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K. Mahesh Krishna

Publications and source records attributed to K. Mahesh Krishna.

At least 19 recordsLinked to original sources
arXiv2026

A Functional Version of the Sparsity Theorem

Celebrated breakthrough sparsity theorem obtained independently by Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]} and Gribonval and Nielsen \textit{[IEEE Trans. Inform. Theory, 2003]} and Fuchs \textit{[IEEE Trans. Inform. Theory, 2004]} says that unique sparse solution to NP-Hard $\ell_0$-minimization problem can be obtained using unique solution to P-Type $\ell_1$-minimization problem. In this paper, we extend their result to abstract Banach spaces. We notice that the `normalized' condition for Hilbert spaces can be generalized to a larger extent when we consider Banach spaces.

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arXiv2026

p-adic Heisenberg-Robertson-Schrodinger and p-adic Maccone-Pati Uncertainty Principles

Let $\mathcal{X}$ be a p-adic Hilbert space over a conjugated non-Archimedean valued field $\mathbb{K}$ with $2\neq0$. Let $A: \mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X}$ and $B: \mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X}$ be possibly unbounded self-adjoint linear operators. For $x \in \mathcal{D}(A)$ with $\langle x, x \rangle =1$, define $ Δ_x(A):= \|Ax- \langle Ax, x \rangle x \|.$ Then for all $x \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\langle x, x \rangle =1$, we show that \begin{align*} (1) \quad Δ_x(A)+Δ_x(B)\geq \max\{Δ_x(A), Δ_x(B)\}\geq \frac{\sqrt{\bigg|\big\langle [A,B]x, x \big\rangle ^2+\big(\langle \{A,B\}x, x \rangle -2\langle Ax, x \rangle\langle Bx, x \rangle\big)^2\bigg|}}{\sqrt{|2|}} \end{align*} and \begin{align*} (2) \quad Δ_x(A)+Δ_x(B)\geq \max\{Δ_x(A), Δ_x(B)\} \geq |\langle (A+B)x, y \rangle |, \quad \forall y \in \mathcal{X} \text{ satisfying } \|y\|\leq 1, \langle x, y \rangle =0. \end{align*} We call Inequality (1) as p-adic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as p-adic Maccone-Pati uncertainty principle.

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arXiv2026

Non-Archimedean Cauchy-Schwarz Angle-Length and Chebyshev Arithmetic Mean Inequalities

Let $\mathbb{K}$ be a non-Archimedean valued field. Let $n \in \mathbb{N}$. For every $(a_j)_{j=1}^n, (b_j)_{j=1}^n \in \mathbb{K}^n$, we show that \begin{align*} \left|\sum_{j=1}^{n}a_jb_j\right|^2\leq \max\left\{\left|\sum_{j=1}^n a_j^2\right|\left|\sum_{k=1}^nb^2_k\right|, \max_{1\leq j<k \leq n}|a_jb_k-a_kb_j|^2\right\}, \end{align*} \begin{align*} \left|\sum_{j=1}^n a_j^2\right|\left|\sum_{k=1}^nb^2_k\right|\leq \max\left\{\left|\sum_{j=1}^n a_jb_j\right|^2, \max_{1\leq j<k \leq n}|a_jb_k-a_kb_j|^2\right\}, \end{align*} \begin{align*} \left|AM(a_j)_{j=1}^n\right|\left|AM(b_j)_{j=1}^n\right|\leq \max\left\{ \left|AM(a_jb_j)_{j=1}^n\right|, \frac{1}{|n|^2}\max_{1\leq j < k \leq n}|a_j-a_k||b_j-b_k|\right\}, \end{align*} \begin{align*} \left|AM(a_jb_j)_{j=1}^n\right|\leq \max\left\{ \left|AM(a_j)_{j=1}^n\right|\left|AM(b_j)_{j=1}^n\right|, \frac{1}{|n|^2}\max_{1\leq j < k \leq n}|a_j-a_k||b_j-b_k|\right\}, \end{align*} where $AM$ denotes the arithmetic mean. First and second are non-Archimedean versions of Cauchy-Schwarz angle-length and third and fourth are Chebyshev arithmetic mean inequalities. Unlike in the Archimedean case, no conditions are required to derive non-Archimedean Chebyshev arithmetic mean inequalities.

