scholarly journals Infinite matrices, wavelet coefficients and frames

2004 ◽  
Vol 2004 (67) ◽  
pp. 3695-3702
Author(s):  
N. A. Sheikh ◽  
M. Mursaleen
Keyword(s):  
Series Expansion ◽  
Infinite Matrix ◽  
Wavelet Coefficients ◽  
Infinite Matrices ◽  
Frame Condition ◽  
Wavelet Series

We study the action ofAonf∈L2(ℝ)and on its wavelet coefficients, whereA=(almjk)lmjkis a double infinite matrix. We find the frame condition forA-transform off∈L2(ℝ)whose wavelet series expansion is known.

scholarly journals Matrix Transformations between Certain Sequence Spaces over the Non-Newtonian Complex Field

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Uğur Kadak ◽  
Hakan Efe
Keyword(s):  
Linear Operator ◽  
Sufficient Conditions ◽  
Sequence Spaces ◽  
General Linear ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
Complex Field ◽  
Infinite Matrices ◽  
The Matrix ◽  
Necessary And Sufficient

In some cases, the most general linear operator between two sequence spaces is given by an infinite matrix. So the theory of matrix transformations has always been of great interest in the study of sequence spaces. In the present paper, we introduce the matrix transformations in sequence spaces over the fieldC*and characterize some classes of infinite matrices with respect to the non-Newtonian calculus. Also we give the necessary and sufficient conditions on an infinite matrix transforming one of the classical sets overC*to another one. Furthermore, the concept for sequence-to-sequence and series-to-series methods of summability is given with some illustrated examples.

scholarly journals Zeros and Periodicity of Functions of Infinite Matrices

1959 ◽  
Vol 11 (4) ◽  
pp. 225-229
Author(s):  
Shafik Asaad Ibrahim
Keyword(s):  
Power Series ◽  
Infinite Matrix ◽  
Infinite Matrices ◽  
Image Position ◽  
Matrix Period

Certain functions of infinite matrices are known to exist.† This gives rise to the following questions:1. Whether the power series of matriceshas a zero in the field ‡ of infinite matrices, and2. If f(A) exists for a certain infinite matrix A, is there an infinite matrix B such thatIn other words, is there a matrix period for f(A)?In this paper theorems concerning zeros and periodicity of functions of semi block infinite matrices § (defined below) are established.

scholarly journals Vector-valued sequence spaces generated by infinite matrices

2001 ◽  
Vol 26 (9) ◽  
pp. 547-560
Author(s):  
Nandita Rath
Keyword(s):  
Sequence Spaces ◽  
Infinite Matrix ◽  
Real Sequence ◽  
Normed Spaces ◽  
Infinite Matrices ◽  
Positive Real ◽  
The Matrix ◽  
Almost All ◽  
Vector Valued

LetA=(ank)be an infinite matrix with allank≥0andPa bounded, positive real sequence. For normed spacesEandEkthe matrixAgenerates paranormed sequence spaces such as[A,P]∞((Ek)),[A,P]0((Ek)), and[A,P](E)which generalize almost all the existing sequence spaces, such asl∞,c0,c,lp,wp, and several others. In this paper, conditions under which these three paranormed spaces are separable, complete, andr-convex, are established.

scholarly journals Generating Functions Attached to some Infinite Matrices

10.37236/492 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Paul Monsky
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Infinite Matrix ◽  
Infinite Matrices ◽  
Positive Integers

Let $V$ be an infinite matrix with rows and columns indexed by the positive integers, and entries in a field $F$. Suppose that $v_{i,j}$ only depends on $i-j$ and is 0 for $|i-j|$ large. Then $V^{n}$ is defined for all $n$, and one has a "generating function" $G=\sum a_{1,1}(V^{n})z^{n}$. Ira Gessel has shown that $G$ is algebraic over $F(z)$. We extend his result, allowing $v_{i,j}$ for fixed $i-j$ to be eventually periodic in $i$ rather than constant. This result and some variants of it that we prove will have applications to Hilbert-Kunz theory.

