Lebesgue Integrable Functions and Sets

2003 ◽  
pp. 183-193
Author(s):  
Jürgen Jost
Keyword(s):  
Integrable Functions

ON ABSOLUTELY HENSTOCK INTEGRABLE FUNCTIONS

1986 ◽  
Vol 12 (2) ◽  
pp. 524
Author(s):  
Chuan-Song ◽  
Peng-Yee
Keyword(s):  
Integrable Functions

EQUIINTEGRABILITY AND CONTROLLED CONVERGENCE OF PERRON-TYPE INTEGRABLE FUNCTIONS

1991 ◽  
Vol 17 (1) ◽  
pp. 110 ◽  
Author(s):  
Kurzweil ◽  
Jarník
Keyword(s):  
Integrable Functions

scholarly journals A New Upper Bound for Sampling Numbers

2021 ◽  
Author(s):  
Nicolas Nagel ◽  
Martin Schäfer ◽  
Tino Ullrich
Keyword(s):  
Upper Bound ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Sampling Procedure ◽  
Compact Embedding ◽  
Recovery Method ◽  
Grid Sampling ◽  
Integrable Functions ◽  
Random Draw ◽  
Square Integrable Functions

AbstractWe provide a new upper bound for sampling numbers $$(g_n)_{n\in \mathbb {N}}$$ ( g n ) n ∈ N associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants $$C,c>0$$ C , c > 0 (which are specified in the paper) such that $$\begin{aligned} g^2_n \le \frac{C\log (n)}{n}\sum \limits _{k\ge \lfloor cn \rfloor } \sigma _k^2,\quad n\ge 2, \end{aligned}$$ g n 2 ≤ C log ( n ) n ∑ k ≥ ⌊ c n ⌋ σ k 2 , n ≥ 2 , where $$(\sigma _k)_{k\in \mathbb {N}}$$ ( σ k ) k ∈ N is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding $$\mathrm {Id}:H(K) \rightarrow L_2(D,\varrho _D)$$ Id : H ( K ) → L 2 ( D , ϱ D ) . The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of $$H^s_{\text {mix}}(\mathbb {T}^d)$$ H mix s ( T d ) in $$L_2(\mathbb {T}^d)$$ L 2 ( T d ) with $$s>1/2$$ s > 1 / 2 . We obtain the asymptotic bound $$\begin{aligned} g_n \le C_{s,d}n^{-s}\log (n)^{(d-1)s+1/2}, \end{aligned}$$ g n ≤ C s , d n - s log ( n ) ( d - 1 ) s + 1 / 2 , which improves on very recent results by shortening the gap between upper and lower bound to $$\sqrt{\log (n)}$$ log ( n ) . The result implies that for dimensions $$d>2$$ d > 2 any sparse grid sampling recovery method does not perform asymptotically optimal.

scholarly journals Certain fundamental properties of generalized natural transform in generalized spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shrideh Khalaf Al-Omari ◽  
Serkan Araci
Keyword(s):  
Inversion Formula ◽  
Acta Math ◽  
Generalized Functions ◽  
General Nature ◽  
Qualitative Behavior ◽  
Fundamental Properties ◽  
Integrable Functions ◽  
Extended Operator ◽  
Definition Of ◽  
Space Of Integrable Functions

AbstractThis paper considers the definition and the properties of the generalized natural transform on sets of generalized functions. Convolution products, convolution theorems, and spaces of Boehmians are described in a form of auxiliary results. The constructed spaces of Boehmians are achieved and fulfilled by pursuing a deep analysis on a set of delta sequences and axioms which have mitigated the construction of the generalized spaces. Such results are exploited in emphasizing the virtual definition of the generalized natural transform on the addressed sets of Boehmians. The constructed spaces, inspired from their general nature, generalize the space of integrable functions of Srivastava et al. (Acta Math. Sci. 35B:1386–1400, 2015) and, subsequently, the extended operator with its good qualitative behavior generalizes the classical natural transform. Various continuous embeddings of potential interests are introduced and discussed between the space of integrable functions and the space of integrable Boehmians. On another aspect as well, several characteristics of the extended operator and its inversion formula are discussed.

Type-2 computability on spaces of integrable functions

2004 ◽  
Vol 50 (45) ◽  
pp. 417-430 ◽  
Author(s):  
Daren Kunkle
Keyword(s):  
Integrable Functions

scholarly journals A generalization of the Itô formula

2002 ◽  
Vol 31 (8) ◽  
pp. 477-496
Author(s):  
Said Ngobi
Keyword(s):  
Sobolev Space ◽  
White Noise ◽  
Itô Formula ◽  
Ito Formula ◽  
Integrable Functions ◽  
Square Integrable Functions ◽  
White Noise Space

The classical Itô formula is generalized to some anticipating processes. The processes we consider are in a Sobolev space which is a subset of the space of square integrable functions over a white noise space. The proof of the result uses white noise techniques.

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