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arXiv2026

Non-Archimedean Massera-Schaffer-Maligranda-Pecaric-Rajic Inequality

Massera and Schaffer [\textit{Ann. Math. (2), 1958}] derived a breakthrough upper bound for the Clarkson angle between two nonzero vectors in a normed linear space, which was later improved by Maligranda [\textit{Am. Math. Mon., 2006}]. Pecaric and Rajic [\textit{Math. Inequal. Appl., 2007}] extended Maligranda's inequality to finitely many nonzero vectors. We derive a non-Archimedean version of Massera-Schaffer-Maligranda-Pecaric-Rajic inequality.

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arXiv2026

Non-Archimedean Tarski-Maligranda Inequalities

In 1930, Tarski observed that \begin{align*} \bigg||r|-|s|\bigg|=|r-s|+ |r+s|-(|r|+|s|), \quad \forall r, s \in \mathbb{R}. \end{align*} In 2008, Maligranda converted the previous equality into inequalities that are valid in every normed linear space. We derive non-Archimedean versions of Tarski-Maligranda inequalities. Difference between Archimedean and non-Archimedean inequalities is surprising.

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arXiv2026

Finite Field Tarski-Maligranda Inequalities

Let $\mathbb{F}$ be a sub-modulus field such that $2 \neq 0$. Let $\mathcal{X}$ be a sub-normed linear space over $\mathbb{F}$. Then we show that \begin{align*} \bigg|\|x\|-\|y\|\bigg|\leq \frac{2}{|2|}\|x+y\|+\frac{2}{|2|}\max\{\|x-y\|, \|y-x\|\}-(\|x\|+\|y\|) \end{align*} and \begin{align*} \bigg|\|x\|-\|y\|\bigg|\leq \|x\|+\|y\|-\frac{2}{|2|}\|x+y\|+\frac{2}{|2|}\max\{\|y-x\|, \|x-y\|\}. \end{align*} Above inequalities are finite field versions of important Tarski-Maligranda inequalities obained by Maligranda [\textit{Banach J. Math. Anal., 2008}].

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arXiv2026

Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle

Let $(\{f_j\}_{j=1}^n, \{τ_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^m, \{ω_k\}_{k=1}^m)$ be p-Schauder frames for a finite dimensional Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align} (1) \quad \|θ_f x\|_0^\frac{1}{p}\|θ_g x\|_0^\frac{1}{q} \geq \frac{1}{\displaystyle\max_{1\leq j\leq n, 1\leq k\leq m}|f_j(ω_k)|}\quad \text{and} \quad \|θ_g x\|_0^\frac{1}{p}\|θ_f x\|_0^\frac{1}{q}\geq \frac{1}{\displaystyle\max_{1\leq j\leq n, 1\leq k\leq m}|g_k(τ_j)|}. \end{align} where \begin{align*} θ_f: \mathcal{X} \ni x \mapsto (f_j(x) )_{j=1}^n \in \ell^p([n]); \quad θ_g: \mathcal{X} \ni x \mapsto (g_k(x) )_{k=1}^m \in \ell^p([m]) \end{align*} and $q$ is the conjugate index of $p$. We call Inequality (1) as \textbf{Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle}. Inequality (1) improves Ricaud-Torrésani uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2013]}. In particular, it improves Elad-Bruckstein uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2002]} and Donoho-Stark uncertainty principle \textit{[SIAM J. Appl. Math., 1989]}.

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arXiv2026

Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty Principle

We derive an uncertainty principle for Lipschitz maps acting on subsets of Banach spaces. We show that this nonlinear uncertainty principle reduces to the Heisenberg-Robertson-Schrodinger uncertainty principle for linear operators acting on Hilbert spaces.

