Full Müntz Theorem inLp[0, 1]

Operstein, V.

doi:10.1006/jath.1996.0039

Cited by 16 publications

(10 citation statements)

References 2 publications

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“…For example, in [9,11,12], Theorem 1 is generalized to C(A) where A ⊆ [0, ∞) is an arbitrary compact set of positive measure, and in [25], it is generalized to C(D) where D is the closed unit disc in the complex plane. In [10,22], Theorem 1 is generalized to L p [0, 1], 1 p ∞. Most recently, Erdélyi and Johnson [17] extended several of these results to L p (A), 0 < p < ∞.…”

Section: Introductionmentioning

confidence: 94%

“…Let { i } be a sequence with inf i i > 0. Then Müntz's theorem and Müntz polynomials have seen a renewed interest in recent years [6][7][8][9][10][11][12]17,19,22,25], especially by Borwein and Erdélyi. There are many generalizations and variations of the Müntz-Szász theorem in these papers. For example, in [9,11,12], Theorem 1 is generalized to C(A) where A ⊆ [0, ∞) is an arbitrary compact set of positive measure, and in [25], it is generalized to C(D) where D is the closed unit disc in the complex plane.…”

Section: Introductionmentioning

confidence: 99%

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Perturbations in Müntz's theorem

Spalsbury

2008

Journal of Approximation Theory

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Perturbations in Müntz's theorem

Spalsbury

2008

Journal of Approximation Theory

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“…Let f ∈ L p (I) and > 0 be arbitrary. From the full Müntz theorem in L p (I) (see [5,16]) it follows that for the sequence Λ :…”

Section: The Map T Is a Contraction And Possesses A Fixedmentioning

confidence: 99%

Associate fractal functions inLp-spaces and in one-sided uniform approximation

Viswanathan

Navascués

Chand

2016

Journal of Mathematical Analysis and Applications

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Fractal interpolation function defined with the aid of iterated function system can be employed to show that any continuous real-valued function defined on a compact interval is a special case of a class of fractal functions (self-referential functions). Elements of the iterated function system can be selected appropriately so that the corresponding fractal function enjoys certain properties. In the first part of the paper, we associate a class of self-referential L p -functions with a prescribed L p -function. Further, we apply our construction of fractal functions in L p -spaces in some approximation problems, for instance, to derive fractal versions of the full Müntz theorems in L p -spaces. The second part of the paper is devoted to identify parameters so that the fractal functions affiliated to a given continuous function satisfy certain conditions, which in turn facilitate them to find applications in some one-sided uniform approximation problems.

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“…Because for an a > 0 K(x+a) K(x) is positive, and it tends to zero, when x tends to − 1 2 , according to Lemma 1, we can suppose that |b(z)| > δ > 0 on ℜz ∈ −a, − 1 2 . Now, because |H * (z + a)| ≤ (CΨ(r)) x+a (x ≥ −a) (see (20)), we have that if C 1 is large enough, than according to (22)…”

Section: Proofmentioning

confidence: 99%

Müntz-type theorems on the half-line with weights

Horváth

2014

Journal of Mathematical Analysis and Applications

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We consider the linear span S of the functions t a k (with some a k > 0) in weighted L 2 spaces, with rather general weights. We give one necessary and one sufficient condition for S to be dense. Some comparisons are also made between the new results and those that can be deduced from older ones in the literature.This classical result was first proved in L 2 [0, 1] and then extended to C[0, 1], as stated above. Also, it was stated only for increasing sequences λ k . Subsequently, this theorem has had several different proofs and generalizations, and there are several surveys in this topic (see for instance the papers of J. Almira and A. Pinkus [1], [23]).On C[0, 1] and L p (0, 1), "full Müntz theorems", i.e. theorems with rather general exponents, were later proved by eg. P. Borwein, T. Erdélyi, W. B. Johnson and V. Operstein ([7], [13], [12], [22]).

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Full Müntz Theorem inLp[0, 1]

Cited by 16 publications

References 2 publications

Perturbations in Müntz's theorem

Perturbations in Müntz's theorem

Associate fractal functions inLp-spaces and in one-sided uniform approximation

Müntz-type theorems on the half-line with weights

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