Spectral properties of diagonally dominant infinite matrices, part I

1989 ◽  
Vol 111 (3-4) ◽  
pp. 301-314 ◽  
Author(s):  
F. O. Farid ◽  
P. Lancaster
Keyword(s):  
Spectral Properties ◽  
Matrix Operator ◽  
Infinite Matrix ◽  
Constructive Approach ◽  
Infinite Matrices ◽  
Approximating Sequence ◽  
Diagonally Dominant

SynopsisTheorems of Gersgorin-type are established for a diagonally dominant, unbounded, infinite matrix operator A acting on lp for some l ≦p≦∞. The results are established using an approximating sequence of infinite matrices An that converges to A in the generalised sense as n → ∞. This constructive approach admits approximation of the spectral properties of A by those of An.

scholarly journals Nonnegative Infinite Matrices that Preserve (p,q)-Convexity of Sequences

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Chikkanna R. Selvaraj ◽  
Suguna Selvaraj
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
Infinite Matrices ◽  
Necessary And Sufficient

This paper deals with matrix transformations that preserve the (p,q)-convexity of sequences. The main result gives the necessary and sufficient conditions for a nonnegative infinite matrix A to preserve the (p,q)-convexity of sequences. Further, we give examples of such matrices for different values of p and q.

ON AN EFFICIENT SPARSE REPRESENTATION OF OBJECTS OF GENERAL SHAPE VIA CONTINUOUS EXTENSION AND WAVELET APPROXIMATION

2010 ◽  
Vol 08 (02) ◽  
pp. 253-269 ◽  
Author(s):  
NAOKI SAITO ◽  
ZHIHUA ZHANG
Keyword(s):  
Sparse Representation ◽  
Continuous Extension ◽  
Wavelet Coefficients ◽  
General Shape ◽  
General Domain ◽  
Wavelet Approximation ◽  
Internal Information ◽  
The Moment ◽  
Wavelet Series

In this paper, we discuss the continuous extension and wavelet approximation of the detected object on a general domain Ω of ℝ2. We first extend continuously the image to a square T such that it vanishes on the boundary ∂T. On T∖Ω, the extension has a simple and clear representation which is determined by the equation of the boundary ∂Ω. We expand the extension into wavelet series on ℝ2. Since the extension tools are polynomials, by the moment theorem, we know that the sequence of wavelet coefficients obtained by us is sparse. Therefore, we can approximate and analyze the internal information of the object very well even if we only store a few wavelet coefficients.

scholarly journals Primitive Rings of Infinite Matrices

1964 ◽  
Vol 14 (1) ◽  
pp. 47-53 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Jacobson Radical ◽  
Final Section ◽  
Matrix Ring ◽  
Primitive Ideal ◽  
Alternative Proof ◽  
Infinite Matrix ◽  
Infinite Matrices ◽  
Matrix Rings ◽  
Primitive Ideals ◽  
Infinite Case

E. C. Posner (5) has shown that a ring R is primitive if and only if the corresponding matrix ring Mn(R) is primitive. From this result he is able to deduce that the primitive ideals in Mn(R) are precisely those ideals of the form Mn(P), where P is a primitive ideal in R. This affords an alternative proof that the Jacobson radical of Mn(R) is Mn(J), where J is the Jacobson radical of R. But Patterson (3, 4) has shown that this last result does not hold in general for rings of infinite matrices and thus that the above result concerning primitive ideals cannot be extended to the infinite case. Nevertheless in this paper we are able to show that Posner's result on primitive rings does extend to infinite matrix rings. Patterson's result depends on showing that if the Jacobson radical J of R is not right vanishing then a certain matrix with entries from J does not lie in the Jacobson radical of the infinite matrix ring. In the final section of this paper we consider a ring R with this property and exhibit a primitive ideal in the infinite matrix ring, which does not arise, as above, from a primitive ideal in R. Finally the Jacobson radical of this ring is determined.

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