)\geq 0$ for any choice of nonnegative increments $h_1, \ldots, h_n$. This statement generalizes, therefore, the Jordan decomposition of a function of bounded variation of one real variable. If a function has bounded Hardy variation, then it also has necessarily bounded Arzelà variation.

References

[Har]	G.H. Hardy, "On double Fourier series and especially those which represent the double zeta-function with real and incommensurable parameters" Quarterly J. Math. , 37 (1905) pp. 53–79.
[Ha]	H. Hahn, "Theorie der reellen Funktionen" , 1 , Springer (1921).