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Sharp discrete Hardy constants in low dimensions
Natanael Alpay
For , let denote the sharp constant in the nearest-neighbor Hardy inequality on with Euclidean inverse-square weight for functions vanishing at the origi…
The sharp discrete Hardy inequality on
Natanael Alpay
We determine the sharp constant in the nearest-neighbor Hardy inequality on with the Euclidean inverse-square weight. For every finitely supported function $u:\Z^3\to\C$, we…
Sharp Ternary Martingale Isoperimetry and -adic Takagi-Type Lower Bounds
Natanael Alpay
Let be the one-variation associated with the regular -adic martingale filtration on . We study the martingale isoperimetric profile \[ V_n(x):= \inf_{\substack{A\su…
Complex Stochastic Gradient Descent and Directional Bias in Reproducing Kernel Hilbert Spaces
Natanael Alpay, Emeric Battaglia
Stochastic Gradient Descent (SGD) is a known stochastic iterative method popular for large-scale convex optimization problems due to its simple implementation and scalability. Some…
Representer Theorem in Complex Reproducing Kernel Hilbert Spaces with Applications to Fock and Hardy Spaces and Superoscillations
Natanael Alpay, Antonino De Martino, Kamal Diki
We introduce a complex-valued counterpart of the representer theorem in machine learning. We study several learning and minimization problems in reproducing kernel Hilbert spaces (…
Fractional Supershifts and their associated Cauchy Evolution problems
Natanael Alpay
In this work, we extend the notion of supershifts and superoscillation sequence to fractional Fock spaces based on Gelfond-Leontiev fractional derivatives. We first introduce the f…