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arXiv2026

p-adic Ghobber-Jaming Uncertainty Principle

Let $\{τ_j\}_{j=1}^n$ and $\{ω_k\}_{k=1}^n$ be two orthonormal bases for a finite dimensional p-adic Hilbert space $\mathcal{X}$. Let $M,N\subseteq \{1, \dots, n\}$ be such that \begin{align*} \displaystyle \max_{j \in M, k \in N}|\langle τ_j, ω_k \rangle|<1, \end{align*} where $o(M)$ is the cardinality of $M$. Then for all $x \in \mathcal{X}$, we show that \begin{align} (1) \quad \quad \quad \quad \|x\|\leq \left(\frac{1}{1-\displaystyle \max_{j \in M, k \in N}|\langle τ_j, ω_k \rangle|}\right)\max\left\{\displaystyle \max_{j \in M^c}|\langle x, τ_j\rangle |, \displaystyle \max_{k \in N^c}|\langle x, ω_k\rangle |\right\}. \end{align} We call Inequality (1) as \textbf{p-adic Ghobber-Jaming Uncertainty Principle}. Inequality (1) is the p-adic version of uncertainty principle obtained by Ghobber and Jaming \textit{[Linear Algebra Appl., 2011]}. We also derive analogues of Inequality (1) for non-Archimedean Banach spaces.

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arXiv2026

Noncommutative Spherical Codes

Spherical codes, with a rich history spanning nearly five centuries, remain an area of active mathematical exploration and are far from being fully understood. These codes, which arise naturally in problems of geometry, combinatorics, and information theory, continue to challenge researchers with their intricate structure and unresolved questions. Inspired by Polya's heuristic principle of "vary the problem," we extend the classical framework by introducing the notion of noncommutative spherical codes, with particular emphasis on the noncommutative Newton-Gregory kissing number problem. This generalization moves beyond the traditional Euclidean setting into the realm of operator algebras and Hilbert C*-modules, thereby opening new avenues of investigation. A cornerstone in the study of spherical codes is the celebrated Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein linear programming bound, developed over the past half-century. This bound employs Gegenbauer polynomials to establish sharp upper limits on the size of spherical codes, and it has served as a fundamental tool in coding theory and discrete geometry. Remarkably, a recent elegant one-line proof by Pfender [\textit{J. Combin. Theory Ser. A, 2007}] provides a streamlined derivation of a variant of this bound. We demonstrate that Pfender's argument can be extended naturally to the setting of Hilbert C*-modules, thereby enriching the theory with noncommutative analogues.

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arXiv2026

Non-Archimedean Brauer Oval (of Cassini) Theorem and Applications

Nica and Sprague [\textit{Am. Math. Mon., 2023}] derived a non-Archimedean version of the Gershgorin disk theorem. We derive a non-Archimedean version of the oval (of Cassini) theorem by Brauer [\textit{Duke Math. J., 1947}] which generalizes the Nica-Sprague disk theorem. We provide applications for bounding the zeros of polynomials over non-Archimedean fields. We also show that our result is equivalent to the non-Archimedean version of the Ostrowski nonsingularity theorem derived by Li and Li [\textit{J. Comput. Appl. Math., 2025}].

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arXiv2025

Continuous Donoho-Elad Spark Uncertainty Principle

Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]} introduced the important notion of the spark of a frame, using which they derived a fundamental uncertainty principle. Based on spark, they also provided a necessary and sufficient condition for the uniqueness of sparse solutions to the NP-hard $\ell_0$-minimization problem. In this nano note, we show that the notion of spark can be extended to linear maps whose domains are measure spaces. Using this generalization, we derive an uncertainty principle and provide a sufficient condition for the existence of sparse solutions to linear systems on measure spaces.

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arXiv2025

Functional Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender Bound

Pfender \textit{[J. Combin. Theory Ser. A, 2007]} provided a one-line proof for a variant of the Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein upper bound for spherical codes, which offers an upper bound for the celebrated (Newton-Gregory) kissing number problem. Motivated by this proof, we introduce the notion of codes in pointed metric spaces (in particular on Banach spaces) and derive a nonlinear (functional) Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender upper bound for spherical codes. We also introduce nonlinear (functional) Kissing Number Problem.

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arXiv2025

p-adic Equiangular Lines and p-adic van Lint-Seidel Relative Bound

We introduce the notion of p-adic equiangular lines and derive the first fundamental relation between common angle, dimension of the space and the number of lines. More precisely, we show that if $\{τ_j\}_{j=1}^n$ is p-adic $γ$-equiangular lines in $\mathbb{Q}^d_p$, then \begin{align*} (1) \quad\quad \quad \quad |n|^2\leq |d|\max\{|n|, γ^2 \}. \end{align*} We call Inequality (1) as the p-adic van Lint-Seidel relative bound. We believe that this complements fundamental van Lint-Seidel \textit{[Indag. Math., 1966]} relative bound for equiangular lines in the p-adic case.

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